Chaos, Complexity and Transport: Theory and Applications [illustrated edition] 9812818790, 9789812818799

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CHAOS, COMPLEXITY AND TRANSPORT

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T-

CHAOS, COMPLEXITY AND

TRANSPORT

Theory and Applications Proceedings of the CCT '07

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CHAOS, COMPLEXITY AND

TRANSPORT

Theory and Applications Proceedingts of the CCT '07 Marseille, France

4 - 8 June 2007

edited by Cr i stel Chandr e CNRS, France

Xavi er Leonci ni

zyx

Aix-Marseille Universite, France

Geor ge Zasl avsky

New York University, USA

World Scientific N E W JERSEY

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LONDON

SINGAPORE

*

BEIJING

- SHANGHAI

*

HONG KONG

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TAIPEI

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CHENNAI

Published by

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World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224

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British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

CHAOS, COMPLEXITY AND TRANSPORT Theory and Applications Proceedings of the CCT'07

Copyright 0 2008 by World Scientific Publishing Co. Re. Ltd.

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ISBN- 13 978-98 1-281-879-9 ISBN- 10 981 -281-879-0

Printed in Singapore by World Scientific Printers

V

PREFACE Chaos and turbulence are ubiquitous features of physical systems. Their manifestations are very diverse and not always well understood. Improving knowledge in this field is not only important for our apprehension of non-linear physics but also essential to tame or control their behaviours. Moreover the interdisciplinary character of such phenomena which were observed notably in fluid dynamics, atomic physics, plasma physics, accelerator physics, celestial mechanics, condensed matter, among others, makes research in this field quite peculiar as any advance in one direction may have strong repercussions and consequences in others (Chaos!). As a shared interest one can notably consider transport properties. These are often characterised by Ldvy-type processes, strange (fractal) kinetics, intermittency, etc. Typically one finds that portions of the trajectories are almost regular for quite a long time. This phenomenon (Ldvy flight) gives rise to strong memory effects. History comes into play, and thus rules out the traditional use of Markov processes to model transport. What makes these properties so special is that they are associated with rare events in time but are crucial for the physical behaviour of the system. One may also emphasise on the important role played by coherent structures and their impact on transport. The classical approach to study transport dynamics has been complemented by various novel approaches, based on either the development of new physical and mathematical ideas and on the implementation of sophisticated numerical codes. The concept of Ldvy processes, fractional kinetics and anomalous transport have proved to be extremely important from a conceptual point of view indicating a new direction in the non-linear dynamics. However, many questions are still open, from both a conceptual and an applied point of view. For example, the role of chaotic advection in complex situations has still to be properly addressed. The aim is to understand which (if any) of the properties commonly attributed to the processes of turbulent dispersion may be accounted for by the basic non-linear mechanisms encountered in chaotic advection. Analogously, the non-perfect nature of the tracers used in geophysical measurements and/or the possible “active” nature of some constituents turns out to be very important, determining a different behaviour of the advected particles and of true fluid particles.

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Further on, it is now clear that there exist regimes of anomalous transport, which may lead to a faster spreading and escape of advected quantities. Such a phenomenon is especially important in plasma dynamics, as well as in turbulent flows due to the action of coherent structures. The regimes of anomalous transport may have a truly asymptotic nature or they may be intermediate regimes encountered in proximity of significant time scales of the system; a better comprehension of these regimes is necessary for various applications. One of them being the transport in magnetised fusion plasmas, understanding transport in these systems as well as characterising the origin of anomalous behaviour is essential not only to define proper control strategies to obtain better confinement, but also to monitor what may happen near the plasma edge, where the energy is collected. All these anomalous phenomena arise once we accept the fact that uniform chaos is not often realistic. In the early days of the study of chaos, ergodic theory provided an adequate support for the kinetic approach. This is no longer the case. If we are to describe new experimental observations, data from simulations, and to develop new applications, a significantly broader notion of transport is required as well as an expanded arsenal of mathematical tools. The phase space is divided between regions where the motion is regular or irregular. Such diversity in the dynamical landscape makes transport properties more subtle than initially anticipated. In fact, many difficulties are already present in the case of few degrees of freedom Hamiltonian systems. Typically the phase space of smooth Hamiltonian systems is not ergodic in a global sense, due to the presence of islands of stability, the rate of phase space mixing in the chaotic sea is not uniform due to the phenomenon of %tickiness”, and the Gaussian nature of transport is generally lost, due to the so-called flights and trappings and the associated powerlaw tails observed in probability distribution functions. This last feature is also shared with most systems dealing with complexity. Understanding the paths from dynamics to kinetics and from kinetics to transport and complexity involves a strong interdisciplinary interaction among experts in theory, experiments and applications. The contributions are the proceedings of the conference Chaos, Complexity and Transport which was held in Marseilles (France) from June 4th to June 8th 2007. Due to the interdisciplinary character of the problem the conference made a point on balancing theoretical, numerical and experimental contributions in order to encourage the interactions between experimentalists and theoreticians in the same fields but also cross-disciplinary contributions.

vii

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This book is organised into two parts. In the first part, we gather what we consider more general or theoretical contributions, while the second part is dedicated to applications. In the first part, some features of the dynamics for large N systems with long range interactions and a large number of degree of freedom giving rise to out of equilibrium phase transitions are presented in detail. One may also discover how stochastic webs in multidimensional systems can be used as a way for tiling the plane with specific symmetries. At the same time one discusses the phenomenon of chaotic transport and chaotic mixing through the course of geodesics or the construction of mixing flows using knots. Then one can learn about entropy and complexity, or about Bose-Einstein condensation of classical waves, as well as transport in deterministic ratchets. Regarding Hamiltonian systems, some new approaches to the theoretical treatment of separatrix chaos are discussed. The possibility of having a giant acceleration and about a control technique in area preserving maps are explored. The second part covers mainly applications. To facilitate reading, we have created two subdivisions. The first one deals with plasmas and fluids, while the second concerns more fields. In the plasmas and fluid subdivision, one will be able to learn in some detail, the implication of topological complexity and Hamiltonian chaos in fusion plasmas, as well as precise experimental studies of advection-reaction diffusion systems. And also nondiffusive transport observed in simulations of plasmas and the problem of solving numerically rotating Rayleigh-BQnardconvection in cylinders. Then there are the experimentally observed self-excited instabilities in plasmas containing dust particles and the clustering properties of plasma turbulence signals as well as intermittency scenario of transition to chaos in plasma. Finally, one can read about magnetic reconnection in collisionless plasmas as well as the complexity of the neutral curve of oscillatory flows. In the second subdivision, an overview of chemotaxis models using an interesting analogy with non-linear mean-field Fokker-Planck equations is presented. Then one shall learn about switchability of a flow, before moving to celestial mechanics and the formation of spiral arms and rings in barred galaxies or learning about LBvy walks for energetic electrons in space. The phenomenon of wave chaos in an underwater sound-channel is then discussed, followed by problem of Fermi acceleration in randomised driven billiards. Finally one shall learn about memory regeneration phenomenon in fractional depolarisation of dielectrics, as well as nodal pattern analysis

viii

for conductivity of quantum ring and the application of the GAL1 method to the dynamics of multidimensional symplectic maps. As already mentioned, this book reflects to some extent the presentations and the resulting discussions carried out during the conference Chaos, Complexity and Transport: Theory and Applications, which was held in the Pharo site of the Universitk de la Mkditerranke, Marseilles, France, in June 2007. In these regard, we would like to thank all participants and express our sincere gratitude to the contributing authors. We also take this opportunity to express our debt and gratitude for the support to sponsors: Centre National de la Recherche Scientifique, the GDR Phenix et GDR Dycoec, the Commissariat B 1’Energie Atomique (CEA), the Conseil Gknkral des Bouches du RhGne, the Ministkre D616guk B la Recherche, the Ville de Marseille, the GREFI-MEFI, the European Physical Society, the University de Provence and University de la Mkditerranke, The US department of Naval Research, the Delegation Generale de 1’Armement and the Centre de Physique Thkorique (UMR 6207). We also would like to thank Mrs A. Elbaz, V. Leclercq-Ortal and M-T Done1 (from the Centre de Physique Thkorique) for their help before, during and after the workshop. We would like also to thank M. Mancis and S. Foulu from Protisvalor Mkditerranke for their help.

zyxwv Cristel Chandre Xavier Leoncini George Zaslavsky Editors

ix

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CONTENTS

Preface

zyxwvut V

THEORY

1

Out-of-Equilibrium Phase Transitions in Mean-Field Hamiltonian Dynamics

P.-H. Chavanis, G. De Ninno, D. Fanelli and S. Ruff0

3

Stochastic Webs in Multidimensions

G. M. Zaslavsky and M. Edelman

27

zyxwvu

Chaotic Geodesics

J.-L. Thiffeault and K . Kamhawi

A Steady Mixing Flow with No-Slip Boundaries R. S. MacKay

40

55

Complexity and Entropy in Colliding Particle Systems

M. Courbage and S. M. Saberi Fathi

69

Wave Condensation

S. Rica

84

Transport in Deterministic Ratchets: Periodic Orbit Analysis of a Toy Model

R. Artuso, L. Cavallasca and G. Cristadoro

106

Separatrix Chaos: New Approach to the Theoretical lleatment

S. M. Soskin, R. Mannella and 0. M. Yevtushenko Giant Acceleration in Weakly-Perturbed Space-Periodic Hamiltonian Systems M . Yu. Uleysky and D. V. Malcarov

119

129

X

zyxw z zyxwvuts

Local Control of Area-Preserving Maps C. Chandre, M. Vittot and G. Caraolo

136

APPLICATIONS

(1) PLASMA & FLUIDS

145

Implications of Topological Complexity and Hamiltonian Chaos in the Edge Magnetic Field of Toroidal Fusion Plasmas 7'. E. Evans

147

zyxwvu zyxw zy

Experimental Studies of Advection-Reaction-Diffusion Systems T. H. Solomon, M. S. Paoletti and M. E. Schwartz

177

Non-Diffusive Transport in Numerical Simulations of Magnetically-Confined Turbulent Plasmas R. Sa'nchez, B. A . Carreras, L. Garcia, J. A . Mier, B. Ph. Van Milligen and D. E. Newman

189

Rotating Rayleigh-Birnard Convection in Cylinders J . J. Sa'nchea-Alvarez, E. Serre, E. Crespo Del Arc0 and F. H. Busse

207

Self-Excited Instabilities in Plasmas Containing Dust Particles (Dusty or Complex Plasmas) M.Mikikian, M. Cavarroc, L. Couedel, Y. Tessier and L. Boufendi

218

Clustering Properties of Confined Plasma Turbulence Signals M. RajkoviC and M. jkoriC

227

Intermittency Scenario of Transition to Chaos in Plasma D. G. Dimitriu and S. A . Chiriac

237

Nonlinear Dynamics of a Hamiltonian Four-Field Model for Magnetic Reconnection in Collisionless Plasmas E. Tussi, D. Grasso and F. Pegoraro

245

On the Complexity of the Neutral Curve of Oscillatory Flows M. Wadih, S. Carrion, P. G. Chen, D. Fougbre and B. Roux

255

zyxw XI

(2) OTHERS

263

Generalized Keller-Segel Models of Chemotaxis. Analogy with Nonlinear Mean Field Fokker-Planck Equations

P.- H. Chavanis

265

On Switchability of a Flow to the Boundary in a Periodically Excited Discontinuous Dynamical System

A. C. J . Luo and B. M. Rapp

287

The Formation of Spiral Arms and Rings in Barred Galaxies

M. Romero- Gdmez, E. Athanassoula, J. J. Masdemont and C. Garcia-Gdmez

300

LBvy Walks for Energetic Electrons Detected by the Ulysses Spacecraft a t 5 AU

S. Perri and G. Zimbardo

zyx 309

Wave Chaos and Ghost Orbits in an Underwater Sound Channel

D. V. Makarov, L. E. Kon’kov, E. V. Sosedko and

M. Yu. llleysky

318

Displacement Effects on Fermi Acceleration in Randomized Driven Billiards

A . K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher Memory Regeneration Phenomenon in Fractional Depolarization of Dielectrics V. V. Uchaikin and D. V. Uchaikin

327

337

Nodal Pattern Analysis for Conductivity of Quantum Ring in Magnetic Field

M. Tomiya, S. Sakamoto, M. Nishikawa and Y. Ohmachi Application of the Generalized Alignment Index (GALI) Method to the Dynamics of Multi-Dimensional Symplectic Maps T. Manos. Ch. Skokos and T. Bountis

346

356

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THEORY

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3

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OUT-OF-EQUILIBRIUM PHASE TRANSITIONS IN MEAN-FIELD HAMILTONIAN DYNAMICS PIERREHENRI CHAVANTS

Laboratoire de Physique The‘orique, Universite‘ Paul Sabatier, 118, route de Narbonne 31062 Toulouse, h n c e E-mail: [email protected] tlse.fr GIOVANNI DE NINNO

Sincrotrone ’Prieste, S.S. 14 K m 163.5, Basovizza, 34012, l?rieste, Italy University of Nova Gon’ca, Vipavska 13, P O B 301, SI-5000, Nova Gorica, Slovenia E-mail: giovanni.deninnoOe1ettra.trieste.it DUCCIO FANELLI

Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom E-mail: Duccio.FanelliOmanchester.ac.uk STEFAN0 RUFF0

Dipartimento di Energetica “S. Stecco” and CSDC, Universitb d i Firenze, and INFN, via S. Marta, 3, 50139 Firenze, Italy E-mail: stefano.ruffoOunafi.it Systems with long-range interactions display a short-time relaxation towards Quasi-Stationary States (QSSs), whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here review Lynden-Bell’s theory of “violent relaxation”. The latter results in a maximum entropy scheme for a water-bag initial profile which predicts the presence of out-of-equilibrium phase transitions separating homogeneous (zero magnetization) from inhomogeneous (non-zero magnetization) QSSs. Two different parametric representations of the initial condition are analyzed and the features of the phase diagram are discussed. In both representations we find a second order and a first order line of phase transitions that merge at a tricritical point. Particular attention is payed to the condition of existence and stability of the homogeneous phase.

Keywords: Quasi-stationary states, Hamiltonian Mean-Field model, Out-ofequilibrium phase transitions.

4

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1. Introduction

Haniiltonian systems arise in many branches of applied and fundamental physics and, in this respect, constitute a universal framework of extraordinary conceptual importance. Spectacular examples are undoubtedly found in the astrophysical context. The process of hierarchical clustering via gravitational instability, which gives birth to the galaxies,' can in fact be cast in a Hamiltonian setting. Surprisingly enough, the galaxies that we observe have not yet relaxed to thermodynamic equilibrium and possibly correspond to intermediate Quasi-Stationary States (QSSs). The latter are in a long-lasting dynamical regime, whose lifetime diverges with the size of the system. The emergence of such states has been reported in several different domains, ranging from charged cold plasmas2 to Free Electron Lasers (FELs),~and long-range forces have been hypothesized to be intimately connected to those peculiar phenomena. Long-range interactions are such that the two-body interaction potential decays at large distances with a power-law exponent which is smaller than the space dimension. The dynamical and thermodynamical properties of physical systems subject to long-range couplings were poorly understood until a few years ago, and their study was essentially restricted to astrophysics (e.g., self-gravitating systems). Later, it was recognized that long-range systems exhibit universal, albeit unconventional, equilibrium and out-of-equilibrium feature^.^ Besides slow relaxation to equilibrium, these include ensemble inequivalence (negative specific heat, temperature jumps), violations of ergodicity and disconnection of the energy surface, subtleties in the relation of the fluid (i.e. continuum) picture and the particle (granular) picture, new macroscopic quantum effects, etc.. While progress has been made in understanding such phenomena, an overall interpretative framework is, however, still lacking. In particular, even though the ubiquity of QSSs has been accepted as an important general concept in non-equilibrium statistical mechanics, different, contrasting, at tempts to explain their emergence have catalysed a vigorous discussion in the l i t e r a t ~ r e . ~ To shed light onto this fascinating field, one can resort to toy models which have the merit of capturing basic physical modalities, while allowing for a dramatic reduction in complexity. This is the case of the so-called Hamiltonian Mean Field (HMF) model which describes the evolution of N rotators, coupled through an equal strength, attractive or repulsive, cosine

5

interaction.' The Hamiltonian, in the attractive case, reads

zyxwvu zyx zyxwv

where B j represents the orientation of the j-th rotator and p j stands for the conjugated momentum. To monitor the evolution of the system, it is customary to introduce the magnetization, an order parameter defined as M = IMI = ICmil/N, where mi = (cos&,sin&) is the magnetization vector. The HMF model shares many similarities with gravitational and charged sheet model^^>^ and has been extensively studiedg as a paradigmatic representative of the broad class of systems with long-range interactions. The equilibrium solution is straightforwardly worked out' and reveals the existence of a second-order phase transition at the critical energy density U, = 3/4: below this threshold value the Boltzmann-Gibbs equilibrium state is inhomogeneous (magnetized). In the following, we shall discuss the appearance of QSSs in the HMF setting and review a maximum entropy principle aimed a t explaining the behaviour of out-of-equilibrium macroscopic observables. The proposed approach is founded on the observation that in the continuum limit (for an infinite number of particles) the discrete HMF equations converge towards the Vlasov equation, which governs the evolution of the single-particle distribution function (DF). Within this scenario, the QSSs correspond to statistical equilibria of the continuous Vlasov model. As we shall see, the theory allows us to accurately predict out-of-equilibrium phase transitions separating the homogeneous (non-magnetized) and inhomogeneous (magnetized) phases.">" Special attention is here devoted to characterizing analytically the basin of existence of the homogeneous zone. Concerning the structure of the phase diagram, a bridge between the two possible formal settings, respectively'O and,12 is here established. The paper is organized as follows. In Section 2 we present the continuous Vlasov picture and discuss the maximum entropy scheme. The properties of the homogeneous solution are highlighted in Section 3, where conditions of existence are also derived. Section 4 is devoted to analyze the stability of the homogeneous phase. A detailed account of the phase diagram is provided in Sections 5 and 6, where the case of a "rectangular" and generic water-bag initial distribution are respectively considered. Finally, in Section 7 we sum up and draw our conclusions.

zyxw zyxw

6

zyxw zyxwvu zyxw z zyxwvut

2. On the emergence of quasi-stationary states: Predictions from the Lynden-Bell theory within the Vlasov picture

As previously mentioned, long-range systems can be trapped in long-lasting Quasi-Stationary-States (QSSs),I3 before relaxing to Boltzmann thermal equilibrium. The existence of QSSs was firstly recognized with reference to galactic and cosmological applications (see7 and references therein) and then, more recently, re-discovered in other fields, e.g.two-dimensional turbulence14 and plasma-wave interactions.8 Interestingly, when performing the infinite size limit N -+ 00 before the infinite time limit, t -+ co,the system remains indefinitely confined in the &SSs.l5 For this reason, QSSs are expected to play a relevant role in systems composed by a large number of particles subject to long-range couplings, where they are likely to constitute the solely experimentally accessible dynamical regimes.2,3 QSSs are also found in the HMF model, as clearly testified in Fig. 1. Here, the magnetization is monitored as a function of time, for two different values of N . The larger the system the longer the intermediate phase where it remains confined before reaching the final equilibrium. In a recent series of p a p e r ~ , ~ i ' an ~ - approximate ~~>~~ analytical theory based on the Vlasov equation has been proposed which stems from the seminal work of LyndenBell." This is a fully predictive approach, justified from first principles, which captures most of the peculiar traits of the HMF out-of-equilibrium dynamics. The philosophy of the proposed approach, as well as the main predictions derived within this framework, are reviewed in the following. In the limit of N -+ 00, the HMF system can be formally replaced by the following Vlasov equation

zy

a f +p-a f - (M,[f] af = 0, sin8 - Mv[f] cose) at 88 aP where f (8,p, t ) is the one-body microscopic distribution function normalized such that M [f] = f d8dp = 1,and the two components of the complex

s

magnetization are respectively given by

M , [f]=

J f cos ededp,

(3)

]

Mv[f] = f sin8dddp. The mean field energy can be expressed as

M:+M; 2

+ -.21

(4)

7

zyxwvuts zyxwvut zyx

Fig. 1. Magnetization M ( t ) as function of time t . In both cases, an initial “violent” relaxation toward the QSS regime is displayed. The time series relative to N = 1000 (thick full line) converges more rapidly to the Boltzmann equilibrium solution (dashed horizontal line). When the number of simulated particles is increased, N = 10000 (thin full line), the relaxation to equilibrium gets slower (the convergence towards the Boltzmann plateau is outside the frame of the figure). Simulations are carried on for a rectangular water-bag initial distribution, see Eq. (13).

z

Working in this setting, it can be then hypothesized that QSSs correspond to stationary equilibria of the Vlasov equation and hence resort to the pioneering Lynden-Bell’s violent relaxation theory17 . The latter was initially devised to investigate the process of galaxy formation via gravitational instability and later on applied to the two-dimensional Euler equation. l8 The main idea goes as follows. The Vlasov dynamics induces a progressive filamentation of the initial single particle distribution profile, i.e. the continuous counterpart of the discrete N-body distribution, which proceeds at smaller and smaller scales without reaching an equilibrium. Conversely, at a coarse grained level the process comes to an end, and the distribution function f ( O , p , t ) , averaged over a finite grid, eventually converges to an asymptotic form. The time evolution of a rectangular water-bag initial distribution is shown in Fig. 2.

8 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

-3 -2 - 1

-3 -2 -1

zyxwvutsrq zyxwvutsrqp zyxwvutsrq 0

1

2

3

-3 -2

-1

0

1

2

3

0

1

2

3

-3 -2

-1

0

?

2

3

Fig. 2. The process of phase mixing is here illustrated, showing four snapshots of the time evolution of an initial rectangular water-bag distribution. The final state (right bottom) is a QSS.

Following Lynden-Bell, one can associate a mixing entropy to this process and calculate the statistical equilibrium by maximizing the entropy, while imposing the conservation of the Vlasov dynamical invariants. More specifically, the above procedure implicitly requires that the system mixes well, in which case, assuming ergodicity (efficient mixing), the QSS predicted by Lynden-Bell, fQss(O,p,t ) ,is obtained by maximizing the mixing entropy. As a side remark, it is also worth emphasizing that the prediction of the QSS depends on the details of the initial condition,lg not only on the value of the mass M and energy U as for the Boltzmann statistical equilibrium state. This is due to the fact that the Vlasov equation admits an infinite number of invariants, the Casimirs or, equivalently, the moments M , = J f ” d 0 d p of the fine-grained distribution function. In the following, we shall consider a very simple initial condition where the distribution function takes only two values fo and 0. In that case, the invariants reduce to M and U since the moments Mn>l can all be expressed in terms of M and fo as M , = J f ” d e d p = J f;-’ x f d e d p = J f;-lTdedp = f;-lM. For the specific case at hand, the Lynden-Bell entropy is then explicitly

zyxwvu

zyx zyxwvut zyx z z zyxw zyxw zy 9

constructed from the coarse-grained DF

f and reads

We thus have to solve the optimization problema max{S[T]

T

I

U[T]= U ,M [ T ]= 1).

(6)

The maximization problem ( 6 ) is also a condition of formal nonlinear dynamical stability with respect to the Vlasov equation, according to the refined stability criterion of Ellis et aL2’ (see also Chavanis21). Therefore, the maximization of S at fixed U and M guarantees (i) that the statistical equilibrium macrostate is stable with respect to the perturbation on the microscopic scale (Lynden-Bell thermodynamical stability) and (ii) that the coarse-grained DF 7 is stable for the Vlasov equation with respect to macroscopic perturbations (refined formal nonlinear dynamical stability). We again emphasize that it is only when the initial DF takes two values fo and 0 that the Lynden-Bell entropy can be expressed in terms of the coarse-grained DF 7, as in Eq. ( 5 ) . In general, the Lynden-Bell entropy is a functional of the probability distribution of phase 1 e ~ e l s . From l~ Eq. (5), we write the first order variations as SS - PSU - abM = 0,

(7)

where the inverse temperature ,B = 1/T and cr are Lagrange multipliers associated with the conservation of energy and mass. Requiring that this entropy is stationary, one obtains the following distributionlO>ll

As a general remark, it should be emphasized that the above distribution differs from the Boltzmann-Gibbs one because of the “fermionic” denominator, which in turn arises because of the form of the entropy. Morphologically, this distribution function is similar to the Fermi-Dirac statistics so that several analogies with the quantum mechanics setting are to be expected. Notice also that the magnetization is related to the distribution function by Eq. (3) and the problem hence amounts to solving an integrodifferential system. In doing so, we have also to make sure that the critical “The momentum P = f p d e d p is also a conserved quantity but since we look for solutions where the total momentum is zero, the corresponding Lagrange multiplier y vanishes trivially” so that, for convenience, we can ignore this constraint right from the beginning.

10

zyx zyxwvu zyxwv

point corresponds to an entropy maximum, not to a minimum or a saddle point. Let us now insert expression (8) into the energy and normalization constraints and use the definition of magnetization (3). Further, defining X = e" and m = (cos 0, sin 0) yields:

where we have defined the Fermi integrals

We have the asymptotic behaviours for t

and for t

-+

0:

zyx

+ +oo

The magnetization in the QSS, MQSS = M[fQss], and the values of the multipliers are hence obtained by numerically solving the above coupled implicit equations. It should be stressed that multiple local maxima of the entropy are in principle present when solving the variational problem, thus resulting in a rich zoology of phase transitions. This issue has been addressed inloill and more recently in,12 to which the following discussion refers to. It is important to note that, in the two-levels approximation, the Lynden-Bell equilibrium state depends only on two control parameters (U,f ~ ) This . ~ is valid for any initial condition with f ( & p , t = 0) E (0, fo}. This general case will be studied in Section 6 where we describe the phase

zyxwvu zyxw zy +

bThese parameters are related to those introduced inlo by U = €14 112, p = 27), fo = g o / N = p/(27r), k = 27r/N, x = As, y = (2/7r)Ap and the functions F inll are related to the Fermi integrals by F k ( l / y ) = 2(k+1)/2yZ(k_l),z(y).

zyxwv zyxwvutsr zy

zyxwvuts 11

diagram in the (fo,U ) plane. Now, many numerical simulations of the N body system or of the Vlasov equation have been performed starting from a family of rectangular water-bag distributions. The latter correspond to assuming a constant value fo inside the phase-space domain D :

D

=

zyxw

( ( 9 , ~E) [-n,nI x

[-m,03]

I

191 < A9,

I P ~ < Ap},

(13)

where 0 5 A9 5 n and Ap 2 0. The normalization condition results in fo = 1/(4A9Ap). Notice that, for this specific choice, the initial magnetization Mo and the energy density U can be expressed as functions of A9 and Ap as

zyx

For the case under scrutiny, 0 5 MO 5 1 and U 2 UMIN(MO) = (l-M;)/2. The variables ( M o ,U )are therefore used to specify the initial configuration and hereafter assumed to define the relevant parameters space. This particular but important case will be studied specifically in Section 5 where we illustrate the phase diagram in the ( M 0 , U ) plane for the rectangular water-bag initial condition. Before that, we analytically study the stability of the Lynden-Bell homogeneous phase: two important limits, namely the non degenerate and the completely degenerate ones, are considered. We also discuss the condition for the existence of a homogeneous, non-equilibrium phase.

3. Properties of the homogeneous Lynden-Bell distribution

If we consider spatially homogeneous configurations (MQSS = 0), the Lynden-Bell distribution becomes

Using Eqs. (9), the relation between the inverse temperature energy U is given in parametric form by

p

and the

This defines the series of equilibria T ( U ) for fixed fo parametrized by X (see Fig. 3 in"). The stable part of the series of equilibria is the caloric curve. Note that the temperature T is a Lagrange multiplier associated with the conservation of energy in the variational problem (7). It also has the interpretation of a kinetic temperature in the Fermi-Dirac distribution

12

zyxwvuts zyxw

(15). If we start from a water-bag initial condition, recalling that fo = 1/(4A0Ap) and ( A P ) ~= 6[U - (1- M,3/2], we can express fo as a function of MO and U by 1 fo2 = 48[(2U - l)(A0)2 sin2 A01 ’

zy zy +

where A0 is related to MO by Eq. (14). Inserting this expression in Eqs. (16), we obtain after some algebra the caloric curve T ( U ) for fixed MO parametrized by A:

(

1 2 ,B = - -I-1/2(X)2 - 2(A0)’ sin2A0 6 Eqs. (16) can be rewritten

where G(X) is a universal function monotonically increasing with X (see Fig. 2 oflo). A solution of the above equation certainly exists provided: 1 (U - -)8n2 2 2 G(0). (20)

fi

To compute G(0) we use the asymptotic expansions (11) and (12) of the Fermi integrals. This yields G(0) = 1/12. Therefore, the homogeneous Lynden-Bell distribution with fixed fo exists only for:1° 1

1

For the rectangular water-bag initial condition, using Eqs. (17) and (21), we here find that the homogeneous Lynden-Bell distribution with fixed MO exists only for:

This result can also be obtained from Eq. (18) by taking the limit X --+ 0. Let us now describe more precisely the asymptotic limits of the FermiDirac distribution (see Fig. 3): Non degenerate limit: In the limit X -, +m, the Lynden-Bell distribution reduces to the Maxwell-Boltzmann distribution

13

zyxwvutsr zyx zyxwvuts zyxwvu P

Fig. 3. Spatially homogeneous Lynden-Bell distribution function for increasing values of X (top to bottom). For X = 0, the distribution reduces to a step function (completely +m, it becomes equivalent to the Maxwell-Boltzmann distribudegenerate) and for X tion (non degenerate). In the figure, we have taken f o = 0.13 and p has been calculated from Eq. (16). ---f

zy zy

Since 7 U > Vi'l'(f0)and stable again for U,"l'(fo) > U 2 U m i n ( f 0 ) . This range of parameters corresponds to a first order phase transition. To make the connection between the phase diagram ( f 0 , U ) obtained inlo and the phase diagram ( M o ,U ) obtained in,12 we can plot the iso-Mo lines in the (fo,U ) phase diagram. If we fix the initial magnetization M o , or equivalently if we fix the parameter AB, the relation between the energy U and fo is

Therefore, the iso-Mo lines are of the form

22 h

zyxwvuts A8=1.5

Homogeneous phase (stable)

0.7

,

/------

r

Homogeneous phase

zyxwvuts zyxwv 0.16

zyx

Fig. 7. Iso-Mo lines in the (fo, U ) phase diagram. This graphical construction allows one t o make the connection between the (fo,U ) phase diagram of Fig. 6 and the ( M o , U ) phase diagram of Fig. 4.We can vary the energy at fixed initial magnetization by following a dashed line. The intersection between the dashed line and the curve U m i n ( f o ) determines the minimum energy Umin ( M o ) of the homogeneous phase. The intersection between the dashed line and the curve Uc(fo) determines the energy U,(Mo) below which the homogeneous phase becomes unstable.

zyxwv i (w)'i,

1 with A(Ad) = goI and B(A0) = which are easily represented in the (fo,U ) phase diagram (see Fig. 7). As an immediate consequence of this geometrical construction, we can recover the minimum energy of the homogeneous phase for a fixed initial magnetization MO (or Ad). Indeed, for a given AO, the homogeneous phase exists iff U A e ( f 0 ) 2 Umin(fo) leading to

This corresponds to U > U min (M0) = U ((f0) ) leading to

u 2 Umin(M0)= -

zyxw 23

Homogeneous phase

Homogeneous phase (unstable) IV

0'58

zyx zyxwvut zyxwv

1 I

0'&05

No homogeneous phase

v

I

0.115

0.11

I

I

12

F

Fig. 8. Zones of metastability in the (fo,V) phase diagram. In region (I), the homogeneous phase is fully stable and the inhomogeneous phase is inexistent. In region (11) the homogeneous phase is fully stable and the inhomogeneous phase is metastable. In region (111) the homogeneous phase is metastable and the inhomogeneous phase is fully stable. In region (IV) the homogeneous phase is unstable and the inhomogeneous phase is fully stable. The three curves separating these regions connect themselves at the tricritical point. In region (V) the homogeneous phase is inexistent. The corresponding caloric of) the inhomogeneous phase curves as well as the absolute minimum energy U M I N ( ~ O will be determined in a future contribution.

zyx

which is identical to (22). Note again that A Q is related to A40 by Eq. (14). Figure 7 is in good agreement with the structure of the phase diagram in the (U,Mo) plane. Indeed, along an iso-A40 line, we find that for large energies U > U,(Mo) the homogeneous phase is stable and for low energies U < UC(Mo)the homogeneous phase becomes unstable. In that case, there is no re-entrant phase. In," only the stability of the homogeneous phase has been studied, i.e. whether it is an entropy maximum at fixed mass and energy or not. The question of its metastability, i.e. whether it is a local entropy maximum with respect to the inhomogeneous phase, has not been considered. However, considering Fig. 7 and comparing with the results of,12we conclude that there must exist zones of metastability in the (fo, U ) phase diagram. They

24

have been represented in Fig. 8. In region (I) the inhomogeneous phase does not exist while the homogeneous phase is fully stable. In region (11), the inhomogeneous phase appears but is metastable while the homogeneous phase is fully stable. In region (111), the inhomogeneous phase becomes fully stable while the homogeneous phase becomes metastable. In region (IV), the homogeneous phase becomes unstable while the inhomogeneous phase is fully stable. The three curves separating these regions connect themselves a t a tricritical point. This is clearly the same as in Fig. 4. Using Eqs. (36), (37), (14) we find that it corresponds t o

zyxw zyx zyxwvu zyx zyxw

(Mo)*= 0.1757... (41) with AQ, = 2.656.... Therefore, the phase diagrams in (fo,U ) and ( M o ,U ) planes are fully consistent. Note, however, that the physics is different whether we vary the energy at fixed fo or a t fixed Mo. In particular, there is no “re-entrant” phase when we vary the energy a t fixed M0l2 while a “re-entrant” phase appears when we vary the energy at fixed fO.lo

U,

= 0.608...,

7. Conclusions In this paper, we have discussed the emergence of out-of-equilibrium Quasi Stationary States (QSSs) in the Hamiltonian Mean Field (HMF) model, a paradigmatic representative of systems with long-range interactions. The analysis refers to a special class of initial conditions in which particles are uniformly occupying a finite portion of phase space and the distribution function takes only two values, respectively 0 and fo. The energy can be independently fixed to the value U . The Lynden-Bell maximum entropy principle is here reviewed and shown to result in a rich out-of-equilibrium phase diagram, which is conveniently depicted in the reference plane (fo, U).l0 When considering a rectangular water-bag distribution the concept of initial magnetization, M o , naturally arises as a control parameter and the different QSSs phases can be represented in the alternative space ( M o ,U).12In both settings first and second order phase transitions are found, which merge together in a tricritical point. These findings have been tested versus numerical simulation in,12 where the adequacy of Lynden-Bell theory was confirmed. A formal correspondence between the two above scenarios is here drawn and their equivalence discussed. It is worth mentioning that swapping from one parametric representation to the other allows us to put the focus on intriguingly different physical mechanisms, as it is the case of the “re-entrant” phases discussed in Section 6.

zyxw

25

zyxwvuts zyxw

Further, in this paper, we have provided an analytical characterization of the domain of existence of the Lynden-Bell spatially homogeneous phase and investigated its stability. Homogeneous QSS are also expected to occur for U > U, = 314, a claim here supported by dedicated numerical simulations. Despite the fact that Lynden-Bell’s theory results in an accurate tool to explain the peculiar traits of QSSs in HMF dynamics, one should be aware of the limitations which are intrinsic to this approach. Most importantly, Lynden-Bell’s recipe assumes that the system mixes well so that the hypothesis of ergodicity, which motivates the statistical theory (maximization of the entropy), applies. Unfortunately, this is not true in general. Several example of incomplete violent relaxation have been identified in stellar dynamics and 2D turbulence (see some references inz6) for which the QSSs cannot be exactly described in term of a Lynden-Bell distribution. Also in this case, however, the QSSs are stable stationary solution of the Vlasov equation and novel analytical strategies are to be eventually devised which make contact with the underlying Vlasov framework. Acknowledgements

D.F. and S.R. wish to thank A. Antoniazzi, F. Califano and Y. Yamaguchi for useful discussions and long-lasting collaboration. This work is funded by the PRIN05 grant Dynamics and thermodynamics of systems with longrange interactions. References

zyx zyxw zyx

1. P.J. Peebles, The Large Scale Structure ofthe Universe, (Princeton University

Press, Princeton, NJ, 1980). 2. C. Benedetti, S. Rambaldi, G. Turchetti, Physica A 364, 197 (2006); P.H. Chavanis, Eur. Phys. J. B 52, 61 (2006). 3. J. B a d , T. Dauxois, G. de Ninno, D. Fanelli, S. Ruffo, Phys. Rev E 69, 045501(R) (2004). 4. T. Dauxois et al., Dynamics and Thermodynamics of Systems with Long Range Interactions, Lect. Not. Phys. 602, Springer (2002). 5. A. Rapisarda, A. Pluchino, Europhysics News 36,202 (2005); F. Bouchet, T. Dauxois and S. Ruffo, Europhysics News 37,9 (2006). 6. M. Antoni, S. Ruffo, Phys. Rev. E 52, 2361 (1995). 7. T. Tsuchiya, T. Konishi, N. Gouda, Phys. Rev. E 50, 2607 (1994). 8. Y . Elskens, D.F. Escande, Microscopic Dynamics of Plasmas and Chaos, IoP Publishing, Bristol (2003). 9. P.H. Chavanis, J. Vatteville, F. Bouchet, Eur. Phys. J. B 46, 61 (2005) and references therein.

26

zyxw zyxwvuts zyx

10. P.H. Chavanis, Eur. Phys. J. B 53,487 (2006). 11. A. Antoniazzi, D. Fanelli, J. Bar&, P.H. Chavanis, T. Dauxois, S. Ruffo, Phys. Rev. E 75,011112 (2007). 12. A. Antoniazzi, D. Fanelli, S. Ruffo, Y. Y. Yamaguchi, Phys. Rev. Lett. 99 040601 (2007). 13. V. Latora, A. Rapisarda, S. Ruffo, Phys. Rev. Lett. 80, 692 (1998). 14. X.P. Huang, C.F. Driscoll, Phys. Rev. Lett. 72,2187 (1994); H. Brands, P.H. Chavanis, R. Pasmanter, J. Sommeria, Phys. Fluids 11, 3465 (1999). 15. V. Latora, A. Rapisarda, C. Tsallis, Phys. Rev. E 64,056134 (2001). 16. A. Antoniazzi, F. Califano, D. Fanelli, S. Ruffo, Phys. Rev. Lett., 98,150602 (2007). 17. D. Lynden-Bell, Mon. Not. R. Astron. SOC.136,101 (1967). 18. P.H. Chavanis, J. Sommeria, R. Robert, ApJ 471,385 (1996); P.H. Chavanis, Ph. D Thesis, ENS Lyon (1996). 19. P.H. Chavanis, Physica A 359,177 (2006). 20. R. Ellis, K. Haven, B. Turkington, Nonlinearity 15,239 (2002). 21. P.H. Chavanis, A&A 451, 109 (2006). 22. Y.Y. Yamaguchi, J. Bar&, F. Bouchet, T. Dauxois, S. Ruffo, Physica A 337, 36 (2004). 23. M. Antoni, S. Ruffo, A. Torcini, Europhys. Lett. 66,645 (2004). 24. P.H. Chavanis, A&A 432,117 (2005). 25. A. Campa, A. Giansanti, G. Morelli, Phys. Rev. E 76,041117 (2007). 26. P.H. Chavanis, Physica A 365,102 (2006). 27. A. W. Francis, Liquid-liquid equilibrium, (Interscience, NY, 1963); C. M. Sorensen, Chem. Phys. Lett., 117,606 (1985).

zyxwvut

27

STOCHASTIC WEBS IN MULTIDIMENSIONS

zyxwv

G. M. ZASLAVSKY

Courant Institute of Mathematical Sciences and Department of Physics, New York University, New York, NY 10012, USA City E-mail: zaslavQcims.nyu.edu

M. EDELMAN Courant Institute of Mathematical Sciences, New York University, New York, NY 10018, USA City E-mail: edelmanQcims.nyu. edu Stochastic web is a thin net that penetrates phase space of dynamical system and has dimensionality less or equal to the dimension of the phase space. Dynamics within the stochastic web is chaotic and separated from the dynamics outside the web, which could be regular. Arnold web or a web with quasicrystal symmetry are examples of the stochastic web. Particles transport can be unbounded along the webs. We review the origin of the web with symmetry in 1 1 / 2 degrees of freedom and present examples of kicked coupled oscillators with the stochastic webs for 2 1/2 and 3 1 / 2 degrees of freedom. In both cases the web has co-dimension two in the phase space. Keywords: Chaos; Stochastic web; Quasicrystal symmetry, Coupled oscillators.

1. Introduction Weak perturbation of integrable system that leads to interaction of the system’s degrees of freedom can generate in the phase space a thin net of channels inside of which trajectories are chaotic. The net is called stochastic web (SW), and its emergence is an important physical phenomenon that imposes the transport properties of systems. Arnold web is a paradigm example of the SW, and the corresponding transport is known as Arnold Arnold web is a universal characteristic of nonlinear systems. It is generated by intersection of the resonance separatrices, and it exists under the condition that the number of degrees of freedom N > 2 and the

28

zyxwvuts zy zyxwvu

system is non-degenerate:

where & ( I ) is unperturbed Hamiltonian and action I E EN.Diffusion along the Arnold web is unbounded but very slow3 and that is confirmed by simulation in.4 Another situation occurs when the condition (1) fails and the unperturbed system is degenerate

or close to the degeneracy. As an example, such situations are typical for perturbed particle dynamics in a strong constant magnetic field.5>6The case (2) for N = 1 1 / 2 (kicked harmonic oscillator) was considered in5 and it was shown that the SW exists and that thin fibers of the SW can form a net with crystal or quasicrystal type symmetry depending on some resonant conditions. Another important feature of the SW with a symmetry is that diffusion along the web is not slow and, under specific values of the parameters, could be superdiff~sive.~ The corresponding dynamical equations can be also considered as dynamical generator of symmetric patterns that appears in different areas and objects: in fluid dissipative systems,12 and condensed matter and c r y s t a l l ~ g r a p h y .Recent ~ ~ ? ~ ~developments on the use of stochastic webs of crystal and quasicrystal symmetries are related to particle dynamics in optical latticesl59l6 and photonic quas i c r y ~ t a l s . l ~Particular -~~ interest is in the atoms’ dynamics and diffusion of Bose-Einstein condensates confined by optical lattices and quasicrystal symmetry20 and the existence of band structure.21i22Different important properties of SW were studied It is easy to see that if at least one degree of freedom is degenerate in the additive Hamiltonian of the system, i.e.

zy

and Hess Ho(I1) = 0, then the condition (2) is valid and full system is degenerate. Some systems of this kind were considered demonstrating fast diffusion process compared to the Arnold diffusion. The goal of this article is to review some results of30 and to continue study of SW for 2 1/2 and more degrees of freedom. The system under consideration consists of few coupled linear oscillators perturbed by a sequence of periodic kicks.

zy zyxwv z zy zyxwvu zyxw zy 29

2. Kicked Two Coupled Oscillators

Consider a Hamiltonian of two coupled linear oscillators perturbed nonlinearly with time-periodic kicks,

00 1 1 H = ~ ( p : t p ~ ) + ~ ( w : z : + w z 2 s ~ ) - K o T c o s ( l c l z l + k 2 z 2 ) b(t-nT),

C

n=-m

(4)

where pi = Xi (i = 1 , 2 ) , KOis a perturbation parameter, and T is a period of kicks. By introducing dimensionless variables and parameters lcl =

1,

IC2

= b;

51

= -v,

XI =w~u,

~2 =

-2,

X2

= w ~ Y , (5)

we can write the equation of motion as a map between two consequent kicks, u,+1 = w, w,+1

sin a1

= v, cos a1

Y,+I = 2, sin a2

Zn+1= 2, cos a2

+ [u,+ K1 sin(v, + bZ,)] cosa1 , - [u,+ K1 sin(v, + bZ,)] sin , + [Y,+ sin(v, + bZ,)] cos - [Y, + K2 sin(v, + bZ,)] sin a1

K2

a2,

(6)

a2,

where

ai = w ~ T , Ki = kiKoT (i = 1 , 2 ) , K2 = bK1. Appearance of the SWs is related to the resonance conditions a 2

= 27r/qi,

(7)

(8)

(i = 1,2)

where qi are integers (actually qi could be rational and we consider integer qi for simplicity). Coupling of two degrees of freedom is due to the phase

9, = 21,

+ b2,

(9)

A special case is for oscillators with equal frequencies = WT= 2

= a2

~/q,

(10)

zyxwvu

when an integral of motion exists. It follows from (6)

bv,) sin a + (Y, - bun) cos a , Zn+l- bw,+l = (2, - bun) cos a + (Y, - h,) sin a ,

(11)

g = (2, - bv,,Y, -bun)

(12)

Y,+1 -

i.e. vector

bu,+l

= (2,

-

rotates by a each step of the map. The invariant is its length

+

Igl2 = (2, - b ~ , ) ~ (Y, -

= inv

(13)

zyxw zy zyxwvuts zyx zyxwvut

30

and, since q is integer, we have in addition

(2, - bv,, Y, - bu,) = const,

(mod q )

(14)

i.e. points of trajectories are displayed in q invariant planes, in which the pattern (phase portrait) should be similar to the one in 2D-phase plane for N = 1 1/2. Simulations confirm the presented analysis. In Fig. 1 we present six planes of different projections of the web for q = 4. Straight lines correspond to the invariant lgI2 and Eq. (6). The web is located on four planes which intersect the (Y,u)and ( 2 , ~planes ) along 4 lines. The location of the lines depends on the initial conditions. Each of four webs, located on the corresponding four planes, can be obtained by a rotation of one of them by 27rk/q k = 0 , 1 , . . . ,q - 1).Similar SW appears for q = 5. In Fig. 2 we present only two projection planes. These figures show that the stochastic web fills 3D space but it has a tape-type structure located in a plane. More information on how the SW evolves in time can be seen from Fig. 3 (q = 8) and Fig. 4. 3. Symmetry of the Stochastic Web

The SW has invariant structure in phase space called web skeleton. For n = 1 1/2 the web skeletons were described in detail in.5 Following the same way, consider the case (10). Let us introduce the polar coordinates u =pcos$1; u = -psin$I; Y = Rcosql~; Z = -Rsin$Z;

I = w1p2/2, J = w2R2/2,

zyx (15)

that brings the Hamiltonian (4) to the form (up to a constant)

H

= 011

+ aaJ - KoTcos(psin41 + bRsin4z) x

c 00

S(T - n)

(16)

,=-m

with dimensionless time the generating function

7.

The first two terms in H can be excluded using

F = ($1

+

- C Y T ) ~( $ 2

- CYT)J.

(17)

The new Hamiltonian is

c 00

Hq = H+dF/dr= -KoTcos[psin($l+a~)+bRsin(&+a~)] x

S(T-~),

n=-m

(18)

31

50

50 I

N

h

0

0

zyxwv i

U

5

-90

0

_1

50

0

V

0

50

1

V

zyxwvut zyxwvu 0

50

9

0

V

0

50

Fig. 1. Six projections of the stochastic web for q = 4 present the four-fold symmetric structure in 3D space (u, v, Z ) ; K1 = 1; KZ = 0.5.

32 1000

zyxwvutsrqp

333

> -333

-loon .__ -1000

zyxwv zyxwvuts zyxwvutsr zyxwvu zyx -333

Fig. 2 .

where

U

1000

333

( u ,w ) and (u,Y )projections for q = 5 ; K1 = 0.44; Kz = 0.22.

(19) are phases in the rotational frame of reference. The invariant structure of the SW can be obtained from (18) if we present H , in the split form

H,

= V,

+ ABg(t)

(20) where V, is time independent, and perform an averaging over time. It gives 1 V, = ---K~

4

C [cos(r.ejl + U) + cos(u + b

~ejz)] .

(21)

zyxwv 2q

j1jz=l

where ejl,zare unit vectors ej1,2

= (cos(27rj1,2/q),

sin(27rj1,2/q),

j1,z

= 1, * * . 14

(22)

r is a vector in (u, u ) space and R is a vector in (Y,2)space. The particular case of q = 5 was considered in.30 Expression (21) defines two coupled degrees of freedom and their interaction could, in general, lead to chaotic trajectories. Nevertheless, the conservation law (14) provides another possibility to present V, in the form

where f . = (u,V),

R = (Y,Z)

(24)

zyxwvutsrq 33

1000

333

500

zyxwvutsrq

> -333

-1000 -1000

-333

U

333

1000

zyxwv zyxw -1000

-333

U

333

1000

500

167

>

-167

-1000

-333

,,

-500

333

1000

-1000

-333

U

v

333

1000

500

167

N -167

-1000

-333

Fig. 3.

v

zyxw zyxw zyxw zyxwv 333

1000

-500

Y

500

167

Six projections of the SW for q = 8; K1 = 0.44;Kz = 0.22

with renormalized components

v = const. v ,

Z = const. z

34

zyxwvuts zyxwvutsrqponm ,

200

%

67

> -67

-200

-200

-67

U

67

200

zyxwv -30

-10

10

30

Y7

20

U

60

20

N -20

-60

-60 -20

Fig. 4.

Y 2o

-20

60

-7

zyxw zyxw zyx zyxwvut z

Four consequent magnifications of the SW from Fig. 3.

with renormalize and the constants are defined from the initial conditions and conservation laws (4). The new form (23) for V, shows additive splitting of V, onto two independent structures in 4D phase space. By structure we consider a topological structure of the phase portrait. Projection of the 4D structures on 3D hypersurface reveals 3D structures. For example, if Y = 0, then one can present P

V, = const.

C cos(r1 .

ej)

j=O

where rl = (u,B, Z),ej for j = 0 is a unit vector along 2, and ej = e j, (jl = 1,. . . , q ) are defined in (22). The contour plots for different projections

35 -100

100

-100

I00

-inn

inn

-100

nn I O(

100

in 100

zyxwvuts zyxwv zyx zyxw -100

Fig. 5 . SW skeleton projections: (Top row) (z, y, 0 ) , and (2,y, 1);(Bottom row) (z, 0, z ) , and ( z 1 2 , z ) . Contour lines correspond to the levels V5(z, y, z ) = - 1 , l , 3,4.

(26) are shown in Fig. 5 for q = 5. The patterns in Fig. 5 corresponds to the 2D projections of the 3D quasicrystal with 5-fold ~ y m m e t r y . ~Different ' other symmetric tilings can be obtained in a similar way by changing q.

4. More Coupled Oscillators

zy zy

The goal of this section is to show that system (6) can be generalized to more degrees of freedom. Here we provide an example for three coupled oscillators ( N = 3 1/2). More oscillators can be considered in a similar way. Let the coordinate and momentum for each oscillator be given as (yi,xi),

36

zyxwvuts

i = 1 , 2 , 3 . The Hamiltonian of the system is similar to (4): .

3

i=l The equation of motion can be reduced to the map similar to (6): zi,n+l

(28)

zyxw zyx

yi,n+i = Yi,n COSQi

where

+ + Ki sin$,,] cos ai - [ X i , n + Ki sin$,] Sinai, (i = 1 , 2 , 3 ) ,

= Yi,n sin ( ~ i [xi,,

3

ai = w ~ T , Ki

= IciKoT,

$n =

C ki~i,~ i=l

(29)

Consider the case of equal frequencies ai =

=2~/q,

(Vi),

(30)

where q is an integer. Symmetry of the equations (28) shows the existence of two independent integrals of motion It follows that K251,n+1 - K 1 ~ 2 , n += i (K2~1,n - Klyz,n) s i n a

+

(K2z1,n - Kiz2,n)C O S ~

K2~1,n+1- K1~2,n+i= (K2~1,n - Kiy2,n) COSQ - (K2z1,n - K 1 ~ 2 ,s ~ i n)a (31) Equations (31) preserves the length of the vector g12 =

( K 2 ~ 1- K 1 ~ 2K2zi , - Kiz2) = inv

(32)

Similarly one can obtain preservation of the vectors g13 and g23 that can be obtained from g12 by a corresponding change of subscripts. Only two of the invariants are independent. The SW in this case looks similar to what is for two coupled oscillators. It is located in 2D layers and the layers are embedded in 6D space. The invariants (32) reduce the effective dimension by 2 where chaotic diffusion is performed. That means that effectively the SW is located in the space of co-dimension 2 if the condition (30) is applied. That is similar to the case of two coupled oscillators. Examples of the SW for the cases q = 4 and q = 5 are given in Figs. 6,7 and the diffusion for q = 4 is presented in Fig. 7.

37

7

zyxwv zy zyx zy -3

XI

Fig. 6.

1

-1

3

Xl

Different projections of the SW for 3 coupled oscillators for q = 4.

60

100

20

rl

h

h“

0-

-20

-!to

-20

XI

20

60 -’YO0

x1

O

100

Fig. 7. Diffusion along the SW for 3 coupled oscillators for q = 4 and q = 5. Projections on ( $ 1 , y1) - plane.

zy zyxwvut

5 . Conclusion

We have demonstrated that SW of the degenerate type can occur in N = 2 1/2 and N = 3 1/2 degrees of freedom. The same type of the SW can be assumed to exist in higher dimensions. The SW persists to have a symmetry or quasi-symmetry inherited from the N = 1 1/2 case in 2D phase space. A particle wanders in a “sandwich type” topological structure and each layer is similar to N = 1 1/2 case with rotations by 27r/q from layer to layer. The

38

main differences of t h e SW from t h e Arnold web are t h e web symmetry a n d fast particles diffusion along t h e web.

Acknowledgments This work was partly supported by t h e Office of Naval Research G r a n t No. N00014-02-0056. P a r t of t h e work was performed at CPT, Aix-Marseille University.

References

zyxwvut zyxw zy z zy zyxwv

1. V. I. Arnold, Dokl. Akad. Nauk SSSR 156,9 (1964) [Sou. Math. Dokl. 5 , 581 (1964)]. 2. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Dynamical Systems 111. Encyclopedia of Mathematical Sciences), 3rd ed. (Springer, New York, 2006). 3. N. N. Nekhoroshev, RUSS.Math. Surveys 32,1 (1977). 4. C. FroeschlC, E. Lega and M. Guzzo, Celest. Mech. Dyn. Astron. 95, 141 (2006). 5. G. M. Zaslavsky, M. Yu. Zakharov, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Sou. Phys. JETP 64,294 (1986). 6. G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Weak Chaos and Quasiregular Patterns (Cambridge University Press, Cambridge, UK, 1991). 7. G. M. Zaslavsky, Phys. Report 371,461 (2002). 8. V. V. Beloshapkin, A. A. Chernikov, M. Ya. Natenzon et al., Nature 337, 133 (1989). 9. J. Gollub, Proc. Natl. Acad. Sci. U.S.A. 92,6705 (1995). 10. W. S. Edwards and S. Fauve, Phys. Rev. A bf 47, R788 (1993). 11. W. S. Edwards and S. Fauve, J. Fluid Mech. 278,123 (1994). 12. M. Field and M. Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (Oxford University Press, Oxford, UK, 1992). 13. J. S. W. Lamb, J . Phys. A 26,2921 (1993). 14. A. Baz&n, M. Torres, G. Chiappe et al., Phys. Rev. Lett. 97,124501 (2006). 15. S. Marksteiner, K. Ellinger, and P. Zoller, Phys. Rev. A 53,3409 (1996). 16. S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79,4790 (1997). 17. B. Freedman, G. Barta, M. Segev et al., Nature 440,1166 (2006). 18. L. Guidoni, B. DBpret, A. di Stefano, and P. Verkerk, Phys. Rev. A 60,R4233 (1999). 19. L. Guidoni, C. TrichB, P. Verkerk, and G. Grinberg, Phys. Rev. Lett. 79,3363 (1997). 20. L. Sanchez-Palencia and L. Santos, Phys. Rev. A 72,053607 (2005). 21. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, Nature 436,993 (2005). 22. M.A. Kalitievsky, S. Brand, T.F. Krause et al., Nanotechnology 11, 274 (2000).

zyxwvu zyxw zyxw zyxwvu 39

I. Dana and M. Arnit, Phys. Rev. E, 51, R2731 (1995). S. Pekarsky and V. Rom-Kedar, Phys. Lett. A 225, 274 (1997). J. H. Lowenstein, Phys. Rev. E 47,R3811 (1993). J. H.Lowenstein, Chaos 5,566 (1995). G. M. Zaslavsky, M. Yu. Zakharov, A. I. Neishtadt, R. Z. Sagdeev, D. A. Usikov, and A. A. Chernikov, Sou. Phys. JETP 69,885 (1990). 28. A. A. Chernikov and A. V. Rogalsky, Chaos 4,35 (1994). 29. V. V. Beloshapkin, A. G. Tretyakov, and G. M. Zaslavsky, Commun. Pure A p p l . Math. 47,39 (1994). 30. G. M. Zaslavsky and M. Edelman, Chaos 17,023127.

23. 24. 25. 26. 27.

40

zyxwvutsrq CHAOTIC GEODESICS

zyxwv

J.-L. THIFFEAULT

Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: [email protected]. edu K. KAMHAWI Department of Mathematics, Imperial College London, London, S W 7 2AZ, UK

When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories ace, to leading order in the layer thickness, geodesics on the twodimensional curved space of the substrate. Since the two-dimensional geodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates. Keywords: shallow water flows; chaotic advection; particle transport

1. Introduction

Many well-known physical systems take the form of geodesic flow on a manifold. For instance, Euler’s equation can be though of as a geodesic flow in the space of volume-preserving diffeomorphisms, and free rigid body motion as geodesic flow in SO(3). In both cases, the metric on the space corresponds to the kinetic energy norm. It is also known that the geodesic deviation equation7 describes the stability of such flows. For instance, a space of negative curvature will lead to divergence of trajectories, and hence to chaos if the space is compact. But compact spaces of strictly negative curvature are hard to come by in the real world, to say the least. If we expect the negative curvature to lead to chaotic geodesics, we are better off looking for spaces with non-sign-definite curvature, but such that the averaging of the curvature over trajectories leads to chaos ( i e . , the negative curvature ‘wins’).

41

In this contribution we will discuss a system which is physicallymotivated and leads to chaotic geodesics. This system is the flow of a shallow layer of ideal, irrotational fluid on a curved substrate. Following Rienstra,8 we will show that, to leading order, the governing equation can be solved in terms of characteristics. Moreover, the characteristics are geodesics on the curved substrate, possibly modified by gravity if it is present. Of course, the chaotic trajectories have a nasty tendency to cross and form caustics everywhere. Hydrodynamically, caustics are usually manifested as hydraulic jumps or so-called ‘mass tube^,'^ visible as a thicker edge region of the fluid in Fig. 1 (top). Edwards et al.g have recently described

zyxw

Fig. 1. Experiment in the kitchen sink, using a cut-open plastic bottle (top). The jet from the faucet impacts the inclined bottle. The pattern is qualitatively well reproduced by following fluid trajectories emanating from a point source on a cylinder (middle). However, if we pursue the trajectories further (bottom), the ideal theory presented here predicts that the flow should crawl back up the side to the same initial height. The discrepancy is clearly due to dissipation.

these mass tubes using the theory of ‘ d e l t a - s h ~ c k s . At ’ ~ ~this ~ point, we are unable to apply this theory to our problem, which means that solutions become dubious after characteristics begin to cross. Unfortunately,

42

zyxwvu zyxw zyxwvuts zyxwv zyxw zyxwvut

since our characteristics are chaotic, they tend to cross a lot. Nevertheless, we believe that studying the basic properties of this geodesic flow is worthwhile as a first stab at describing the transport properties of flows on curved substrates. In addition, the geodesic flow we present is an interesting mathematical system, with rich dynamics that deserve to be studied on their own. Having gotten these disclaimers out of the way, let us proceed with the analysis. We hall do this in stages. In Section 2 we introduce a curved non-orthogonal coordinate system to describe the substrate, singling out a direction normal to the substrate. In Section 3 we use this direction to expand our fluid equations and derive a shallow-layer form. We show that the resulting equation can be solved in terms of characteristics, which are geodesics on the two-dimensional curved substrate, modified by gravity. In Section 4 we look at specific numerical solutions for particle trajectories, and in Section 5 we speculate on their chaotic nature. We offer some closing comments in Section 6. 2. Coordinate System

2.1. Separating the Shallow Direction

In our problem, fluid motion occurs over a curved substrate of arbitrary shape. The direction normal to the substrate is special in that it defines the direction in which the fluid layer is assumed 'shallow.' Hence, it is convenient to locate a point r in the fluid as

r ( x 1 , x 2y) , = x ( z 1 , z 2+ ) y63(z1,z2)

(1)

where X ( d ,x 2 ) is the location of the substrate, 63 is a unit vector normal to the substrate, and y is the perpendicular distance from r to the substrate. The coordinates x1 and x 2 are substrate coordinates used to localise points on the substrate. For example, in Section 2.2 we will use the Monge T parametrisation, X = (dx 2 f(d, x 2 ) ) , where f gives the height of the substrate. The tangent vectors to the substrate are

e , := d,X

(2)

where 8, := d/dx".The coordinate vectors associated with the coordinate system are found from (l), cZa := d,r

= e,

+ U(y) ,

dr 63 := - . dY

(3)

zy zyxwvu zyx zyxw zyxwv zyxw

zyx zy 43

where Greek indices only take the value 1 or 2. Note that the e, are not necessarily orthogonal or normalised. We adopt the convention that quantities with a tilde are evaluated in the ‘bulk’ (away from the substrate), and thus depend on y, whilst those without the tilde are ‘substrate’ quantities and do not depend on y. Thus, &(d, x2,0) = e,(&, x2). The three-dimensional metric tensor g a b has components

Baa where

:= I?,

-

. Ep = Gap ,

-

Gap :=

-

ija3

e,

Gap := e,

:= e,

. e3 = 0 , A

= Gap

833

:= &, .&3 = 1,

+ U(y) ,

. ep .

(4)

The three-dimensional metric tensor is thus block-diagonal, and the y coordinate is unstretched compared to the Cartesian coordinate system. It measures the true perpendicular distance from the substrate to a point in the fluid. Given the substrate vectors e,, it is easy to solve for the covectors eQ, which are such that eQ . ep = &pa. Then the bulk covectors are dQ = e0l

+ (3(y),

(5)

to leading order in y. From (5), we find the inverse metrics,

Higher-order terms in y will not be needed. 2.2. Substrate Coordinates For most applications in the literature of thin films and shallow layers, orthonormal coordinates have been the coordinates of choice. This is because the main substrate shapes that have been treated are planes, cylinders, and spheres, where orthonormal coordinates are readily available. For a general substrate shape, orthonormal coordinates are difficult to construct and require numerical integration. Singularities (umbilics) also cause problems. l4 For our application-flow down a curved substrate-the Monge representation of a surface15 is the most convenient. The Monge representation is a glorified name for a parametrisation of the substrate by

44

in three-dimensional Cartesian space. Following standard notation,15 we define

zyxw zyx zyxw zyxw zyxwv zyxwv

The unnormalised, nonorthogonal tangents el and el = a1X = (1 O P )

T

,

e2

e2

are

= ~ Z X= (0 1 4 )

T

,

and their normalised cross product gives the normal to the substrate, 1

T

( - p -4 1)

&3 = W

,

The corresponding covectors are 1

e1 = 3 ((1+ q 2 ) -P4 P )

T

7

1 T e2 = 3 (-P4 (1+ P 2 ) 4 )

and 63 is its own covector. The metric tensor of the substrate and its inverse are

with determinant

w

=

(det G a p ) 1’2 =

zyx

d

W

.

Finally, we will need the Christoffel symbols r;,, defined by

rzp = pY ( a a ~ r+pa&, and in Monge coordinates given by

-

The Christoffel symbols arise when taking covariant derivative^.^^^^^^^ Note that in (10) we used the usual convention that repeated indices are summed. We write the normalised gravity vector as

a

g = (sin cos 4 sin a sin 4

- cos

,

(12)

zyx

zyxwv zyxwvuts zy zyxwv 45

so that the inclination angle 0 is zero for gravity pointing downwards, and for $ E (-7r/2,7r/2) positive 6 induces flow in the positive z1 direction. Then we have the components

+ q 2 )sinecos$) /w2 , gs2 = g . e2 = - ( q cose + p q sinecos 4 - (1+ p 2 ) sinesin$) /w2 , gsl = g . el = - ( p cose + p q s i n e s i n 4 - (1

(13)

The specific parametrisation of the substrate introduced in this section will not be needed in the derivation of the equations of motion (Section 3), only in their solution. Hence, a different parametrisation could be used if called for by the geometry of the substrate. For instance, flow down a curved filament is better parametrised by cylindrical coordinates, or if the substrate has overhangs (making f multivalued) coordinates based on arc length are preferable.

3. Equations of Motion

Now that we have set up an appropriate coordinate system on our curved substrate, we need some dynamical equations of motion for the fluid. We assume an inviscid, irrotational fluid with a free surface at y = r](zl,x 2 ) , with slip boundary conditions at the substrate y = 0. The pressure on the free surface is assumed constant (zero). We also assume the flow is steady and irrotational, so that the the velocity can be written in terms of a scalar potential, u = Vp. The equations satisfied by the fluid are then

v2p= 0, 1

IVpI2 + P- - g . r P

=H,

mass conservation;

(144

Bernoulli's law;

(14b)

where H is a constant, with boundary conditions

ayp= 0 a t y = 0, V p . Vr] = ayp a t y = r]: p =0 at y = r],

no-throughflow a t substrate; (15a) kinematic condition at free surface; (15b) constant pressure at free surface.

In terms of our curvilinear coordinates, equation (14b) becomes

(15c)

46

zyxwvuts zyxwvuts

zyxwvut zyxw zyxwvuts zyxwvutsrqponmlkj zyx z

3.1. Small-parameter Expansion

Now we assume that the fluid layer is shallow, so that y is proportional to E . After replacing y by E Y , Eq. (14b) becomes

+

2P + ~ - ~ ( d ~+c p ) ~29. ( X + ~ P

(GOip O ( E ) ) a,cpapcp

-

We also expand cp in powers of cp(z

1

~

6 =32H. )

(17)

E,

,32 2 Y) = V(0) + E P(1)+ 2 P(2) 3- . . . . 7

(18)

The leading-order term in (17) occurs a t order E - ~ , and gives d , ~ p ( ~=) 0. Hence, we have cp(o) = @ ( x 1 , x 2 independent ) of y, The next nontrivial terms are a t order E O , GapaOi(a dp(a

2P + (d,(p(1))2+ P

-

2 g . X = 2H.

(19)

We evaluate the whole of (19) at y = 0, and use the boundary conditions (15a) and (15c),

~@a,@ ap(a- 2 9 . x = 2

~ .

(20)

This is the equation that we need to solve to find the leading-order velocity potential @(xl,z2).We discuss the method of solution in the next section. Note that we will not need to solve the mass conservation equation (14a): at leading order, it only serves to find the fluid height once the velocity field is obtained.

zyxw

3.2. Solution in Terms of Characteristics

As pointed out by Rienstra,' the trick to solving Eq. (20) is to use the method of characteristics. To do this, we differentiate (20) with respect t o x,, which gets rid of the constant H, 2GOiOaa@a,ap@

+ a,G@aa@ a p @ = 29 a,x. *

The horizontal components of velocity are k a denotes a time derivative; hence,

a&@

= ap(G&6)

= G,sdpP

From the chain rule, we have x ' = dpi?@; find, after dividing by two,

G,~z'

= G@dp@,where the

+ dpG,sP.

(21) overdot

(22)

using this and ( 2 2 ) in (21),we

+ d p ~ , s @ i + + i i ~ , ~ ~ p ~ , s ~ p , i =? kg p. ey.

(23)

mjzyxwvutsrqponm f1

47 zyxwvuts

10

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC

,

fo=OZ

0 -20

-10

10

0

20

11

30

zyxwvutsrq

zyx zyxwv -20

-10

0

10

20

21

30

zyxwvutsrqponmlkjihgfedcbaZYXW -20 -10 0 10 20 30 7'

Fig. 2. A pencil of 30 trajectories starting from the origin at different angles, each with initial kinetic energy 1/2. The substrate has shape f ( z l , s2)= fo coszl coss2, and the plots are for different values of fo. Gravity is turned off g = 0. The grey background shows the periodicity of the substrate.

Now we multiply by bit of manipulation

GUY,

and obtain after an integration by parts and a

$0

+ rZ0kak0

=9.

(24)

where the rgp are defined by (11).Equation (24) describes geodesics in the curved coordinates of the substrate, under the influence of gravity. In the absence of gravity, the fluid trajectories are essentially going in straight lines in the curved substrate coordinates. (In general relativity, unlike here, the gravity determines the curvature of space.) If we define the covariant derivative of a vector V" along the trajectory,7,16-18

zyx

where T is the time (to avoid confusion with t in Eq. (8)),then the geodesic equation (24) takes the more intuitive form u -xu

= g . e' (26) Or which looks a lot like Newton's second law, but here it incorporates the constraint that fluid particles remain on the substrate. Equation (24) is a two degree-of-freedom autonomous Hamiltonian system, with the energy H defined by equation (20) as an invariant. Hence,

z

48

any other invariant will make the system integrable, and rule out chaos. In particular, a surface with a translational symmetry cannot exhibit chaos. 4. Fluid Particle Trajectories

To get a feel for the possible range of behaviour of fluid trajectories, we now solve the geodesic equation (24) for a range of substrates. We shall always use a set of fluid trajectories starting at the same spatial point, with the same initial kinetic energy but different direction. This models a point source, or a thin jet impacting the substrate. First, following Rienstra8 we solve the equations on a cylindrical substrate. Figure 1 (middle) shows some trajectories, all emanating from the same point. Qualitatively, the pattern captures well the observed behaviour of a jet (from a faucet) impacting the inside of a cut-out plastic bottle (Fig. 1, top). However, if we pursue the trajectories further (Fig. 1, bottom), we see that they crawl back up the side of the cylinder, with no loss

60

50 50

40

40

30

zyxwv

"h 30

20

20

10

10

0

0

0

10

20

30

0

20

51

40

zyxwv 2.1

60

60

40

40

+

N

* 20

20

0

0 -20

0

20

40

-50

0

50

Fig. 3. Same parameters as in Fig. 2, but with gravity turned on: g = 1. The trajectories exhibit chaos-like behaviour for much lower substrate height, since they begin at the top of a bump and thus have potential energy to draw upon.

zy zyxwvutsrqpon 49

1

35

30 25 20 15

zyxwvutsrqpo

10 5

0

10

20

30

0

10

20

10

20

zyxwvutsrqpon 10

20

0

5

10

-10

-

0

-20 -10

0 -5

-10

zyxwvut zyxwvu -15

-30

-10

n

-20

-10

0

:?

-10

20

30

0

10

20

I'

Fig. 4. Same parameters as in Fig. 3, but with initial condition (z',z2) = ( ~ / 2 , 0 ) .For larger substrate amplitudes, the system is dominated by 'rimming,' where particles skim a depression before moving t o the next one, or sometimes undergo long flights.

of energy, in contrast to the experimental picture. This comes from neglecting the hydraulic jumps that occur, as well as v i s c ~ s i t y Observe .~ that the trajectories follow a very ordered pattern, and are definitely not chaotic. This is as expected, since there is a symmetry direction, and so the motion is integrable (Section 3). Next we move on to more complex substrates. Since there is basically an infinity of choices here, we limit ourselves to periodic substrates with shape 1 2 f(z1,z2) = fo sin z sin z

(27)

for various values of fo. The other variables in the system are the strength of gravity (which can be chosen as unity if it is not zero, by rescaling the substrate height) and its orientation (as given by the angles 0 and q5 in Eq. (12)). Figure 2 shows a pencil of 30 trajectories starting from the origin a t different angles, each with initial kinetic energy 1/2, for different values of fo, in the absence of gravity. The first two cases display regular behaviour, but for substrate heights fo = 0.7 there is chaotic-like behaviour. These are, however, fairly extreme values of fo, corresponding to heavily-deformed

50

I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED 0 -1

0

1

2

3

4

zyxwvutsrqponmlkjih

21

Fig. 5. Same parameters as in Fig. 3, but with initial condition (x1,x2)= ( n , O ) . The particles begin at the bottom of the potential well, and they do not have the energy to escape.

zyx zyxwvu

substrates. Our expansion should be able to accommodate this, since the variations in the substrate height are not assumed small (only those in the fluid thickness are). For extreme heights (fo = 1.2, last case in Fig. 2), some trajectories actually backfire and come around the initial point. Figure 3 shows results for the same parameters as Fig. 2, but with gravity g = 1. The inclination is nil ( 0 = 0). It is clear that chaotic-like behaviour sets in for much smaller values of fo, even showing backscatter for fo = 0.5 in the last frame. This is because the fluid elements can now draw on the potential energy they inherit from starting at the top of the bump. This suggests that, in the presence of gravity, the results should be substantially different if we start elsewhere on the substrate. Figure 4 shows simulations with the same parameters as in Fig. 3, but starting at (xl, x2) = (n/2,0),some way down the bump. The motion is then confined to narrow channels for moderate fo. But for larger substrate amplitudes, the system is dominated by ‘rimming,’ where particles skim a depression before moving to the next one, or sometimes undergo long flights. This is a similar situation to basketball (or golf), where the ball turns around the hoop (or cup) a while before deciding to go in or out. If we take an initial condition at the bottom of the bump, (x1,x2)= (n,O),then the trajectories do not have enough energy to escape the potential well (Fig. 5). Finally, in Fig. 6 we illustrate the effect of inclining the substrate at an angle B = nl8. With q5 = -n/2 the trajectories flow ‘downhill,’ in the negative x2 direction, modified by the bumps. The system still appears to becomes chaotic for larger fo. Larger bumps induce a ‘shadow’effect, where they prevent fluid from flowing behind them (in particular for fo = 0.6).

zyxwv zyxwvu zyxw

51

-601 -20

0

-20

zyxwvu zy zyxwvutsr 20

0

-20 -10

3.1

0

0

-10

-10

0

10

20

il

0 -10

-20

-20 -20

'-30

--30 -40

-50 -60

-30

zyx zyxwvutsrqponmlk zyxwvutsrqp -40

-40

-50

-20

0 Ll

20

zyxw -50

-20

0

20

11

-20

0

9

20

8 q5 = -7r/2. For Fig. 6. Same parameters as in Fig. 3, but with incline 0 = ~ / and larger bumps there is a 'shadow' effect, particularly for fo = 0.6.

5 . Lyapunov Exponents and Chaos

zy

We now investigate whether the behaviour described in the previous section is chaotic or not. The first variation of the geodesic Eq. (24) gives

6X'

+ 2rzp~ ~ 6 j +. Payrgpj . a ~ f 1 6 ~=y -9.er r,",dxy

(28)

where we have used ayeu = -I?& er. This equation can be massaged into the geodesic deviation e q z ~ a t i o n , ~ ~ ~ ~

D2 -6s" Or2

+ RapyaX a X p 6 ~ ' = 0

(29)

where DID7 is defined in ( 2 5 ) , and

R~~~~:= ayrZp- a a q p+ r;&

-

rgAr&

(30)

is the Riemann curvature tensor. For two-dimensional surfaces, the curvature tensor simplifies to Rupya = 9 ( y ' S

Gpa - 6"a Goy)

(31)

52

zyxwv zy zyxw zyxwvutsr zyxw

where 9 = (rt - s2)/w4 is the Gaussian curvature and Eq. (9). Hence, a simplified form of (28) for surfaces is 0 2

-axu

0 7 2

Go, is given by

+ 9 ((i,i)ax" - (i,ax) i") = 0

where the inner product is defined by (V,W ) := G,pV"Wo. Note that the gravitational term does not enter Eq. (28) directly, though it does through (24). For the rest of this discussion we will assume g = 0, since it simplifies the discussion considerably. If that is then case, then it is easy to show that

D Or

-@,ax)

D

= (i,-ax), 07

0 2

zyx

-(k,dx) Or2

=0

(33)

which means that if we choose the initial axQ such that (x,bx) = (5,D b x / D r ) = 0, then (i, ax) remains zero for all time. With this choice initial condition, the geodesic deviation equation (32) finally takes the form

D2 -ax" 0 7 2

+ S ( i , i )62" = 0.

(34)

Now we can ask under what condition the substrate shape will be favourable to chaotic geodesics. Since Eq. (34) resembles an oscillator equation, and (5,i)1 0, we see that negative Gaussian curvature will favour divergence of trajectories. We have not yet solved Eq. (28) for bx", but a comparison of the distance between two initially very close trajectories is shown in Fig. 7, for the same parameters as in Fig. 3 (fo = 0.5). Unsurprisingly, the plot confirms exponential growth, demonstrating at least numerically that chaos is indeed present.

zyxwvu

6 . Discussion

We have shown that the flow of a shallow layer of inviscid, irrotational fluid on a curved substrate leads to particle trajectories that follow geodesics in the curved space, subject to gravity. We have displayed the range of behaviour that these geodesics can exhibit, from regular t o chaotic. As Fig. 1 shows, the theory is not likely to be valid much beyond the point where characteristics cross, and viscosity also causes important corrections. Another effect we ignored is the possibility that centrifugal forces can cause the fluid to spin out and detach from the ~ u b s t r a t e .Experi~ ments are needed to determine to what extent the chaos observed here is

z zyxwvuts 53

1oo

zyxwvu

1o-2.

H

Lo

-

1o-6

zyxwvu 20

40

60

80

Fig. 7. The Cartesian distance 16x1 between two trajectories, for the same parameters as in Fig. 3 with fo = 0.5. The trajectories diverge extremely rapidly, consistent with chaotic behaviour.

reproduced in reality. If chaos is indeed prevalent, then perhaps chaotic advection can be exploited in some applications to enhance mixing in shallow layers. We made the case in the introduction that chaos in the geodesic equations was a subject worthy of study on its own. The emergence of chaotic behaviour as a function of Gaussian curvature, as embodied by Eq. (32)) should be a rich subject of study, in particular because of the simple form this equation takes on a surface. We note in closing that a similar study can be made for viscous thin films.'' However, trajectories there are much less prone to chaotic behaviour. because of the diminished role of inertia.

References

zyxwvu zyxwv

1. V. I. Arnold, Ann. Inst. Fourier 16,319 (1966). 2. V. I. Arnold, Usp. Mat. Nauk. 24, 225 (1969). 3. V. I. Arnold, Mathematical Methods of Classical Mechanics, second edn. (Springer-Verlag, New York, 1989). 4. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry

(Springer-Verlag,Berlin, 1994). 5. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics (Springer-Verlag,New York, 1998). 6. Y . Watanabe, Physica D 225, 197 (2007). 7. R. M. Wald, General Relativity (Universityof Chicago Press, Chicago, 1984).

54

zyxwvut zyxwv zyxw

8. S. W. Rienstra, Z A M M 7 6 , 423 (1996). 9. C. M. Edwards, S. D. Howison, H. Ockendon and J. R. Ockendon, I M A J . Appl. Math. (2007), in press. 10. F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, ed. B. Perthame, Advances in Mathematics for Applied Sciences, Vol. 22 (World Scientific, 1994). 11. J. Li, T. Zhang and S. Yang, Two-dimensional Riemann Problems in Gas Dynamics (Chapman & Hall/CRC Press, Boca Raton, FL, 1998). 12. H. Yang, J. Diff. Eqns. 159,447 (1999). 13. J. Li, Appl. Math. Lett. 14,519 (2001). 14. I. Kreyszig, DifSerential Geometry (University of Toronto Press, Toronto, 1959). 15. H. Flanders, Differential Forms with Applications to the Physical Sciences (Dover, New York, 1990). 16. J. L. Synge and A. Schild, Tensor Calculus (Dover, New York, 1978). 17. B. Schutz, Differential Geometry (Cambridge University Press, Cambridge, U.K., 1980). 18. J.-L. Thiffeault, J . Phys. A 34,5875 (2001). a step-by-step derivation of the geodesic deviation 19. For equation see http://io.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/ GeodesicDeviation.htm.

20. J.-L. Thiffeault and K. Kamhawi, http: //arXiv. org/abs/nlin/0607075 (2006).

55

zyxwv zyxwv zyxwv

A STEADY MIXING FLOW WITH NO-SLIP BOUNDARIES R.S. MACKAY

Mathematics Institute, University of Warwick, Coventry CVq 7AL, U.K. E-mail: R.S. MacKayOwamick.ac.uk

A steady mixing (even Bernoulli) smooth volume-preserving vector field in a bounded container in W3 with smooth no-slip boundary is constructed. An interesting feature is that it is structurally stable in the class of C3 volumepreserving vector fields on the given domain of W3 with smooth no-slip boundary, thus if one could think how to drive it then it would be physically realisable. It is pointed out, however, that no flow with no-slip boundaries can mix faster than l / t 2 in time t .

1. Introduction The possibility that the motion of ideal particles in a steady or time-periodic fluid flow could be chaotic was proposed by A r n ~ l ’ dstudied ,~ by people like HBnonlg and Zel’dovich, and was part of the standard training a t Princeton Plasma Physics Lab in 1978. It was found in convectiong by 1983, but did not come to the attention of the fluid mechanics community at large until the article of Aref,’ who christened the phenomenon “chaotic advection” . The subsequent development of the subject has been reviewed in Ref. 3. The ultimate in chaotic advection would be a mixing flow, in the ergodic theorist’s sense: a flow q!~ : R x M -+ M , ( t ,x) H &(x) on a manifold M preserving finite volume p is (strongly) mizing if for all measurable A , B c M , then

zyx zyxw zyxw

(no molecular diffusion is involved). Yet as far as I am aware, no-one has made an example of a fluid flow which is proved to be mixing. Most examples in the literature have, or are suspected t o have, tiny unmixed “islands” (at fixed phase for a time-periodic 2D flow) or long thin invariant solid tori (for a steady 3D flow).

zyxwv zy

56

Furthermore, to be realistic for engineering purposes such a flow should be constructed in a container in R3 with no-slip boundary (an alternative for a physicist could be a flow in a gravitationally or surface-tension bounded ball, but let us restrict to the case of a no-slip container). So, a further 23 years on from Ref. 2, this paper constructs a steady mixing volume-preserving flow in a bounded container in R3 with no-slip boundary. Interestingly, the tools have been available in the pure mathematics literature since 1975. The paper leaves open the question of how one might drive such a flow, but makes two further significant points. Firstly, the flow can be proved structurally stable within the class of C3 volume-preserving vector fields in the interior of the given container with no-slip boundary. Thus all “nearby” flows are topologically equivalent to the given one, and any such flow is mixing. This robustness gives the hope that such an example can be realised physically. The proof will be published elsewhere. Secondly, it is proved that no C2 volume-preserving vector field with C2 no-slip boundaries mixes faster than l/t2 in time t , in a sense to be made precise. The paper concludes with a discussion of possible variants and additional results.

z zyxwvu zy

2. The construction

I begin from a steady vector field which I call s (Figure l),proposed by Arnol’d5 (who showed it to be irrotational Euler for a Riemannian metric to be recalled in (2)). It is the suspension vector field of the automorphism

A =

[;i]

of the 2-torus T2 = R2/Z2. This means it is the vector field

(O,O, 1) in components where

(2, y,

z ) on the quotient space M = (T2 x [O,l])/a

a(x,1) = (Ax, 0)

for x = (x, y) E T2 (meaning that points (x,1) and (Ax,O)are t o be considered as the same). M is a C” manifold and s is a C” vector field on it. It preserves volume dx A dy A dz and has exponentially contracting and backwards contracting subbundles E* leading (by direct sum with the vector field) to invariant foliations .F* by the “planes” y = -yx -t c+ and y = z/y c-, respectively, where y = (1 &)/2 is golden ratio and c* denote arbitrary constants, as a result of which it is ergodic and

zyxw +

+

z 57

\

I

[::I

zyxwvu zyx zyxw zyxw zyx zyxw Fig. 1. The suspension flow s.

C1-structurally stable. It is not physically realisable, however, because the suspension manifold M can not be embedded in R3. The orbit of ( 0 ,0,O)is periodic and hyperbolic. Blow it up to a cylinder by the inverse of the mapping from [0,E ) x S1 x [0,1] 4 M (S' is the unit circle with angular coordinate 8) defined by ( T , 8, Z )

H

(T

cos(8

+ p),

T

sin(8

+ p),

Z)

for some E < 1/4, where p = arctan(l/y) (the inclusion of ,B is not essential but simplifies the next formulae). The identification (Y becomes a(r,8 , l ) = ( T I , 8', 0) with TI

=r

m

,

f(e) = y4C O S ~e + y4sin2 8,

(1)

tan 8' = y-4 tan 8, where 8' in the third equation is chosen from the same quadrant of the circle as 8; it defines a C" map $J of the circle (whose derivative is l / f ) . Denote the blowup manifold by N . It is a C" manifold with boundary 8 N diffeomorphic to T2. Use coordinates ( T , ~ , z near ) the boundary and (z, y, z ) elsewhere, taking into account the identifications (Y and the horizontal integer translations. If one wants to be a stickler for rigour, one can

58

zyxwvu zyxw zyxwvu zyxw zyxwv zyx zyxwvu

make a cover of N by 10 charts, based on these two coordinate systems and the gluing map a. The vector field s on M induces one on N that I call t. It looks like s but the periodic orbit along x = 0 is blown up into an invariant torus with coordinates (0, z ) (modulo gluing by $), representing horizontal directions of approach t3 to points z of the periodic orbit. On this boundary torus the vector field has two attracting periodic orbits 8 = 0 , and ~ two repelling ones 6 = f7r/2, separating four annuli on which all orbits come from a repelling one at large negative time and go to an attracting one at large positive time (this comes out of the gluing $). The vector field t preserves the volume form d x A d y A d z in (x,y , z ) coordinates and r d r A d e A d z in ( r ,8, z ) , and inherits invariant foliations from s. Next, for a C3 function g : N -+ IR+ = {x E IR : x > 0) to be chosen later, let u = gt on N . Note that g is bounded away from 0 and +cc because N is compact. The vector field u preserves volume form i d x A d y A d z and has the same invariant foliations as t. Then, choose a C 3 function p : N -+R which is positive in fi = N \ d N (the interior of N ) and asymptotic to distance to d N near d N , measured with some C3metric. Specifically, I choose Arnol’d’s Riemannian metric

zyxwvu +

ds2 = ~ - ~ ” d x 2 +y4”dx5

+dz2,

(2)

where

d x - = cos p d x

+ sin p d y , + cos p d y ,

(3)

dx+ = - sinp d x and require p

-

7-47-42

Let v = p u . Then v is

sin2 e

+ 7 4 cos2 ~ e as r + 0.

C3,preserves volume form r

wg = -dr PS

1 A dB A d z = -dx PS

A d y A da

(4)

in the two coordinate systems, and is zero on a N . A remarkable fact is that N is C”-diffeomorphic to the exterior of a figure-eight knot in the 3-sphere S3 (“exterior” means the closure of the complement of a closed tubular neighbourhood). Although this is stated in I have found it hard to locate a proof in the many places,11~15~16*27~28~38-40 literature, partly because the interest in many of these references focusses on the additional fact that it can be endowed with a hyperbolic metric. Even then, most topologists are happy with existence rather than an explicit

zyxw zy zyx zyx zyxwvut zyxwv 59

diffeomorphism. In an Appendix I briefly survey the proofs of which I am aware. As the final step, I transfer the example from S3 into R3:choose the figure-eight knot to pass through the “North pole” (0, 0, 0 , l ) of S3 (considered as the unit sphere in R4)and map the rest of S3 stereographically t o the plane tangent to the “South pole”, i.e. (x,y, z , w) H The result is a C” diffeomorphism h from N to the closure of a bounded domain R of EX3 which looks like Figure 2. Think of it as an apple through the core of which a worm has eaten a tubular hole in the form of a figure-eight knot. The domain R is the remaining flesh of the apple.

w.

Fig. 2.

The domain R for the vector field w and four orbit segments.

zy

The desired vector field is w = h*v, the image of v under h. It is C3, vanishes on the boundary of R and preserves volume h*ug.To make it preserve a pre-ordained C3 volume form vol on R (e.g. the Euclidean volume from R3),it suffices to choose the function g = where w1 is the special case of (4) with g = 1 (since all volume forms a t a point are multiples of a given one, this ratio makes sense; also it is a C3 positive function as required).

9,

60

zyxw zyxwvu zyxwvut zyxwv zyxwvu zyxw

To give some idea of what the vector field w looks like on R, Fig. 2 also indicates orbit segments approaching or departing from the four periodic orbits of the skin frzction field on the boundary ( r being distance from the boundary): they alternately attract and repel along the boundary and repel and attract from the interior. The fact that the periodic orbits go the %hart" way around the boundary is a consequence of a nice argument explained to me by Luisa Paoluzzi which I summarise in the Appendix (see also Ref. 38). Fig. 3 shows a slightly different view in which the bottom lobe of the knot has been rotated round the back to enable visualisation of the image of the cross-section z = 0 in N by h. It is based on Fig.11 of Ref. 39. Convince yourself that the cross-section is indeed diffeomorphic to a torus minus a round open disc, a space I’ll denote by To,and that it can be swept round in R, following a given co-orientation and keeping the boundary on dR, and that the action on the surface induced by sweeping once round is homotopic to A’, the blowup of the toral automorphism A. R is said to fibre over the circle, with fibre (or “Seifert spanning surface”) To and monodromy A’. More pictures of this can be found in Refs 15,27.

2

Fig. 3. The domain R and a cross-section to the vector field w, with direction of flow indicated by f.

zyxwvu zyx zy zyxwvut zyx z zyxwvu 61

A similar construction was used in Ref. 11 to make an example of a flow in R3 where the possible knot and link types of periodic orbits could be shown to be very rich (indeed Ref. 16 proved it contains all knots and links, and the same for any flow transverse to the fibration). Their vector field, however, is not volume-preserving. It was obtained from s by a DA (“derived from Anosov”) construction, perturbing the gluing map a near the fixed point (0,O) to replace it by a repelling fixed point and two saddles and then excising the repelling orbit. 3. Mixing

The point of the example w is the following theorem.

Theorem 3.1. All vector fields topologically equivalent to w on R within the class of vector fields on R preserving given volume form vo1, C3 on and vanishing on the boundary are mixing.

Proof. The first return map $ to the cross-section { z = 0) minus its boundary is mixing for the area form given by the flux of vol under w, by the standard Hopf argument using the existence of the invariant foliations for $J (e.g. see Ref. 12 for a nice exposition). By Anosov’s alternative1 (rediscovered in Ref. 32), the only obstacle to the flow being mixing would be if the return time function r : ‘ko -+ R+ for were a constant plus a coboundary. A “coboundary” for a map $ is a function r : $0 -+ lR of the form r(x) = a($(x)) - a(x) for some function a,so its sum along an orbit of 1c, telescopes. This is a somewhat exceptional situation. Indeed, in our case the return time goes to infinity at the boundary, so can not be a constant plus a c ~ b o u n d a r y . ~ ~ 0 $J

Actually, from mixing and a general argument of Ref. 31, it follows that the flow is Bernoulli. A nice feature of the example which makes it potentially physically realisable is that it is robust.

Theorem 3.2. w is structurally stable within the above class of vector fields. A vector field is structurally stable if all small perturbations are topologically equivalent to it. Since the proof involves many technicalities, it will be published elsewhere. How fast does the example mix? To answer this requires first a discussion about how to define rate of mixing.

62

zy zyxwv z zyxwv zy zy

A standard way to define the rate of mixing of a flow 4 on a manifold M preserving a volume form p is to choose a class F of functions f : M -i R and ask how fast the correlation C f g ( t = ) J f($t(z))g(z)dp(z) for f , g E F decays to the product of the means off and g, in comparison to the product of the sizes of f and g using a notion of size appropriate to the function class (or f , g can come from different function spaces). The answer depends strongly on the chosen class of functions, however. For example, if F is L2 then there is no uniform decay estimate: g could be chosen to be f o 4~ for some large T and then C f g ( T )= 11 f IILzllgIIL2.For some mixing systems, exponential decay can be proved for Holder continuous functions, but the decay rate depends in general on the Holder exponent a. An alternative is to use a metric on a space of probability measures on M and ask how fast the push-forward of an initial measure converges to p. A natural metric is the total variation metric, but for a volume-preserving flow this metric is invariant, so gives no information about mixing. A better one is the transportation metric

z

where Pp,qis the set of probability measures on R x R with marginals p , on the first and second factors. It is the minimum average distance that mass from one measure has to be moved to turn it into the other measure. A nice result of Ref. 21 is that

z

over non-constant Lipschitz functions f , where p ( f ) is the expectation of f in measure p and 11 f I I L ~ is~ the smallest Lipschitz constant for f . So the two views come close when Holder is specialised to Lipschitz ( a = 1).In particular, given an initial measure v absolutely continuous with respect to p , it can bewrittenasgpfor afunctiong E L 1 ( p ) .Then ( 4 r v ) ( f ) - p ( f )= C f g ( t )so , D(+r (v),p ) = supf c ( t ) and any upper bound on the correlation function

-

proportional to 11 f I I L ~ gives ~ a corresponding upper bound on the transportation distance. It is not clear to me, however, whether lower bounds transfer so easily, because t o obtain an accurate lower bound for the transportation distance one may have to change the choice of f as time progresses. In any case, I choose to use transportation metric.

zy zyxwv zyxwv 63

Theorem 3.3. No C2 volume-preserving vector field with compact no-slip C2 boundary mixes faster than l / t 2 in time t .

Proof. Let v be a C2 volume-preserving vector field with no-slip boundary, p a C2 positive function asymptotic to distance to the boundary, and u = v/p. Then a simple calculation shows that u is tangent to the boundary. Let C = sup in a neighbourhood r 5 r1 of the boundary. Then [u,I 5 Cr for r 5 r1. Thus 1q.l 5 Cr2 for r 5 r1. It follows that fluid from outside r 5 r1 can get t o a t most distance l / ( l / r l - C t ) of the boundary in time t. Take an initial “dye” density 1 in r 5 T I and 0 outside. Then the subset r < l / ( l / r l - C t ) remains of density 1. It is of thickness of order l / t , so has volume of order l / t and the average distance that dye must be moved to achieve the average density is at least half the thickness. Thus the transportation distance to the uniformly mixed state is a t least of order 1 p . 0

%

zyx

The fact that some flows with no-slip boundaries mix like a power law was noted numerically in Ref. 18, albeit with molecular diffusion added and a different notion of mixing rate. An open question is t o determine an upper bound on the transportation distance as a function of time. This would require some study of the returntime function t o a cross-section, among other things. If one switches attention to correlation functions, there is some literature on systems with power law decay, e.g. Ref. 13 for upper and Ref. 36 for lower bounds. It seems likely to me that the correlation of many pairs of function decays like l / t for our flow. This would give rise to anomalous diffusion. Corresponding to the coordinate z of s is a quantity one can continue t o denote by z which measures how many times (plus fractional part) trajectories have crossed the cross-section of Fig. 3. Then one can examine the deviation from the mean rate of increase of z with time. If the autocorrelation function for i is integrable then the deviation would spread like normal diffusion, but if its integral is infinite then the deviation should spread anomalously. One way to obtain a handle on this would be t o use the fact that the flow has a Markov partition and compute the large deviation rate function for the increment in z (cf. Ref. 26). 4. Discussion

z

zy zyx

At the physical level, there remains the question of how to drive the flow. It suffices to compute w.Vw - YAW,where Y is the kinematic viscosity,

64

zyx zyxwvut zyxw zyxwvu zyxwvu

subtract off its gradient part, and apply a body force equal to the remainder. It might not be easy to implement, however. One can contrast results of Ref. 14 making an Euler flow on S3 containing all knots and links. Being an Euler flow it requires no forcing at all, but the catches are that it also requires zero viscosity, the Riemannian structure could not be specified in advance, and it is not claimed to be mixing: indeed the knots and links are supported on a proper subset. I believe it is possible to make a similar construction of a flow with stressfree boundaries, by using symplectic polar blowup instead. This ought to be C2 structurally stable. To obtain mixing, however, one would need to ensure that the speed function is nontrivial. One can ask whether the flow is a fast dynamo. The dynamics of a magnetic field in a steady conducting fluid flow may have a positive growth rate. The flow is said to be a fast dynamo if the growth rate has a positive lower bound as the magnetic diffusivity goes to zero (in principle this depends on the Riemannian metric assumed for the magnetic diffusion) (see survey in Ch.V of Ref. 6). A r n ~ l ’ dproved ~ ~ ~ that s is a fast dynamo with respect to metric (2). It would be interesting to investigate whether w is a fast dynamo. To make this problem well posed one has to specify what the magnetic field does outside R. One can ask whether there are alternative constructions of robust mixing fluid flows. I believe one would be the “pigtail stirrer”. Start from s on A4 but quotient by o ( x , y , z ) = (-x,--y,z) and blowup the orbits of both (0, 0,O) and f , 0 ) to tori. This gives a vector field in a solid torus minus a tubular neighbourhood of a knot which goes three times round the solid torus making the closure of a pigtail braid (as sketched in Ref. 25 for example). The monodromy goes back to Latt&s.22The analysis is slightly different from the example w, because the blown-up orbits are 1-prong singularities rather than regular orbits, but I think the same structural stability result should be possible. Furthermore, this example opens the possibility to make the outer boundary axisymmetric and to rotate it about its axis, so that the no-slip condition gives a non-zero field on the outer boundary. Equivalently (though different for the fluid dynamics), one could rotate the 3-braid and examine the flow in the rotating frame. Another starting point is geodesic flow on the unit tangent bundle of a surface of negative curvature, which is mixing Anosov. Birkhoff showed that blowup of 6 periodic orbits of the genus 2 case produces a suspension of a hyperbolic toral automorphism with 12 points blown up,” and I expect this can be mapped into R3.

(i,

65

What if one abandons the structural stability requirement but just asks for robust mixing, i.e. all nearby volume-preserving flows are also mixing? I believe this can be achieved by what I call a “baker’s flow” by analogy with the well known baker’s map. It is a volume-preserving flow in a container whose boundary is a surface of genus 2. The 2D stable manifold of a reattachment point on the boundary separates the volume into orbits which go round one loop from ones which go round the other loop. These two sets glue together again along the 2D unstable manifold of a separation point on the boundary. If the two manifolds are designed to intersect transversely, the eigenvalues of the skin-friction field satisfy certain inequalities at the separation and reattachment points, and the flow round the loops rotates trajectories suitably, then the return map to a transverse section in the middle is a nonlinear version of the baker’s map. The system is a volumepreserving analogue of the Lorenz system. The flows are not structurally stable, but are probably robustly mixing (just as for the Lorenz system in the good parameter regime24). Lastly, one can ask about time-periodic 2D flows. I think it might be possible to make a codimension-3 submanifold of C2 area-preserving maps of the torus, isotopic to the identity (so realisable by time-periodic flows), looking perhaps a bit like Zeldovich’s alternating sine-flow, which are mixing and topologically conjugate. The idea is to start from a pseudo-Anosov example (maybe a variant of Ref. 26), then smooth it and show topological conjugacy for all small smooth perturbations preserving the singular orbits.

zyx

Appendix

Here I survey what I have found about the diffeomorphism between the blow-up of the suspension manifold and the exterior of a figure-eight knot. The starting point is to notice that they have isomorphic fundamental groups, with isomorphism respecting the subgroup for the boundary. Then a result of Ref. 37 applies to give a homeomorphism (alternative proofs are in Ref. 30 using Ref. 29, and Cor 6.5 of Ref. 41). Stallings’ paper worries me, however, because he ends by saying that it is not clear whether fibred manifolds with isotopic monodromy are homeomorphic. All of these proofs involve various cutting and gluing operations that make it difficult to see an explicit homeomorphism and they do not address the question of smoothness (but Ref. 23 redoes it in the differentiable category). More explicit are three approaches which involve viewing the manifold as a quotient of hyperbolic 3-space W3 by a discrete group of i s o m e t r i e ~ ~ ~ ~ (see also Ref. 40).

66

zyx zyxwvu zyx zyxw zy zyxw zyxwvu zyxw

Another strategy17 (also described in 10.J of Ref. 35) is to notice that the figure-eight knot has a Z2 symmetry by a half-rotation about some unknot (this was used also by Ref. 11).Quotienting by the symmetry reduces it to the closure of the pigtail braid relative to the unknot symmetry axis. Since any braid-closure is fibred, so is the figure-eight knot, and the monodromy can be seen to act like A’, the blowup of

1; ;]

to +a

L

A

With an explicit diffeomorphism it would be easy to verify the claim of Section 2 about the homotopy class of the periodic orbits on the boundary, but the following argument of Luisa Paoluzzi answers the question anyway. Choose as base point for the fundamental group n l ( N ) the point at z = 0 on b” with 0 = 0. Choose the following generators for n1(N): a translates by (1,0,0) passing over the second tube, b translates by (0,1,0) passing to the left of the second tube, c translates by (O,O, 1) up the periodic orbit at T = 0 = 0 (and glues by A ) . Then a generating set of relations is c-lac = a2b,c-lbc = ab. Also, going once round the tube anticlockwise in the plane 2 = 0 is achieved by K = b-la-lba. The preimage under the diffeomorphism h : fi -+ R of the homotopy class of a closed curve going the short way around the knot in R, cutting the Seifert surface positively, is CK? for some integer n. We want to show n = 0. The quotient of n l ( N ) by is trivial, since it is equivalent to reinserting the knot and its tubular neighbourhood into S3. The quotient of n1(N) by c is indeed trivial (the relations then imply a = a2b,b = ab, so a = b = el the identity). In contrast, one can argue that for any n # 0, the quotient of n l ( N ) by is non-trivial.

Acknowledgements

I first presented the idea at a “Mixing in fluid flows” meeting in Bristo1 in May 2004 supported by the London Mathematical Society. I thank Luisa Paoluzzi for proving the homotopy class of the periodic orbits on the boundary, Sebastian van Strien, Christian Bonatti, Yakov Pesin, Enrique Pujals, Charles Pugh and Mike Shub for believing the proof of structural stability would be possible, Jean-Christophe Yoccoz for pointing out an error in a draft, and Mark Pollicott and Lai-Sang Young for comments about mixing rates. I did the serious work on the structural stability during visits to IMPA (KO),the Fields Institute (Toronto), COSNet (Australia) and IHES (France) from July 05 to June 06. I thank all of them for their hospitality.

67

zy

References 1. Anosov DV, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat Inst Steklov 90 (1967) 210. 2. Aref H, Stirring by chaotic advection, J Fluid Mech 143 (1984) 1-21. 3. Aref H, The development of chaotic advection, Phys Fluids 14 (2002) 131525. 4. Arnol’d VI, Sur la topologie des Bcoulements stationnaires des fluides parfaites, Comptes Rendus Acad Sci Paris 261 (1965) 17-20. 5. Arnol’d VI, Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field, J Appl Math Mech 36 (1972) 236-242. 6. Arnol’d VI, Khesin BA, Topological methods in hydrodynamics (Springer, 1998). 7. Arnol’d VI, Korkina EI, The growth of a magnetic field in a three-dimensional steady incompressible flow, Moscow Univ Math Bull 38 (1983) 50-4. 8. Anrol’d VI, Zel’dovich YaB, Ruzmaikin AA, Sokolov DD, A magnetic field in a stationary flow with stretching in Riemannian space, Sov Phys JETP 54 (1981) 1083-6. 9. Arter W , Ergodic streamlines in three-dimensional convection, Phys Lett A 97 (1983) 171-4. 10. Birkhoff GD, Dynamical systems with two degrees of freedom, Trans Am Math SOC18 (1917) 199-300. 11. Birman JS, Williams RF, Knotted periodic orbits in dynamical systems 11: knot holders for fibered knots, in Low-dimensional topology, ed Lomonaco SJ, Contemp Math 20 (Am Math SOC,1983) 1-60. 12. Burns K, Pugh C, Shub M, Wilkinson A, Recent results about stable ergodicity, in: Smooth ergodic theory and its applications, eds Katok A, Llave R de la, Pesin Ya, Weiss, Proc Symp Pure Math 69 (2001) 327-66. 13. Chernov N, Zhang H-K, Billiards with polynomial mixing rates, Nonlin 18 (2005) 1527-53. 14. Etnyre J, Ghrist R, Contact topology and hydrodynamics 111:knotted orbits, Trans Am Math SOC352 (2000) 5781-94. 15. Francis GF, Drawing Seifert surfaces that fiber the figure-8 knot complement in S3 over S’,Am Math Month 90 (1983) 589-599; and Chapter 8, A topological picture book (Springer, 1987). 16. Ghrist RW, Branched two-manifolds supporting all links, Topology 36 (1997) 423-448. 17. Goldsmith DL, Symmetric fibered links, in: Knots, groups and 3-manifolds, ed Neuwirth LP, Ann Math Studies 84 (Princeton, 1975) 3-23. 18. Gouillart E, Kuncio N, Dauchot 0, Dubrulle B, Roux S, Thiffeault J-L, Walls inhibit chaotic mixing, Phys Rev Lett, in press, 2007. 19. HBnon MR, Sur la topologie des lignes de courant dans un cas particulier, Comptes Rendus Acad Sci Paris 262 (1966) 312-314. 20. Jorgensen T, Compact 3-manifolds of constant negative curvature fibering over the circle, Ann Math 106 (1977) 61-72.

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68

21. Kantorovich LV, Rubenstein G Sh, On a space of totally additive functions, Vestnik Leningrad Univ 13:7 (1958) 52-59. 22. Lattits S, Sur l’iteration des substitutions rationelles et les fonctions de PoincarB, Comptes Rendus Acad Sci Paris 16 (1918) 26-8. 23. Laudenbach F, Le theoreme de fibration de JStallings, Seminaire Rosenberg, Orsay (1969) M13.369. 24. Luzzatto S, Melbourne I, Paccaut F, The Lorenz attractor is mixing, Commun Math Phys 260 (2005) 393-401. 25. MacKay RS, Postscript: Knot types for 3-D vector fields, in: Topological Fluid Mechanics, eds Moffatt HK, Tsinober A, IUTAM Conf Proc, Aug 89 (CUP, 1990), 787. 26. MacKay RS, Cerbelli and Giona’s map is pseudo-Anosov and nine consequences, J Nonlin Sci 16 (2006) 415-434. 27. Miller SM, Geodesic knots in the figure-eight knot complement, Exp Math 10 (2001) 419-436. 28. Milnor 3, Hyperbolic geometry: the first 150 years, Bull Am Math SOC6 (1982) 9-24. 29. Neuwirth L, The algebraic determination of the topological type of the complement of a knot, Proc Am Math SOC12 (1961) 904-6. 30. Neuwirth L, On Stallings fibrations, Proc Am Math SOC14 (1963) 380-1. 31. Ornstein DS, Weiss B, Statistical properties of chaotic systems, Bull Am Math SOC24 (1991) 11-116. 32. Plante JF, Anosov flows, Am J Math 94 (1972) 729-754. 33. Riley R, A quadratic parabolic group, Math Proc Camb Phil SOC77 (1975) 281-8. 34. Robbin JW, On the existence theorem for differential equations, Proc Am Math SOC19 (1968) 1005-6. 35. Rolfson D, Knots and links (Publish or Perish, 1976). 36. Sarig 0, Subexponential decay of correlations, Invent Math 150 (2002) 629653. 37. Stallings J, On fibering certain 3-manifolds, in: Topology of 3-manifolds and related topics, ed Fort MK (Prentice Hall, 1962), 95-100. 38. Thurston WP, The geometry and topology of 3-manifolds, Chs 3 and 4 (preprint, 1978). 39. Thurston WP, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull Am Math SOC6 (1982) 357-381. 40. Thurston WP, Three-dimensional geometry and topology, vol 1, ed Levy S (Princeton U Press, 1997). 41. Waldhausen F, On irreducible 3-manifolds which are sufficiently large, Ann Math 87 (1968) 56-88.

69

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COMPLEXITY AND ENTROPY IN COLLIDING PARTICLE SYSTEMS M. COURBAGE and S. M. SABER1 FATHI

Laboratoire Matibre et Systbmes Complexes ( M s c ) UMR 7057 CNRS et Universitt Paris 7- Denis Diderot Case 7020, Tour 24-14.5bme ttage, 4, Place Jussieu 75251 Paris Cedex 05 / FRANCE emails : [email protected], majid.saberiQparis7.jussieu.fr

We develop quantitative measures of entropy evolution for particle systems undergoing collision process in relation with various instability properties.

Keywords: Lorentz gas, hard disks, entropy, Lyapunov exponents, H-theorem

1. Introduction

There are two concepts of entropy in the theory of dynamical systems: the first one is the famous Kolmogorov-Sinai' entropy introduced by Kolmogorov in 1958. Kolmogorov, who was familiar with the Shannon entropy for random process, designed this concept and used it in order to solve the isomorphism problem of Bernoulli systems. In 1959, Sinai modified and extended the ideas and the results of Kolmogorov to any dynamical system (DS) with an invariant probability measure (also called measurable DS). It is important to note that the measure theoretical entropy is a number that characterizes the family of isomorphic dynamical systems. It is one of the main tools to classify all measurable dynamical systems. Although this theory provided considerable information about their structure, many problems are still open. On the other hand, the non-equilibrium entropy, introduced by Boltzmann in kinetic theory of gases, can be defined in the case of measurable DS. Recall that the Boltzmann H-theorem defines the entropy for the one particle probability distribution ft (z) as

70

Boltzmann showed that this quantity is monotonically increasing for all solutions of his celebrated equation. During many years until the beginnings of the twentieth century the Boltzmann H-theorem was the object of many discussions and controversies. Later on Ehrenfest proposed the urn Markov chain model for the approach to equilibrium with an H-theorem. The model consists of n = 2N balls distributed inside two halves of a box : left and right. On account of collisions between particles, Ehrenfest postulated that a t regular time interval a particle can leave the right half or to join it . So if the state space of the system is described by the number X of particles in the right hand side, the dynamics of the system would be a Markov chain where the allowed transitions are from X = m to X = m - 1, with probability m / 2 N , or from X = m to X = m 1 with the complementary probability. Mark Kac gave an exhaustive solution of this model in his book.’ Briefly speaking, it is possible to find a unique stationary probability distribution { p i } , i = 0,1,. . . ,n = 2N; such that any initial distribution { ~ ( t con)} verges t o { p i } . The non-equilibrium entropy of the distribution {vi(t)}with density ft = !!& is given by the Boltzmann like formula: Pi

zy z zyxwvut +

The variable X is a “macroscopic variable” which means that a given value of X corresponds to a region in the phase space of 2N dimensions. Distinct values of X correspond t o distinct regions, {’Pi}. The set of ’Pi’s form a partition of the phase space. However, it is obvious that strictly speaking the process X ( t ) is not Markovian although some time it is claimed to be approximately Markovian. Independently, Gibbs imagined the dynamical mixing property as a mechanism of the approach to equilibrium for systems out of equilibrium. His ideas are based on the consideration of the phase space of an isolated system of N particles where the equilibrium is described by the microcanonical ensemble as an invariant measure. The system will approach the equilibrium if any initial probability distribution will converge to equilibrium under the Hamiltonian flow. According to Gibbs this will happen if the shape of any subset will change boldly under the flow, although conserving a constant volume, winding as a twisted filament filling, proportionally, any other small subset of the phase space. The famous image of this mechanism is the mixing of a drop of ink in a glass of water. Later on, Hopf found a whole class of mixing DS: the differentiable hyperbolic

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DS where to each trajectory is attached two manifolds expanding and dilating in transversal directions. So, any domain of the phase space will be

z

squeezed an folded filling densely any region of the phase space. For example in the baker transformation, the expanding and contracting manifolds are horizontal and vertical respectively, so that any small horizontal segments will be uniformly distributed in the phase space after few iterations of the transformation (see Fig. 1).The importance of mixing and exponential instability of trajectories for obtaining H-theorem has been discussed by Krylov.2

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The H-theorem for measurable dynamical systems describes the approach to equilibrium, the irreversibility and entropy increase for measurable deterministic evolutions. That is dynamical transformation T on a phase space X with some probability measure p, invariant under T , i.e. p(T-'E) = p ( E ) for all measurable subsets E of X. Suppose also that there is some mixing type mechanism of the approach to equilibrium for T , i.e. there is a sufficiently large family of non-equilibrium measures v such

72

that

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(3) Then, the H-theorem means the existence of a negative entropy functional S(vt)which increases monotonically with t to zero, being attained only for v = p. The existence of such functional in measure-theoretical dynamical systems has been the object of several investigations during last decades see8-12,14i17i18,21). Here we study this problem for the Lorentz gas and hard disks. Starting from the non-equilibrium initial distribution v, and denoting by P a partition of the phase space formed by cells (Pl,P2,..., Pn)and by vi(t) = v o T W t ( P i )the , probability at time t for the system t o be in the cell Pi and such that .(Pi) # p(Pi) for some i, the approach to equilibrium implies that vi(t) -+ pi as t -+ oc) for any i . The entropy functional will be defined by:

which we simply denote here after S ( t ) .The H-functional (4) is maximal when the initial distribution is concentrated on only one cell and minimal if and only if vt(Pi) = p(Pi),Vi. These properties are shown straightforwardly. This formula describes the relative entropy of the non-equilibrium measure vt with respect to p for the observation associated to P. It coincides with the information theoretical concept of relative entropy of a probability vector ( p i ) with respect to another probability vector ( q i ) defined as follows: - lnpi being the information of the ithissue under the first distribution, pi In( is equal t o the average uncertainty gain of the experience ( p i ) relatively t o ( q i ) . A condition under which formula (4) shows a monotonic increase with respect t o t is that the process .,(Pi) = v o T t ( P i )verifies the ChapmanKolmogorov equation valid for Markov chains and other infinite memory chains.lOill For a dynamical system, this condition is hardly verified for given partition P . However, the very rapid mixing leads t o a monotonic increase of the above entropy, a t least during some initial stage, which can be compared with the relaxation stage in gas theory. In this paper, we will first compute the entropy increase for some remarkable non-equilibrium distributions over the phase space of the Sinai’ billiard. The dynamical and stochastic properties of the Lorentz gas in two dimensions which we consider here was investigated by Sinai and Bunimovich as an ergodic dynamical ~ y s t e m Other . ~ ~ transport ~ ~ ~ ~ properties

xi

E),

73

have been also studied numerically eel^,'^). This is a system of non interacting particles moving with constant velocity and being elastically reflected from periodically distributed scatterers. The scatterers are fixed disks. On account of the absence of interactions between particles the system is reduced to the motion of one billiard ball. We shall investigate the entropy increase under the effect of collisions of the particles with the obstacles. For this purpose, we consider the map T which associates to an ingoing state of a colliding particle the next ingoing colliding state. The particle moves on an infinite plane, periodically divided into squares of side D called "cells", on the center of which are fixed the scatterers of radius a ( Fig. 2). The ingoing colliding state is described by an ingoing unitary velocity arrow a t some point of the disk. To a colliding arrow Vl(P1) at point PI on the boundary of the disk the map associates the next colliding arrow Vz(P.2) according t o elastic reflection law. Thus, the collision map does not take into account the free evolution between successive collisions.

zyxwv zy

Fig. 2.

The motion of the particle on a toric billiard.

The billiard system is a hyperbolic system (with many singularity lines) and, in order to have a rapid mixing, we will consider initial distributions supported by the expanding fibers. Such initial measures have been used in.8i12>21For the billiard the expanding fibers are well approximated by particles with parallel arrows velocity. We call this class of initial ensemble beams of particles. We first compute the entropy increase under the collision map for these initial distributions. We will consider finite uniform partitions of the phase space as explained below. The entropy functional will be defined through (4).For this purpose, the phase space of the collision

zyxwv

zyxw zy zy zyxwvu zyxw zy zyxw

74

map is described using two angles (p,$), where p is the angle between the , [0, f [, and 1c, E [0,7r] outer normal a t P and the incoming arrows V ( P ) ,f3 is the angle between s-axis and the outer normal a t P . Thus, the collision map induces a map: (pi,1c,i) --+ (pz,1c,2) (see Fig. A.2) and we shall first use a uniform partition of the (p,1c,) space. The computation shows that whatever is the coarsening of these partitions the entropy has the monotonic property in the initial stage. It is clear that, along mixing process, the initial distribution will spread over all cells almost reaching the equilibrium value. Physically, this process is directed by the strong instability, that is expressed by the positive Lyapunov exponent. It induces a relation between the rate of increase of the entropy functionals and Lyapunov exponents of the Lorentz gas. Our computation shows that this relation is expressed by an inequality max(S(n

+ 1)

-

~(n= ) )AS 5

C

A120

zy (5)

where the “max” is taken over n, which means that the K-S entropy is an upper bound of the rate of increase of this functional. Next, we shall consider the entropy increase of a system of N hard disks. Here the space in which moves a particle is a large torus divided into rectangular cells. Denoting the total number of cells by n and the number of particles initially distributed in only one region, by N , and following them until each executes t collisions with obstacles, we compute the probability that a particle is located in the ith cell as given by:

Pi(t) =

Number of particles in cell i having made t collisions

N

The equi-distribution of the cells leads to take, as equilibrium measure, pi = ,: so that this ((spaceentropy” is defined by:

The maximum of absolute value of this entropy is equal t o -Inn. So we normalize as follows:

(7) We shall also do some comparisons of the H-theorem with the sum of normalized positive Lyapunov exponents.

75

2. Entropy for collision map The entropy for the collision map is computed for a beam of N particles on a toric checkerboard with n cells. We start to calculate the entropy, just after all particles have executed the first collision. In this computation, all particles have the same initial velocity and are distributed in a small part of one cell. For each particle we determine the first obstacle and the angles (PI, $1) of the velocity incoming vector V1 ( P I )( see the figures given in the Appendix). For a uniform partition P of the space of the variables (p,$), the entropy S(t)is computed iteratively just after all particles have executed

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10) and then the instability stops (figure 7(a)).

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Fig. 6 . Last beats of the instability: more and more failed peaks appear before the instability stops. The transition between one and two failed peaks is shown.

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4

6

8

10

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Fig. 7. Slow evolution of the heartbeat instability toward its end: (a) more and more failed peaks until the stop around 18 s, (b) 3D and (c) 2D phase spaces.

As observed for other instabilities (figure 4), this behavior underlines the threshold dependence of this instability and the way it evolves toward its end. This alternation of failed and main peaks seems to correspond to mixed-mode oscillations often encountered in chemical systems or neuronal dynamics. The 3D and 2D phase spaces corresponding to the time series presented in figure 7(a) are plotted in figure 7(b)-(c).These attractors have been obtained by using an appropriate time delay calculated using the mutual information method.

225

Acknowledgments T h e PKE-Nefedov chamber has been made available by t h e Max-PlanckInstitute for Extraterrestrial Physics, Germany, under the funding of DLR/BMBF under grants No.50WM9852. We would like t o thank S. Dozias for electronic support and J. Mathias for optical support. This work was partly supported by CNES under contract 02/CNES/4800000059.

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References

1. A. Bouchoule, Dusty Plasmas :Physics, Chemistry and Technological Impacts in Plasma Processing (Wiley, Chichester, 1999). 2. P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics (Institute of Physics, Bristol, 2002). 3. R. M. Roth, K. G. Spears, G. D. Stein and G. Wong, Appl. Phys. Lett. 46, p. 253 (1985). 4. Y . Watanabe, M. Shiratani, Y. Kubo, I. Ogawa and S. Ogi, Appl. Phys. Lett. 53,p. 1263 (1988). 5. A. Bouchoule, A. Plain, L. Boufendi, J.-P. Blondeau and C. Laure, J . Appl. Phys. 70,p. 1991 (1991). 6. A. Howling, C. Hollenstein and P. J. Paris, Appl. Phys. Lett. 59,p. 1409 (1991). 7. A. Bouchoule, L. Boufendi, J. Hermann, A. Plain, T. Hbid, G. Kroesen and W. W. Stoffels, Pure Appl. Chem. 68, p. 1121 (1996). 8. G. S. Selwyn, J. Singh and R. S. Bennett, J. Vac. Sci. Technol. A 7,p. 2758 (1989). 9. M. Cavarroc, M. Mikikian, G. Perrier and L. Boufendi, Appl. Phys. Lett. 89, p. 013107 (2006). 10. P. Roca i Cabarrocas, P. Gay and A. Hadjadj, J . Vac. Sci. Technol. A 14, p. 655 (1996). 11. A. Dutta, S. P. Lee, Y . Hayafune, S. Hatatani and S. Oda, Jpn. J. Appl. Phys. 39,p. 264 (2000). 12. C. Deschenaux, A. Affolter, D. Magni, C. Hollenstein and P. Fayet, J . Phys. D: Appl. Phys. 32,p. 1876 (1999). 13. S. Hong, J. Berndt and J. Winter, Plasma Sources Sci. Technol. 12, p. 46 (2003). 14. I. StefanoviC, E. KovaEevic, J. Berndt and J. Winter, New. J. Physics 5,p. 39 (2003). 15. K. De Bleecker, A. Bogaerts and W. Goedheer, Phys. Rev. E 73,p. 026405 (2006). 16. J. Robertson, Mater. Sci. Eng. R 37,p. 129 (2002). 17. D. Zhou, T. G. McCauley, L. C. &in, A. R. Krauss and D. M. Gruen, J. Appl. Phys. 83,p. 540 (1998). 18. C. Szopa, G. Cernogora, L. Boufendi, J. J. Correia and P. Coll, Planet. Space Sci. 54,p. 394 (2006).

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19. B. Ganguly, A. Garscadden, J. Williams and P. Haaland, J. Vac. Sci. Technol. A 11,p. 1119 (1993). 20. G. Praburam and J. Goree, Phys. Plasmas 3,p. 1212 (1996). 21. M. Mikikian, L. Boufendi, A. Bouchoule, H. M. Thomas, G. E. Morfill, A. P. Nefedov, V. E. Fortov and the PKENefedov Team, New J . Phys. 5, p. 19 (2003). 22. J. Winter, Plasma Phys. Control. Fusion 40, p. 1201 (1998). 23. C. Amas, C. Dominique, P. Roubin, C. Martin, C. Brosset and B. PBgouriB, J . Nucl. Mater. 353,p. 80 (2006). 24. B. Walch, M. Horanyi and S. Robertson, IEEE Trans. Plasma Sci. 22, p. 97 (1994). 25. C. Arnas, M. Mikikian and F. Doveil, Phys. Rev. E 60, p. 7420 (1999). 26. A. A. Samarian and S. V. Vladimirov, Phys. Rev. E 67,p. 066404 (2003). 27. L. Couedel, M. Mikikian, L. Boufendi and A. A. Samarian, Phys. Rev. E 74, p. 026403 (2006). 28. D. Samsonov and J. Goree, Phys. Rev. E 59, p. 1047 (1999). 29. M. Mikikian, M. Cavarroc, L. Couedel and L. Boufendi, Phys. Plasmas 13, p. 092103 (2006). 30. M. Cavarroc, M. C. Jouanny, K. Radouane, M. Mikikian and L. Boufendi, J. Appl. Phys. 99, p. 064301 (2006). 31. M. Mikikian and L. Boufendi, Phys. Plasmas 11,p. 3733 (2004). 32. M. Mikikian, L. Couedel, M. Cavarroc, Y . Tessier and L. Boufendi, New J . Phys. 9, p. 268 (2007).

227

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CLUSTERING PROPERTIES OF CONFINED PLASMA TURBULENCE SIGNALS MILAN RAJKOVIC

Institute of Nuclear Sciences VinEa, P. 0. Box 522, 110001 Belgrade, Serbia E-mail: milanrovin. bg.ac.yu MILOS SUKORIC

National Institute for Fusion Science, 322-6 Oroshi-cho, Toki, Giju, Japan E-mail: [email protected]

Intermittency in turbulent boundary layers of fusion devices is studied by considering only the zero-crossing information of signals for various confinement regimes. Certain common features with neutral fluid turbulence are recognized in the low confinement regime (L-mode) of fusion plasma while completely different characteristics are noticed for the dithering H mode (L/H-mode) and the high confinement regime (H-mode). Spectral characteristics of approximate signals (containing only the zero-crossing information) in each regime are compared with each other and with the spectral scalings of the velocity signals of neutral fluid turbulence. A scaling exponent characterizing the tendency of small scales to cluster is introduced and its relationship with large scale clustering is investigated.

1. Introduction

Study of turbulence in magnetic confinement devices represents one of the most important issues in the pursuit of fusion energy production since turbulence hinders confinement, suppresses reactions as it causes particle and energy losses. Control of turbulence requires a thorough knowledge of its dynamics in the core of magnetic confinement devices as well as on the edges beyond the last closed flux surface, in the region known as the scrape-off layer (SOL). A major breakthrough in the confinement improvement occurred by the end of the eighties, when the high confinement (H-mode) was discovered in contrast to the well known L- or low confinement mode.' The H- or high confinement mode manifests itself by self-organization of a region just inside the poloidal field separatrix where the transport coefficients are reduced by up to an order of magnitude compared with the L-mode forming a pedestal in the plasma pressure. As a consequence a n improvement in

228

the global confinement is usually increased by a factor of two in the case of toroidal devices while it is in the order of -20% in the case of large helical devices (LHD). The thickness of this barrier-type region is about equal to the ion poloidal gyroradius or the width of an ion banana orbit. Properties of intermittency both in the case of neutral fluids and plasmas are usually deduced from the analysis of temporal and/or spatial fluctuations of one or several relevant quantities. In the case of neutral incompressible fluids one or all three components of the fluid velocity represent the basic quantity from which all other relevant quantities, such as dissipation, may be derived. In the case of confined plasmas these quantities are usually the ion saturation current, recorded at one or more spatial locations, from which plasma density fluctuations may be inferred and floating potential recorded at different poloidal positions from which radial velocity fluctuations may be determined. In spite of many universal features these two types of turbulence have important differences. Nonlinearites in plasma turbulence are more numerous having different spectral cascade directions in addition to the most important E x B nonlinearity, leading to more complex fluctuating characteristics. Also, time and space measurements in plasmas lead to different information on the structure of turbulence. Turbulence in confined plasmas is created and damped at the same spatial location where the measurements are taken so that spatial and temporal informations are interwoven and Taylor’s frozen flow hypothesis cannot be applied, a common practice in neutral fluid turbulence studies. For the same reason the inertial range14may exist only locally in space or in time, and the extent of this range changes along the temporal scale as well as along space, for example along poloidal direction. Two features of intermittency are its clustering property and the variability in amplitude. Namely, different events cluster together creating uneven density in space and time, and events reflected in the highly variable amplitude are dispersed in space and time disproportionately. Much insight into the nature of intermittency may be gained from the study of approximations of turbulent signals which neglect amplitude aspect , 2 3 In this approach amplitude variations and local frequency of oscillations are separated by retaining only the zero-axis crossings (frequency) information. In a related approximation conveying somewhat different information about the original signal, turbulent signal is approximated by a zero-mean process constructed from the original signal x ( t ) as

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229

[ ( t ) is a binary approximation (BA) of the original signal and hence may assume values of 1 and 0 only. In Section I1 we examine spectral charac-

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teristics of approximated neutral fluid turbulence signal and approximated intermittent signals recorded in different plasma confinement regimes. In Section 111 the clustering properties of these signals are studied based on variance properties of the zero-crossings number in a given time period. 2. Power spectra of approximate signals In Fig. 1 we present power spectra of local velocity fluctuations measured in the turbulent region of a round free jet5 and its binary approximation. Binary approximations of velocity fluctuations, which have been studied previously and as explained recently in,2 are characterized by a spectral slope exhibiting “-4/3” scaling in contrast to the Kolmogorov “-5/3” scaling in the inertial range of the original velocity fluctuations signal. In2 a general relation relating the slope uwof the complete turbulent signal and the slope CJBA of its BA is proposed and heuristically proved, namely

zyxwz

Plasma intermittency in this presentation consists of the ion saturation current ( I S A T )fluctuations recorded by the movable Langmuir probe located at the outboard midplane on MAST device.6 Sampling rate was 1 MHz and during the discharge the distance from the plasma edge to the probe changed slowly. Three signals, each recorded during different confinement regimes are used: 6861 L-mode, 9031 dithering H-mode (denoted as L/H mode and representing an unstable state during which plasma switches intermittently from low to high confinement) and 5738 H-mode. In Fig.2 the spectra of 6861 L-mode and of its BA approximation are presented and inspection of these spectra supports the claim about similarity of neutral fluid and low confinement plasma t ~ r b u l e n c e . ~ In Figs. 3 and 4 the same type of spectra are presented for the L/Hmode and the H-mode and the slope values for each spectrum are indicated. Evidently, relationship (1) is not valid for these confinement regimes. The most striking feature of these spectra is increasing difference between slopes of the signal and its BA approximation as confinement increases. Also the slope of BA approximation approaches the slope of Gaussian white noise spectrum in the H-mode, suggesting perhaps similarity in certain aspects of H-mode with white Gaussian noise. However, rn will be seen in the next Section this is far from being the case.

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230

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. . . . 10-

k (m-1)

Fig. 1. Spectra for incompressible fluid turbulence.

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Fig. 2.

Fig. 3.

Spectra for the 6861 L-mode.

Spectra for the 9031 L/H mode.

231

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Fig. 4. Spectra for the 5738 H-mode.

3. Clustering information in the approximate signal

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Zero-crossing (or for that matter crossing of any particular level of interest) may offer important insight into the underlying process whose temporal variations are studied. The average number of zero-crossings of stationary Gaussian process in a specific time interval may be analytically determined and is given by the celebrated Rice formula,' which we here present in the following form:

zyxwvutsrqp J, PT

N ( T )= lim

G(rc(t)) (a(rc(t)/at( dt,

?%-+a

where limn--ta S(z) is the Dirac delta function. Important information on the clustering properties of the signal is however contained in the expression for the variance of the number of zero crossings. The expression for variance again may be derived analyti~ally,~ and is directly proportional to the time interval T,i.e.(N2(T)) T . Based on this expression for Gaussian process out goal is to contrast clustering properties of turbulent signals with the white Gaussian noise. For this purpose a running average within a time interval T of the number of spikes of BA approximation is constructed; this quantity is actually equal to N ( T ) ,the average number of zero-crossings in T . Then fluctuations of the running average are ~ N ( T= ) N ( T )- ( N ( T ) ) , where the brackets denote long-time average, possibly the time of the whole signal. We are interested in the scaling of the variance

-

232

zyx zyxwvuts

Fig. 5 . Standard deviation of the running density fluctuations vs. fluid turbulence.

T,

for incompressible

For a white Gaussian noise p = 1/2. We call p the clustering exponent and since white noise has no clustering, the value of 1/2 indicates lack of clustering. In Fig. 5 we show the standard deviation of the running density fluctuations for a neutral fluid. Two scaling intervals of the type (3) appear dividing the scales of interest into two groups which we interpret using the Taylor's frozen flow hypothesis. The scaling interval with exponent 0.5 suggests that there are no clustering effects for scales larger than the integral scale of the flow. This is an indication that large scales behave as white noise. The scaling on the left, in the range corresponding to the dissipative and inertial range scales with an exponent value less than 1/2 showing tendency of small scales to cluster. Analogous quantities are presented in Fig. 6 for the L-mode signal. In comparison with the turbulent neutral fluid, it is evident that clustering in plasma turbulence takes place on a lesser number of small scales, i.e. the extent of scales on which clustering takes place is smaller. Also, the clustering exponent is somewhat larger (-0.335 in comparison with -0.36). The extent of large scales is, on the other hand, greater and this is due to the large structures of confined plasma turbulence known as blobs or avaloids. These structures do not exhibit clustering since they behave as white noise. Note that the attribute of scales being large or small should be taken in restricted sense, since Taylor's frozen flow hypothesis may not be applicable in the case of confined plasma turbulence. In the next figure, Fig. 7, standard deviation of the running density fluctuations is presented for the L/H mode which again shows white noise scaling for large scales ( p = 0.5), however temporal extent (and possibly N

233

Fig. 6.

Standard deviation of the running density fluctuations vs.

100

zyx 7,for

6861 L-mode.

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9331 UH

zyx

Fig. 7. Standard deviation of the running density fluctuations vs. T , for 9031 L/H mode.

spatial) of this region is smaller than for the L-mode. The implication is that the presence of blobs is diminished in this regime The clustering exponent (exponent corresponding to smaller scales) increases i.e. the slope decreases indicating increasing tendency to cluster. Increased confinement, resulting in the H-mode, may generate at certain points in time huge coherent structures known as edge localized modes (ELMs) whose temporal evolution is presented in Fig. 8. On the bottom the signal with large coherent structures, known as edge localized modes (ELMs) is presented while a part of the signal without ELMs is presented on the top. Corresponding standard deviations of the running density fluctuations are presented in Fig. 9 (H-mode without ELMs) and Fig. 10 (with ELMs). In the absence of ELMs the clustering

234

zyxwvuts zyxwvutsrqpon zyxwvutsrqponm zyxwvutsrqpo 10

5-

:

- L o - - -

5d -5

-5-

-10-

-15r -20

!

2

zyxwvutsrqpo 3

4

5

time (m)

6-

3

d

op -5-

-2

'

'

'5738H-i&d.e

10

1

I

1

-10-15

6

-

-20

Fig. 8. Ion saturation current (plasma density) of 5738 H-mode as a function of time. In the top figure (first 10ms) the ELMs are absent, while in the bottom figure ELMS are dominant events.

Fig. 9. Standard deviation of the running density fluctuations vs. (ELMs absent).

zyx T,

for 5738 H-mode

is evident on all scales with an exponent 0.36 and there are no structures without clustering effects. Introduction of ELMs causes intense clustering (small slope, exponent -0.2), which involves large scales. In comparison with the L-mode which shows no clustering related to large structures as blobs (or avaloids), the large scale structures of H-mode (ELMs) are conN

zyx zyx 235

100 .

.

,

. . . . . .,

,

,

,

. , ., , ,

. . . , . . , .,

,

. . . . . .. I

5738bH-made

-

s

10-3

'

'

"""'

'

' " " " I

'

' " " " I

'

Fig. 10. Standard deviation of the running density fluctuations vs. (ELMs present).

'

zyx " " "

T,

for 5738 H-mode

centrated sets formed by particles clustering and possibly by accumulation of vorticity. The physics behind clustering in plasma turbulence is rather difficult to express in a form amenable to rigorous analysis but present analysis offers some interesting conclusions and opens up new areas for understanding plasma turbulence. First, blobs (large scale structures of L-mode) have no clustering properties and are very much different from edge localized modes which are produced by clustering effects. Even small ELMs have different temporal (and most likely) spatial characteristics from blob filaments. Moreover, the overall extent of scales corresponding to blobs surpass the scales corresponding to large scale structures of incompressible fluid turbulence. In the H-mode clustering effects are present on all scales relating this effect to the formation of transport barrier and zonal flows. Since transport is to a large extent suppressed in the H-mode, the value of the clustering exponent can be related to the transport coefficient.1° Finally, clustering effects may offer new insight about the hierarchy of length scales and their role in the creation of coherent structures

Acknowledgment The authors are grateful to Ben Dudson for providing the MAST data and to Ruchard Dendy for stimulating discussions.

References 1. The ASDEX team, Nucl. Fusion. 29 (1989) 1959. 2. K. R. Sreenivasan, A. Bershadskii, J. Stat. Physics, 125 (2006) 1145-1157.

236

zyxwvuts zyxwvuts

3. A. Bershadskii, J. J. Niemela, A. Praskovsky, K. R. Sreenivasan, Phys. Rev. E 69 (2004) 056314. 4. U. Frisch, Turbulence, the Legacy of A . N. Kolmogorov, Cambridge University Press, Cambridge, 1995. 5. C. Renner, J. Peinke and R. Friedrich, J. Fluid Mech., 433 (2001) 383-409. 6. B. D. Dudson, R. 0. Dendy, A. Kirk, H. Meyer and G. F. Councel, Plasma Phys. Control Fusion 47 (2005) 885-901. 7. M. RajkoviC, M. SkoriC, K. S ~ l n aand G. Antar, Nucl. Fusion (2008). 8. S. 0. Rice, Bell Syst. Techn. J. 23 (1944) 282-332, 24 (1945) 46-156; these papers are also in Selected Papers on Noise and Stochastic Processes, ed. Nelson Wax, Dover, New York, 1954. 9. M. R. Leadbetter and J. D. Gryer, Bull. Am. Math. SOC.71 (1965) 561. 10. M. Rajkovid, M. SkoriC (in preparation).

237

zyxwv zyxw zyxw zyx

INTERMITTENCY SCENARIO OF TRANSITION TO CHAOS IN PLASMA D. G. DIMITRIU,S . A. CHIRIAC Faculty of Physics, Alexandru loan Cuza University, I 1 Carol I Blvd. Iasi, RO-700506, Romania

zyx

Experimental results are presented that clearly line out a scenario of transition to chaos in plasma by type I intermittency in connection with the nonlinear dynamics of a double layer structure. The intermittencies were recorded in the time series of the current through the plasma conductor as random bursts that interrupt regular oscillations. The oscillations are triggered by the nonlinear dynamics of the double layer structure.

1. Introduction

Chaotic evolution is a frequent phenomenon in filament-type discharge plasma, occurring in relation to sheath in~tabilitiesl-~. Plasma is a nonlinear system where a wide variety of transition from ordered to low and high dimensional chaotic states were identified through different types of scenarios: period d ~ u b l i n g ’ ~ ~ , intennitten~ies~’~, q~asiperiodicity~ and torus breakdown6. In plasma devices, the chaotic states were observed by time series analysis of the ac components of the discharge c ~ r r e n t ” floating ~, potential of a probe3, or the current collected by a positively biased electrode immersed in the p l a ~ m a ~ ~ ~ . ~ . Type I intennittency is associated with a saddle-node bifurcation (tangent bifurcation in one-dimensional maps). Since Pomeau and Maneville did pioneering work on the analysis of low dimensional systems’ transition to chaos7, different types of intennittencies were classified. For type I intermittency, the theory is developed on a quadratic map, based on which we can numerically or theoretically derive other characteristic features such as the probability distribution of the laminar length and the llfpower spectrum. The duration of the periodic state (so-called laminar length) seems to be at random due to the stochastic occurrence of bursts, which lead to intermittent states. Here we report on type I intermittency in plasma, related to the nonlinear dynamics of a double layer structure. Double layers are localized nonlinear potential structures’ consisting of two adjacent positive and negative space

238

charge sheaths, sustaining a potential difference equal to or higher than the ionization potential of the background gas, depending on the gas pressure and plasma density. One common way to obtain a double layer structure is to positively bias an electrode immersed into a plasma being in equilibrium. In this case, a complex space charge structure in form of a quasi-spherical intense luminous body attached to the electrode is obtained. Experimental investigations revealed that such a complex space charge structure consists of a positive nucleus (an ion-enriched plasma) surrounded by a nearly spherical double layer9*". The stability of the structure is ensured by the balance between the charges lost by recombination and diffusion and the charges created by ionizations and accumulated in the adjacent regions. By increasing the potential applied on the electrode, the rate of the ionization processes increases and the balance needed for the double layer existence is perturbed. Consequent on this, the structure disrupts, passing into a dynamic state. This dynamic state is a periodic one and determines the appearance of strong oscillations of the plasma parameters, such as the plasma density or the current collected by the electrode. Our results indicate that this dynamics evolves chaotically under certain experimental conditions. For gradually increasing the voltage on the electrode, we recorded the time series of the ac components of the electrode current. By statistical analysis of these time series, we identified a scenario of transition to chaos by type I intermittency.

2.

Experimental results and discussion

The experiments were performed in a hot-filament discharge plasma diode, schematically shown in Fig. 1. The plasma is created by volume ionization processes between energetic electrons from the hot filament (marked by F in Fig. 1) and gas atoms. The chamber wall acts as anode (marked by A in Fig. 1) and is made from non-magnetic stainless steel, being grounded. The discharge current is Zd = 40 mA. The plasma parameters, measured by mean of emissive and cold probes, were plasma density npl E 5 ~ 1 ~0 m ~ -electron ~, temperature T, G 2-3 eV, for an argon pressure p = 5x10" mbar. The plasma diffuses into the chamber, were an additional electrode of 3 cm in diameter (marked by E in Fig. 1) is introduced and positively biased in respect to the plasma potential (and also to ground). Figure 2 shows the static current-voltage characteristic of the electrode, obtained by gradually increasing and subsequently decreasing the potential on the electrode E, V,. The sudden jumps of the current collected by E, marked by ZE, are related to the generation and dynamics of the double layer structureg and

zyxwvu zyx zyxwv z

239

u1 uz

A

EP

1 E

I

zy

zyx X.

+Y

-

/A9

-

Figure 1. Experimental setup (F - filament, E - additional electrode, A - anode, U1 power supply for heating the filament, U2 - power supply for discharge, PS - power supply for the electrode bias, R, R2 - load resistors, EP - emissive probe, PP - plane probe, X, Y - to the oscilloscope).

Figure 2. Static current-voltage characteristic of the additional electrode (the small letters mark the positions on the characteristic where the behavior of the plasma system changes).

show hysteresis". After the first sudden jump, marked c-d in the static currentvoltage characteristic of the electrode in Fig. 2, a quasispherical luminous structure appears in front of E (photo in Fig. 3). Its appearance implies a process by which thermal energy of the electrons extracted from the surrounding plasma

240

zyxwvutsr

Figure 3. Photo of the complex space charge structure obtained in front of the positively biased additional electrode.

is converted into the electric field energy of the double layer at the border of the structure”. The second jump of the current through the electrode E, marked by e-f in the Fig. 2, is related to the transition of the structure in the dynamic state. The periodical disruptions and re-aggregations of the double layers cause a modulation of the current collected by the electrode, as shown in Fig. 4a. The fast Fourier transform (FFT) amplitude graph of these oscillations and the reconstructed 3D state space (by time delay m e t h ~ d ’ ~of) the plasma system dynamics are shown in Figs. 5a and 6a, respectively. y increasing the voltage applied on the additional electrode from 55 V to 64 V, we recorded the time series of the ac component of the current collected by the electrode E (Fig. 4), with a sampling rate of 500 kHz.From these time series we calculated the FFT amplitude graphs (Fig. 5) and we reconstructed the 3D state spaces of the plasma system dynamics (Fig. 6). From the figure 4 we observe a transition to a chaotic state, due to intermittencies.The FFT amplitude graphs indicate the evolution to a chaotic state by embedding the fundamental frequency in broadband noise, associated with the onset of the intermittencies. The reconstructed 3D state spaces indicate the loss of stability of a periodic attractor (limit cycle) through a succession of bursts. The mechanism of reinsertion of trajectories in the closed loop of the attractor is relevant for proving the intermittency route to chaos’.

zy

4-

3-

4

3

-

-= 2

zy

zyxwvutsrqponm zyxwvutsrq

zyxwvut 5-

2-

2

1-

-w

1

1-

54

1

0

0I 0

2

I

1

2

4

3

i

5

.

0

E

2

-w

1

,

2

3

4

5

1

zyxwvutsr

Figure 4. Oscillations of the current collected by the additional electrode, for different values of the voltage applied on it: (a) 55 V, (b) 58 V, (c) 62 V, (d) 64 V.

Frequency (kHz)

0.3

7

Frequency (kHz)

1 0.2

0.2

-?

IJ.0.1

t

-+:

c t

0.0

zyxwvu (d)

0.1

zyxwvu 0.0

0

Frequency (Wz)

Frequency ( W z )

Figure 5. FFT amplitude spectra of the corresponding signals from Fig. 3.

242

zyxwvutsrq zyxwv

Figure 6 . 3D reconstructed state spaces of the plasma system dynamics from the corresponding signals from Fig. 3.

A typical fingerprint of type I intennittency is the presence of a tangent bifurcation (saddle-node point of bifurcation), represented in the return map shown in Fig. 7. We reconstructed this map by plotting the maxima and minima of the time series.

3. Conclusion The evolution to a chaotic state of a complex space charge structure dynamics in plasma through intennittency is experimentally investigated. The results from the time series analysis, spectral analysis, 3D reconstructed state space of the current oscillations and the return map confirm the existence of the type I intennittency.

zy 243

z

a4-

a3 -

-: ?

a2-

-

2 ai-

I

I

0.0

0.1

0.2

0.3

0.4

In(a.u.)

Figure 7. Return map, reconstructed by plotting the maxima and minima of the time series.

Acknowledgments This work was financially supported by the National Authority for Scientific Research - Romanian Ministry for Education, Research and Youth, under the Excellence Grant No. 1499/2006, cod ET 69.

References

zyxwv zy

1. P. Y. Cheung and A. Y. Wong, Phys. Rev, Lett. 59,551 (1987). 2. P. Y. Cheung, S. Donovan and A. Y. Wong, Phys. Rev. Lett. 61, 1360 (1988). 3. J. Qin, L. Wang, D. P. Yuan, P. Gao and B. Z. Zhang, Phys. Rev. Lett. 63, 163 (1989). 4. S. Chiriac, D. G. Dimitriu and M. Sanduloviciu, Phys. Plasmas 14, 072309 (2007). 5. W. Ding, W. Huang, X. Wang and C. X. Yu, Phys. Rev. Lett. 70, 170 (1993). 6. S. Chiriac, M. Aflori and D. G. Dimitriu, J. Optoelectron. Adv. Mater. 8, 135 (2006). 7. Y. Pomeau and P. Maneville, Commun. Math. Phys. 74, 189 (1980). 8. C. Charles, Plasma Source. Sci. Technol. 16, R1 (2007). 9. M. Sanduloviciu and E. Lozneanu, Plasma Phys. Control. Fusion 28, 585 (1986).

244

zyxw zyxw zy

10. B. Song, N. D’Angelo and R. L. Merlino, J. Phys. D: Appl. Phys. 24, 1789 (1991). 11. S. Chiriac, E. Lozneanu and M. Sanduloviciu, J. Optoelectron. Adv. Muter. 8, 132 (2006). 12. E. Lozneanu and M. Sanduloviciu, Chaos, Solitons and Fractals 30, 125 (2006). 13. D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989.

245

NONLINEAR DYNAMICS OF A HAMILTONIAN FOUR-FIELD MODEL FOR MAGNETIC RECONNECTION IN COLLISIONLESS PLASMAS

zy

zyxwv zyxwv zyxw zyxwv

E. TASSI' and D. GRASS0

Dipartimento di Energetica, Politecnico di Torino, Torino, 10189, Italy * E-mail: [email protected]

F. PEGORARO

Dipartimento di Fisica, Universitd di Pisa, Pisa, 56127, Italy

In this contribution we present some aspects of the nonlinear dynamics of a model which describes the phenomenon of magnetic reconnection (MR) occurring in plasmas where particle collisions can be neglected. The concept of MR is introduced in the framework of the singlefluid description of a plasma. The model under consideration is then reviewed with focus on its non-canonical Hamiltonian structure. Numerical solutions of the model equations show that in the nonlinear phase a secondary instability of Kelvin-Helmholtz type occurs at moderate values of the parameter p, which indicates the ratio between thermal and magnetic pressure, and also in the presence of finite electron compressibility. This represents a novel feature, in the nonlinear dynamics of reconnection, with respect to previously investigated models valid only for very low values of p. Keywords: Magnetic reconnection, non-canonical Hamiltonian systems, KelvinHelmholtz instability

1. Introduction Astrophysical and laboratory plasmas often exhibit nonlinear behaviors which manifest themselves, for instance, through processes such as selforganization, chaos and turbulence.' Plasmas therefore represent a fertile ground for the investigation of nonlinear phenomena. Among these, the process of MR plays an important role, also due to its implications for space and fusion plasma^.^^^ In the former context MR has indeed been suggested to play a key role in triggering solar flares and magnetic substorms

246

zyxw zyxw

whereas in the latter case it is believed to be one of the processes causing the so-called sawtooth oscillations observed in tokamak fusion devices. In the framework of a single-fluid description of a plasma, MR can be thought of as result of the violation of the so-called frozen-in condition, according to which, in a perfectly conducting plasma, if two infinitesimal volumes of fluid are threaded a t some time by the same magnetic field line, then they will always be joined by the same magnetic field line a t any later time. The mathematical justification for this condition is a consequence (see e.g.’l4) of the equation

aB at

--

Vx

(V x

B ) = 0,

which is valid for an infinitely conducting plasma. In (1) B is the magnetic field and v is the plasma fluid velocity. In many laboratory and astrophysical contexts (1) is satisfied to a good extent. However, there can be localized spatial regions where effects such as collisional resistivity, electron inertia or turbulence can prevent the plasma from behaving as a perfect conductor and break the constraint of the frozen-in condition. Infinitesimal plasma volumes initially joined by a single magnetic field line can then disconnect from that line and connect to two distinct field lines a t a later time (see Fig. 1). This phenomenon is accompanied by a conversion of magnetic energy into plasma kinetic and thermal energy, thus explaining, for instance, why MR is believed to be responsible for the huge releases of energy occurring during solar f l a r e ~ . Among ~t~ the various physical effects determining the violation of the frozen-in condition and the consequent possibility for MR to take place, electron inertia has been suggested t o be a relevant one in particular in the context of fusion plasmas, for it provides characteristic reconnection time scales comparable with those observed during sawtooth

-

magncticfisld lins

zyxwvut . . . . . . 3

pbrnavclofityfisld line

Fig. 1. Figure sketching the mechanism of violation of the frozen-in condition: pairs of small plasma volumes (the black and the white squares) are initially joined by the same magnetic field line. At a later time, due t o reconnection, the black and the white squares, initially disjoined, are connected by the same magnetic field line.

247

oscillations in t o k a m a k ~ Models .~ in which MR is caused by electron inertia and where dissipative effects are negligible (which are for instance applicable namely t o the high temperature, weakly collisional tokamak plasmas), possess the remarkable property of admitting a Hamiltonian formulation, which, in addition to its formal elegance, makes it possible to take advantage, in the analysis, of the huge amount of results available for Hamiltonian systems. In this contribution we present results related to the nonlinear dynamics of a model for MR driven by electron inertia. In Sec. 2 we briefly introduce the model and in Sec. 3 we present its non-canonical Hamiltonian formulation. In Sec. 4 we present the results from the numerical simulations and we focus on the onset of secondary fluid-like instabilities taking place after the initial instability that led to reconnection. The final section is devoted to conclusions.

zyx

zyx

2. The model

The dynamical model for MR that we consider here is the four-field model derived in Ref.6v7 Such model describes MR processes caused by electron inertia in a plasma where particle collisions can be neglected. To describe the geometry of the problem a Cartesian coordinate system (z, y, z ) is adopted, with z as ignorable coordinate. The dynamics is then two-dimensional with reconnection of magnetic field lines taking place in a slab lying in the zy plane. The model equations, written in a standard dimensionless form, read

zyxwvu zyxwv dU

+ [%VI

(5)

-

In the above equations the flux function +, the stream function p and the fields 2 and w are related to the magnetic field B and to the plasma average velocity field v by the following relations: B(z,y, t ) = V+(z, y, t) x i (B(O) c p Z ( z , y , t ) ) i and v ( z , y , t ) = - V p ( z , y , t ) x z v ( z , y , t ) i . The constant B(O) is the so called guide field which mimics the toroidal

+

+

+

zyxw zyxwv zyxwvu

248

field present in tokamak devices. The parameter cp is defined as cp = where j3 indicates the ratio between the plasma background pressure and the magnetic pressure based on the guide field B(O). The other parameters present in the system are the electron skin depth d , and d p = dicp, with di indicating the ion skin depth. Finally the Poisson bracket is defined by [ f , g ]= ( O f x V g ). z. The above model can be derived starting from the standard two-fluid description of a plasma. In particular (2) comes from the electron momentum equation, (3) from the electron vorticity equation, (4) from the average plasma vorticity equation and (5) from the average plasma momentum equation. Notice that the term d2aV2+/at, in Eq. (2), is proportional to the electron inertia and it is namely the presence of this term that breaks the frozen-in condition allowing MR to take place. If electron mass were neglected the magnetic flux would be a scalar quantity advected by the stream function (p dpZ and its contour lines (corresponding to the projections of magnetic field lines in the xy plane) could not reconnect.

d

m

+

+

3. Hamiltonian structure

As many dissipation-free models, the system (2)-(5) admits a non-canonical Hamiltonian formulation.8 In short this means that the above model can be written in the form

xi

where ci {, } is an antisymmetric bilinear form satisfying the Jacobi identity. For the model under consideration it has been showng that the dynamical equations for the field variables = - dZV2@,(p, Z and u can be obtained from the Hamiltonian

+, +

's

H =2

d 2 z ( d Z J 2+ IV(pI2+ u2 + lV+I2+ Z 2 )

(7)

and from the bracket defined by

{ F ,GI =

1

d2x ( ~ [ F uGu] ,

z

zyx zy zyx

-at ={&,H}, i = l , . . ., 4 (6) are field variables, H is a Hamiltonian functional and the bracket

+ +e(

[F$,7 Gu]

+

+ P U , G$J - dP([FZ,G$,I+ [F?be, GZI) + cp([FvrGzl [ F z ,GI)) + ~ ( [ F Z , G+ U [Fu,Gz] ] -dpde2[F~,,G+,,] +cpde2([Fv,G$,] [ F QG ~ v,] )

+

+

Gvl - a [ F z , Gzl - C P Y P v , GI)+ V([Fv,GU] [Fu,

+ [F+,,Gzl)- c p ~ ( [ F v , G z+] [ F z , G ~ ] ) ) ) , +~pde~([Fz,G$,I

(8)

zyxwvuts zyxwvu zyxw zy z zyxwvu 249

+

where F and G are two functionals of the dynamic variables, a = d p cpde2/di,y = d e 2 / d i and subscripts indicate functional differentiation. It has also been shown that the bracket (8) possesses four infinite families of so-called Casimir invariants, i.e. functionals C of the field variables such that {C, G} = 0,

for every functional G.

(9)

Casimir functionals are constants of motion and can provide important information about the nonlinear dynamics of the system. For the four-field model the four families of Casimirs are given by

C1 =

C,

where D

=

/

d2zwF(D),

/d2zg+ ( D

-

“pw

ff

-

C2 =

/

d2z’H(D),

E)/ -Z

d2zg+(T+),

=

(11)

w = V 2 p + Z / a ,T&= &(l/2a)&&4D-(a/cp)u~ whereas F , N , g+ and g- are arbitrary functions. When written in terms of the variables D , w , T* the model equations (2)-(5) take the form

d*Z),

= $e+div,

aD at

- = -[P,DI,

aw

- = -[p,w] at

+ de2 +1 di

2

P7

$13

This representation shows that the fields D , T* turn out to be Lagrangian invariants advected by corresponding velocity fields. In particular the field D gets advected by the actual average plasma velocity whereas T* get advected by “virtual” velocity fields associated to stream functions composed by a fluid (associated to cp) and a magnetic (associated to $) component.

zyxwvutsr zyxwvut

250

4. Numerical simulations

zyx zy zy zyx zy zyx

The model equations (2)-(5) have been solved numerically, adopting a finite volume scheme, over the rectangular domain {(x, y) : -2n 5 5 5 2n, -7r 5 y 5 n} discretized over 1024x512 grid-points. The initial condition corresponds to the choice: $eq = 1/ cosh2(x), peg = 0, Z q = 0,V e g = 0 and double periodic boundary conditions have been imposed.

@

-0

t=O

zyxwvutsr

5

10

-7

-1

0 x/n

1

2

0

-1

1

x/.

Fig. 2. Contour plot of the flux function at two different times showing how initially disconnected magnetic field lines get joined at a later time due t o MR.

Figure 2 shows contour plots of the 1c, function obtained from the simulations at two different times. The comparison of the plots shows how initially disconnected magnetic field lines get connected a t later time due t o the on-going MR process. In addition t o the investigation of the MR process itself the numerical simulations make it also possible to observe and study further nonlinear processes which occur as a by-product of MR. In the analysis presented in this contribution we focus on the observation of plasma vorticity structures and on their dependence on the parameters cp and d p , while keeping d, fixed to the value 0.24. Looking a t the vorticity field is significant in order t o detect the presence of fluid-like instabilities and allows the comparison with previous work based on low-p models. Indeed, numerical simulations of the four-field model run in the low-p regime, whose results are not presented here, correctly tend to reproduce the vorticity and current density behavior already observed in the analysis of low-/3 models. In particular for values of cp, dp lop3, in which limit the four-field model reduces to the two-field model investigated in one observes the formation of two vorticity and current density jets directed along the vertical direction. Such jets N

zy zyxwvu zyxwvut zyxw 251

subsequently propagate toward each other until they collide and undergo a Kelvin-Helmholtz-type instability. On the other hand if cp l o v 3 and dp 10-1 the simulations of the four-field model reproduce the behavior observed in the low-p two-field model investigated in.l3?l4In this limit vorticity and current density tend to concentrate along the separatrix lines of the magnetic island formed after reconnection, and to create filamented structures. This behavior was explained in terms of the constraints imposed to the dynamics by the presence of two Lagrangian invariants in the model equations. One of the novel aspects introduced by the four-field model is that it makes it possible to investigate nonlinear dynamics also in higher-/? regimes. In Fig. 3 it is possible to see the evolution of the vorticity field V2cp through four different times (time here is normalized with respect to the Alfvkn time L,/ij/Bo where L,Bo and p are characteristic length scales and intensity of the magnetic field, and plasma mass density, respectively) for cp = 0.2 and dp = 0.48. The contour plot a t t = 50 shows the presence of a localized vertical structure a t the center of which a quadrupolar structure is formed. Contour plots a t earlier times show that such structures form due to the headon collision between two vertical jets moving in opposite directions. The vertical structure is enclosed by a region where vorticity is filamented, and whose boundary turns out to coincide with the contour of the magnetic island formed due t o reconnection,. At subsequent times pairs of vortices of the quadrupolar structure start to drift in opposite directions along the y = 0 axis. At t = 65 the vortices reach the filamented region and one can also observe that the vertical structure starts to break up indicating the onset of a Kelvin-Helmholtz type instability. In the higher-P regime, where w and 2 are not decoupled from the system, the vorticity is given by N

N

zyxw zyxwvuts I

U = L&,/----(T+ 1 2d,

dz+d:

- T-)

+w ,

The vorticity can then be interpreted as the superposition of two contributions. One contribution corresponds to the terms with T+ and T-. These fields behave similarly to two Lagrangian invariants already investigated in the very low-/? limit. Indeed such fields get advected by the fields cp+ = c p f and are responsible for the appearance of filamented

2 J$- / ,

vorticity structures observed in the contour plots. As explained in Ref.13 such filamented structures are the result of the stretching of the fields T*

252

zyx

zyxwvuts zyxwv $

0.5 0.0

---0.5 -1.0 2

1

0

1

Z , h

05 I=

00

\

3

-0 5 10

a

1

0

1

X/7T

0.5 0.0

s ---

0.5 1 .li

z

1

0

1

.c/n

zyxwvuts zyxwvu

Fig. 3. Contour plots of the vorticity field at different Alfvkn times. The values of the parameters are: cp = 0.2 (corresponding to /3 = 0.04), do = 0.48, d, = 0.24.

zyxw zyx zyx zyxw zy zyxwv 253

operated by the “virtual” velocity fields 96 which rotate in opposite directions. The localized vertical structure is then due t o the presence of the remaining contribution to U , i.e. the one coming from w , which had no corresponding term in the expression for U in the low-p limit. Contour plots of U at subsequent times show that such vorticity structures undergo a KelvinHelmholtz type instability as it is the case in the very low-p regime when the value of the electron compressibility parameter ps = a / e B is much smaller than dele . Recalling that for the low+ regime under consideration cp f l ,so that dp = p s , our results show that the Kelvin-Helmholtz instability, which is inhibited in the presence of finite ps a t very low-p, can take place even if ps # 0 , if the value of p is increased. The effect of increasing p introduces a shift between -Z/a and U , which yields a finite generalized vorticity w . Indeed, as above explained, it is the generalized vorticity that is related to the appearance of the vorticity jets undergoing Kelvin-Helmholtz instability. Moreover, at higher ,f3 values, an effective coupling with Eq. (5) occurs, whereas w decouples from the system for very small values of p.

-

5. Conclusions In this contribution we presented new results concerning the nonlinear dynamics of a collisionless reconnection process described by a four-field Hamiltonian model. Numerical simulations reproduce already known results in the regime of very low-p. At higher p values a new regime is found, where both filamentation and formation of vorticity jets are observed. It is shown that in this new regime a secondary Kelvin-Helmholtz instability can take place even if ps is finite.

References

zyxwvu zyx

1. W. Horton and Y.-H. Ichikawa, Chaos and Structures in Nonlinear Plasmas (World Scientific Publishing Co., 1996). 2. E. R. Priest and T. G. Forbes, Magnetic Reconnection (Cambridge University Press, 2000). 3. D. Biskamp, Magnetic Reconnection in Plasmas (Cambridge University Press, 2000). 4. G. Hornig and K. Schindler, Phys. Plasmas 3,781 (1996). 5. J. Wesson, Nucl. Fusion 30,p. 2545 (1990). 6. R. Fitzpatrick and F. Porcelli, Phys. Plasmas 11,4713 (2004). 7. R. Fitzpatrick and F. Porcelli, Phys. Plasmas 14,p. 049902 (2007). 8. P. J. Morrison, Phys. Plasmas 12,058102 (2005). 9. E. Tassi, P. J. Morrison and D. Grasso, Hamiltonian structure of a collisionless reconnection model valid for high and low @ plasmas, in PTOC.Workshop

254

10.

11. 12. 13. 14.

zyxwvutsrqpo

zy zy zy

Collective Phenomena in Macroscopic Systems, eds. M. R. G. Bertin, R. Pozzoli and K. Sreenivasan (World Scientific, Como, Italy, August 2007). D. Del Sarto, F. Califano and F. Pegoraro, Phys. Plasmas 12, p. 012317 (2005). D. Del Sarto, F. Califano and F. Pegoraro, Mod. Phys.Lett. B 2 0 , 931 (2006). D. Grasso, D. Borgogno and F. Pegoraro, Phys. Plasmas 14,p. 055703 (2007). E. Cafaro, D. Grasso, F. Pegoraro, F. Porcelli and A. Saluzzi, Phys. Rev. Lett. 80, 4430 (1998). D. Grasso, F. Califano, F. Pegoraro and F. Porcelli, Phys. Rev. Lett. 86, 5051 (1994).

255

zyx

ON THE COMPLEXITY OF THE NEUTRAL CURVE OF OSCILLATORY FLOWS M.WADIH, S.CARRION, P.G.CHEN, D.FOUGeRE, B.ROUX MSNM-GP, UMR 6181 CNRSKJniversitksAix-Marseille, France

zyxwv

In this paper, the linear stability property of oscillatory flow in a circular pipe under a periodic pressure gradient is examined. Our results of neutral curve show the existence of a much more remarkable finger-like protrusions structure. In this case, we may speak about two neutral curves: one inside which corresponds to a rather late critical threshold and the other one which makes this threshold weaker but closer to experiment observations. The underlying mechanism is explained both from mathematical point of view and from physical point of view.

1. Introduction The linear stability property of oscillatory flows such as flat Stokes layers usually exhibits regular and smooth enough neutral curves, suggesting a light influence of unsteadiness of the basic flow; the unsteady effect when compared to classic steady flows is often limited itself to advance or enhance transition. Many studies, whether theoretical or numerical in nature even failed to predict transition thresholds onto less orderly regimes as observed in some experimental works. From large body of literature dealing with the linear stability of various types of oscillatory flows one can distinguish a lot of works, started from the pioneer work in 1930’s by Schlichting [l],followed by Stuart [2] and Riley [3], who analyzed the steady streaming induced by an oscillatory flow. Venezian [4] considered BCnard problem with periodic temperature gradient and obtained solutions by assuming the amplitude temperature gradient be small and expanding all quantities in powers of this parameter. The same problem was studied by Rosenblat & Herbert [5] with a WKB method. Analytical studies of purely oscillatory flows begun with the paper of von Kerzeck & Davis[6] for a high frequency oscillatory, the study was followed by Hall [7] in the case of the classical flat Stokes layer.

256

The recent works of Blennerhassett & Bassom [8, 91 on the linear stability of Stokes layers provide a new insight into the investigation of oscillatory flows for a better prediction of various transition modes. The most intriguing feature of their results is the fine structure of the neutral curve, which has thin fingerlike protrusions from an essentially smooth curve in some range of wavenumber. The present work concerns the linear theory of neutral stability curve for the oscillatory flow in a circular pipe under a periodic pressure gradient. The calculations of stability based on Floquet theory lead to a relationship between Reynolds number and wavenumber of the first instability mode. The finger-like protrusions appear on the neutral curve in such a way that we can speak of two neutral curves. The formulation of the problem is described in Section 2. Our results are presented and discussed in Section 3. Conclusions are given in Section 4.

2. Formulation of the problem

zyx zyxw zyx

An oscillatory flow in a circular pipe with infinite length, radius R is considered with periodic flow with frequency w and amplitude Q,, . Only the axisymmetric case is considered herein with non-dimensional coordinates (z, r) where the zaxis is in direction of the pipe axis. Lengths, velocities are respectively scaled with the radius R and $ / ( n R 2 ) ,the time being z = ot .

zyxwvu

Figure 1. Schematic of flow configuration.

The dimensionless parameters usually introduced are the Reynolds number R e = Q o / (nRv) and the frequency 52 = R2co/v where v is the kinematic viscosity. The basic motion is described by the equation of continuity and the Navier-Stokes equations:

v.v=o 51a,v+R,v.Vv=-R,Vp+V2V completed with a non-slip condition at the wall.

(1)

zyxwvut zyxwvuts z zyxw 257

As the basic flow V is generated in the direction of longitudinal axis, the solution of ( 1 ) takes the form:

-

, v = w = o , (2)

where J o and J 2 respectively denote Bessel functions of order zero and two. The linear stability analysis is made by introducing an axisymmetric perturbation u ' e , + v ' e , . Then, by the mean of the introduction of stream function disturbance of the form:

with a the real wavenumber of the disturbance, the use of Galerlun method as described by Siouffi et al. [lo] leads to ordinary differential equations for fm

(4

f DKf, = p42 r

,

(4)

zyxw

where D , K are operators defined by (see also Siouffi et al. [ l o ] ) : d2

1d

+---

1 d2

I d

) ' K = -r ( d-----a2 r2 r d r

-+d

D=-dr2 r dr (r12 with the boundary conditions:

Then following the usual Galerkin technique, the first-order linearized differential system is obtained:

c-dG dz

zyx

=AG+B(~)G,

where C, A and B are matrices representing, for N basic functions of the stream function, linear operators which depend only on Ll and a :

258

zyxwvut zyxw zyx zyxwvu

where operator L is defined by L =

3. Results

3.1. Numerical method

Matrices A, B and C are first calculated for fixed parameters R and a . Then the numerical solution of the system ( 5 ) was obtained by using a fourth-order Runge-Kutta method. For a fixed parameter R e , the number of equations N is increased until to find a minimal number N to ensure the convergence of the solution.

3.2. Strong stability

The strong stability corresponds to the resolution of (5) for one initial condition as it was studied by Siouffi et al. [lo] for R=100. They found a critical Reynolds number of 3500 for a = 2.5 which is in agreement with experimental results [ l l ]and [12]. This study gives a sufficient condition of stability.

3.3. Weak stability The use of Floquet theory implies that the stability is ensured in the sense where some disturbances can grow inside a cycle but be globally attenuated in the next cycles as it is showed in figure 2. This stability is insured if the real part of the Floquet exponents is zero. Starting from N initial conditions such as G(z = 0) = I (I is the identity matrix), we found that for SZ =loo, N = 50 is necessary to ensure the convergence of the Floquet exponents.

259

lcll 70 60

-

-

^

-

_

_

-

Re=9000

~

Re=9745

-

z

Re=9780

zyxwvu zyxwvutsrq zyxwvutsrq 0,OO

3,14

6,28

9,42

12,56

15,70

z

18,84

Figure 2. Temporal evolution of the magnitude of the fundamental matrix of system (5) for c;2 =lo0 and a=2. Re = 9000: stable, R, = 9745: critical value, Re = 9780: unstable.

The neutral curve ( a ,Re ) in figure 3 which corresponds to the case of zero value of the real part of the greatest exponent Floquet shows protrusions. The inner points (on the right) correspond to complex conjugate Floquet exponents while the other points correspond to real Floquet exponents. We may define two curves: an outer curve along the extremities of the fingers and another one along inner points. We obtain the critical values aC= 1.997 and Re, = 9631.34,

zyx

ff

2,20

2,oo

1,80

1,70 9600

9700

zyxwv 9800

9900

Re

10000

Figure 3. Neutral curve for 51 =loo.

It is seen in figure 4 that the two greatest Floquet exponents a, and O, become alternatively complex conjugate or real as R, is increased.

260

0,060

0,000

-0,060

zyxwvuts zyxwvut zyx zyxw zyx zyxwvu zy zyxw

-0,120 9400

9500

9600

9700

9800

R

Figure 4. Real part ofthe two greatest Floquet exponents as a function of R e for S2 =lo0 and a=2.

When a is increased, fluctuations appear on the Floquet exponent (figure 5) so that it becomes numerically impossible to obtain the neutral curve for value of a greater than 2.7. 0,005 3

0,000

-0,005

zyxwvuts qr

-

-

d

,v

I,

I

-0,010

4

-0,020

!

13768

t

I

I

13770

4

I

I\

I I

13772

---- 9 r

A

U

13774

Re

13776

Figure 5 . Variation of the real part of the greatest Floquet exponents with Re for S2 =lo0 and ~~=2.62.

4.

Conclusions

Our results of neutral curve show the existence of a much more remarkable finger-like protrusions structure. Therefore, we may speak of two neutral curves: one inside which corresponds to a rather late critical threshold and the other which makes this threshold weaker but closer to experiment observations.

261

The mechanism of the generation of fingers on the neutral curve can be explained from a mathematical point of view by an alternation of zones for real or complex Floquet exponent, and from a physical point of view by the coalescence of two propagating waves in opposite directions. In addition, this mechanism is generic and when the transition takes place in a zone with real value, a finger appears on the neutral curve which decreases the critical threshold and introduces a second curve closer to a smaller Reynolds number. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

zyxwvu zyxwvu zyxwvu zyxwvu

H. Schlichting, Phys. Z. 33, 327 (1932). J. T. Stuart, J. FluidMech. 24, 673 (1966). N. Riley, J. Inst. Math. Appl. 3,419 (1967). G. Venezian, J. Fluid Mech. 35,243 (1969). S. Rosenblat and D. M. Herbert, J. Fluid Mech. 43,385 (1970). C. von Kerzeck and S. Davis, J. FluidMech. 62,753 (1974). P. Hall, Proc. R. SOC.Lond. A 359, 151 (1978). P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 464,393 (2002). P. J. Blennerhassett and A. P. Bassom, J. Fluid Mech. 556, 1 (2006). M. Siouffi, S. Carrion and M. Wadih, C.R. Me'canique 330,641 (2002). M. Hino, M. Savamoto and S. Takasu, J. Fluid Mech. 75, 193 (1976). D. M. Eckmann, J. B. Grotberg, J. Fluid Mech. 222, 329 (1991).

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(2) OTHERS

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265

zy

GENERALIZED KELLER-SEGEL MODELS OF CHEMOTAXIS. ANALOGY WITH NONLINEAR MEAN FIELD FOKKER-PLANCK EQUATIONS

z zyxwvu

PIERREHENRI CHAVANISt

Laboratoire de Physique The'orique, Universite' Paul Sabatier, 118 route de Narbonne 31 062 Toulouse, France $E-mail: chavanis0irsamc. ups-tlse.fr We consider a generalized class of Keller-Segel models describing the chemotaxis of biological populations (bacteria, amoebae, endothelial cells, social insects,...). We show the analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. As an illustration, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). We also discuss the analogy between biological populations described by the Keller-Segel model and self-gravitating Brownian particles described by the Smoluchowski-Poisson system.

1. Introduction

The name chemotaxis refers to the motion of organisms induced by chemical signals.' In some cases, the biological organisms (bacteria, amoebae, endothelial cells, ants...) secrete a substance (pheromone, smell, food, ...) that has an attractive effect on the organisms themselves. Therefore, in addition to their diffusive motion, they move preferentially along the gradient of concentration of the chemical they secrete (chemotactic flux). When attraction prevails over diffusion, the chemotaxis can trigger a self-accelerating process until a point at which aggregation takes place. This is the case for the slime mold Dictyostelium discoideum and for the bacteria Escherichia coli. This is referred to as chemotactic collapse. A model of slime mold aggregation has been introduced by Patlak2 and Keller & Sege13 in the form of two coupled differential equations. The first equation is a drift-diffusion equation describing the evolution of the concentration of bacteria and the second equation is a diffusion equation with terms of source and degradation describing the evolution of the concentration of the chemical. In the simplest model, the diffusion coefficient D and the mobility x of the bac-

zyxwvut zyx zy

266

teria are constant. This forms the standard Keller-Segel model. However, the original Keller-Segel model allows these coefficients t o depend on the concentration of the bacteria and of the chemical. If we assume that they only depend on the concentration of the bacteria, the general Keller-Segel model becomes similar to a nonlinear mean field Fokker-Planck equation. Nonlinear Fokker-Planck (NFP) equations have been introduced in a very different context, in relation with a notion of generalized thermodynami c ~ As . ~ far as we know, the connection between the general Keller-Segel model and nonlinear mean field Fokker-Planck equations has been first mentioned in Chavanis5 and developed in subsequent papers (see6 and references therein). This analogy makes possible t o interpret results obtained in chemotaxis in terms of a generalized thermodynamics. At the same time, chemotaxis becomes an example of great physical importance for which a notion of (effective) generalized thermodynamics is justified. The standard Keller-Segel (KS) model has been extensively studied in the mathematical literature (see the review by Horstmann7). It was found early that, above a critical mass, the distribution of bacteria becomes unstable and collapses. This chemotactic collapse leads ultimately to the formation of Dirac peaks.8-20 Recently, it was shown by Chavanis, Rosier & Sire21 that, when the equation for the evolution of the concentration is approximated by a Poisson e q ~ a t i o n the , ~ standard ~ ~ ~ ~ Keller-Segel ~ ~ ~ (KS) model is isomorphic to the Smoluchowski-Poisson (SP) system describing self-gravitating Brownian particles. The chemotactic collapse of biological populations above a critical mass is equivalent to the gravitational collapse of self-gravitating Brownian particles below a critical temperature.22Assuming that the evolution is spherically symmetric, Chavanis & Sire21-28were able to describe all the phases of the collapse (pre-collapse and post-collapse) analytically in d dimensions, including the critical dimension d = 2. Recently, some authors have considered generalizations of the standard Keller-Segel (KS) model. Two main classes of generalized Keller-Segel (GKS) models of chemotaxis have been introduced: (a) Models with filling factor: Hillen & Painter29>30considered a model with a normal diffusion and a density-dependent mobility x ( p ) = x(1 p/oo) vanishing above a maximum density 00. The same model was introduced independently by C h a ~ a n i s in ~ ?relation ~~ with an “exclusion principle” connected to the Fermi-Dirac entropy in physical space. In these models, the density of bacteria remains always bounded by the maximum density: p(r, t ) I no. This takes into account finite size effects and filling fac-

zyxwvu

zyx zyx

zyxw z zyxwv zy zyxwv 267

tors. Indeed, since the cells have a finite size, they cannot be compressed indefinitely. In this generalized Keller-Segel model, chemotactic collapse leads ultimately to the formation of a smooth aggregate instead of a Dirac peak in the standard Keller-Segel model. This regularized model prevents finite time-blow up and the formation of (unphysical) singularities like infinite density profiles and Dirac peaks. Therefore, the Dirac peaks (singularities) are replaced by smooth density profiles (aggregates). (ii) Models with anomalous diffusion: Chavanis & Sire3' studied a model with a constant mobility and a power law diffusion coefficient D ( p ) = DpY-' (with y = 1 l / n ) . This lead to a process of anomalous diffusion connected to the Tsallis entropy.33 For 0 < n < n3 = d / ( d - 2), the system reaches a self-confined distribution similar to a stable polytrope (e.g. a classical white dwarf star) in astrophysics. For n > n3, the system undergoes chemotactic collapse above a critical mass (the classical chemotactic collapse related to the standard Keller-Segel model is recovered for n -+ + w ) . ~ ' In the pre-collapse regime, the evolution is self-similar and leads to a finite time singularity. A Dirac peak is formed in the post-collapse regime. For n = 723, the dynamics is peculiar and involves a critical mass similar to the Chandrasekhar limiting mass of relativistic white dwarf stars in a s t r o p h y ~ i c s .The ~ ~ case of negative index n < 0 is treated in35 with particular emphasis on the index n = -1 leading to logotropes. In the present paper, we discuss a larger class of generalized Keller-Segel models and interpret these equations in relation with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. For illustration, we present for the first time a model incorporating both a filling factor and some effects of anomalous diffusion.

+

2. The generalized Keller-Segel m o d e l 2.1. The dynamical equations

The general Keller-Segel model3 describing the chemotaxis of bacterial populations consists in two coupled differential equations

dP

at = v . (DzVp) - v . (DlVC) ,

that govern the evolution of the density of bacteria p ( r ,t ) and the evolution of the secreted chemical c ( r , t ) . The bacteria diffuse with a diffusion

268

zyxwvut z zyxw zyx zyxw z

coefficient D2 and they also move in a direction of a positive gradient of the chemical (chemotactic drift). The coefficient D1 is a measure of the strength of the influence of the chemical gradient on the flow of bacteria. On the other hand, the chemical is produced by the bacteria with a rate f(c) and is degraded with a rate k(c). It also diffuses with a diffusion coefficient D,.In the general Keller-Segel model, D1 = D ~ ( p , cand ) 0 2 = D2(p1c) can both depend on the concentration of the bacteria and of the chemical. This takes into account microscopic constraints, like close-packing effects, that can hinder the movement of bacteria. If we assume that the quantities only depend on the concentration of bacteria and write Dz = D h ( p ) , D1 = x g ( p ) , k(c) = k 2 , f ( c ) = X and D, = 1, we obtain

dP

d t = V . (Dh(P)VP- X S ( P ) V C ) dC

E-

at

= AC- k2c

1

+ Xp.

(3)

(4)

For c = 0, Eq. (4) becomes the screened Poisson equation

AC- k2c = -Xp.

(5)

Therefore, we can identify k-' as the screening length. If we assume furthermore that k = 0, we get the Poisson equation

AC= -Xp.

(6)

The generalized Keller-Segel (GKS) model (3) can be viewed as a nonlinear mean-field Fokker-Planck (NFP) e q ~ a t i o n .Written ~ in the form dtp = V . ( V ( D ( p ) p )- x ( p ) p V c ) , it is associated with a stochastic ItoLangevin equation

with

where R(t) is a white noise satisfying ( R ( t ) )= 0 and (Ri(t)Rj(t'))= & S ( t - t') where i = 1,...,d label the coordinates of space. The standard Keller-Segel model is obtained when the mobility x and the diffusion coefficient D are constant. This corresponds to h(p) = 1 and g ( p ) = p. In that case, the stochastic process (7) and the Fokker-Planck equation (3) are similar to the ordinary Langevin and Smoluchowski equations describing the diffusion of a system of particles in a potential @ ( r , t )= - c ( r , t )

zy

zyxw zy 269

that they produce themselves through Eq. (4). For example, when Eq. (4) is approximated by Eq. ( 6 ) , the system becomes isomorphic to the Smoluchowski-Poisson system describing self-gravitating Brownian particles.21122The steady state of the standard Keller-Segel equation is p e 5 ‘ . This is similar to the Boltzmann distribution p e-’@IT of statistical equilibrium provided that we introduce an effective temperature T through the Einstein relation T = D / x . In the present study, we shall consider more general situations and allow the mobility x ( p ) and the diffusion coefficient D ( p ) to depend on the local concentration of particles p ( r , t ) . This is an heuristic approach to take into account microscopic constraints that affect the dynamics of the particles at small scales and lead to non-Boltzmannian distributions at equilibrium. Indeed, it is not surprising that the mobility or the diffusive properties of a particle depend on its environment. For example, in a dense medium its motion can be hampered by the presence of the other particles so that its mobility is reduced.

-

-

zyx zyxwvut

2 . 2 . Generalized free energy and H-theorem

We define the energy by

E=-1 /

2x

For

E

+

[ ( V C ) k2c2] ~ dr -

(9)

= 0, this expression reduces to

E=

‘s

pcdr.

--

2

On the other hand, we define the temperature by

D T=X’ Therefore, the Einstein relation is preserved in the generalized thermodynamical framework. We also set p = 1/T. We introduce the generalized entropic functional

S=-

J’

C(p)dr,

(12)

where C ( p ) is a convex function (C” 2 0) defined by

s

This defines the entropy up to a term of the form A M + B where M = p d r is the mass (which is a conserved quantity). We can adapt the values of the

zyxwvut zyxwvu

270

zyx

constants A and B in order to obtain convenient expressions of the entropy. Finally, we introduce the generalized free energy

F=E-TS. (14) The definition of the free energy (Legendre transform) is preserved in the generalized thermodynamical framework. The free energy is the correct thermodynamical potential since the system is dissipative. Thus, it must be treated within the canonical e n ~ e m b l e . ~ > ~ ~ A straightforward calculation shows that

F

XE

For

zyxwvutsrq

‘J

= --

E

= 0,

( - A c + k2c - Xp)2dr -

- Xg(p)Vc)2dr.

zyxwvu

this equation reduces to

Therefore, F 5 0 (in all the paper, we assume that E , X , X , D , h~ ,are positive quantities). This forms an H theorem in the canonical e n ~ e m b l e ~ ~ ~ ~ for the nonlinear mean field Fokker-Planck equation (3). This also shows that the free energy F [ p ,c] is the Lyapunov functional of the generalized Keller-Segel model (3)-(4). It is sometimes useful to introduce the Massieu function

z

J=S-pE, (17) which is related to the free energy by J = -pF. Clearly, we have J 2 0. We can now consider particular cases: if D = 0 (leading to T = 0), we get F = E so that E 5 0. If x = 0 (leading to p = 0), we have J = S so that s 2 0. 2.3. Stationary solution

The steady state of Eq. (3) satisfies F leads to

AC - k2c = -Xp,

=

0. According to Eq. (15), this

Dh(P)VP

- XS(P)VC = 0.

(18)

Using Eqs. (11) and (13), the second equation can be rewritten

C”(p)Vp

- p v c = 0,

(19)

which can be integrated into C ’ ( p ) = pc - a,

(20)

zyxwvut zyxwv 271

where a is a constant of integration. Since C is convex, this equation can be reversed to give p ( r ) = F(-Dc(r)

+a),

(21)

where F ( z ) = (C’)-l(-z) is a monotonically decreasing function. Thus, in the steady state, the density is a monotonically increasing function p = p(c) of the concentration. We have the identity n

Substituting Eq. (21) in Eq. (5), valid for a stationary state, we obtain a mean-field equation of the form

+ k2c = XF(-Dc + a ) .

-Ac

(23)

The constant of integration a is determined by the total mass M (which is a conserved quantity). Finally, we note that the generalized entropy (12) is related to the distribution (21) by:

zyx zyxwv

zy

C(p)= -

/

P

F-l(z)dz.

(24)

Equation (21) determines the distribution p(r) from the entropy S and Eq. (24) determines the entropy from the density. 2.4. M i n i m u m of free energy

The critical points of free energy at fixed mass are determined by the variational problem

SF

+ TaSM = 0,

(25)

where a is a Lagrange multiplier. We can easily establish that

SE

‘s

= --

x

(Ac - k2c + Xp)Scdr -

65’= -

J’

(26)

C’(p)Spdr.

The variational problem (25) then leads to

zyxwvut A c - k2c = -Xp,

C ’ ( p ) = DC - a.

(28)

Comparing with Eq. (20), we find that a stationary solution of Eq. (3) is a critical point of F at fixed mass. On the other hand, we have established that

FIO,

F=O*&p=O.

(29)

272

zyxwvut z z zy zyx zy

According to Lyapunov’s direct m e t h ~ d this , ~ implies that p(r) is linearly dynamically stable with respect to the NFP equation (3)-(4) iff it is a (local) minimum of F a t fixed mass. Maxima or saddle points of F are dynamically unstable. In conclusion, a steady solution of the GKS model/NFP equation (3)-(4) is linearly dynamically stable iff it satisfies (at least locally) the minimization problem: min

I

{F[p,cl

P>C

M[Pl = MI.

(30)

In this sense, dynamical and generalized thermodynamical stability in the canonical ensemble coincide. Furthermore, if F is bounded from below a, we can conclude from Lyapunov’s theory that the system will converge to a stable steady state of the GKS model for t + +co. Finally, we note that the GKS model can be written ap =

at

v.

[

El

xg(p)V-

,

where b/Sp is the functional derivative. This shows that the diffusion current J = -Xg(p)V(bF/bp) is proportional to the gradient of a quantity SF/Sp that is uniform a t equilibrium ( ( ~ 5 F l b p=) ~-Ta ~ according to Eq. ( 2 5 ) ) . This corresponds to the linear thermodynamics of Onsager. The same result can also be obtained from a generalized Maximum Free Energy Dissipation (MFED) principle which is the variational formulation of Onsager’s linear thermodynamics. 2.5. Particular cases If we take h(p) = 1 and g ( p ) = l/C”(p), the NFP equation (3) becomes

In that case, we have a constant diffusion D ( p ) = D and a density dependent mobility x ( p ) = x/(pC”(p)).If we take g ( p ) = p and h(p) = pC”(p), the NFP equation (3) becomes

’_ a - V . (DpC”(p)Vp- x p V c ) . -

at

(33)

note that for the standard Keller-Segel model, or for the Smoluchowski-Poisson system, the free energy is not bounded from below. In that case, the system can either relax toward a local minimum of F at fixed mass (when it exists) or collapse t o a Dirac peak,24 leading to a divergence of the free energy F ( t ) + -co. The selection depends on a complicated basin of attraction. The same situation (basin of attraction) happens when there exists several minima of free energy at fixed mass.

zyxw zyx zyxwvut zy zyxw zyxwv 273

In that case, we have a constant mobility x ( p ) = x and a density dependent diffusion D ( p ) = D p [ C ( p ) / p ] ’ . Note that the condition D ( p ) 2 0 requires that [ C ( p ) / p ] ’ 2 0. This gives a constraint on the possible forms of C ( p ) . Finally, if we multiply the diffusion term and the drift term in the NFP equation ( 3 ) by the same positive function X(r,t ) (which can be for example a function of p(r, t ) ) ,we obtain a NFP equation having the same free energy (i.e. satisfying an H-theorem P 5 0) and the same equilibrium states as the original one. Therefore, for a given entropy C ( p ) , we can form an infinite class of NFP equations possessing the same general properties. 2.6. Generalized Smoluchowski equation

The NFP equation (33) can be written in the form of a generalized Smoluchowski (GS) equation

2 =v at

’

[ x ( V p-

(34)

with a barotropic equation of state p ( p ) given by

P’(P) = TPC”(P).

(35)

Since C is convex, we have p’(p) 2 0. Integrating Eq. (35) twice, we get

zyx

Therefore, the free energy (14) can be rewritten

F = _I 2x

/

[ ( V C+) ~ k2c2] d r - / p c d r

+ /p/”#dp’dr.

With these notations, the H-theorem becomes

‘s

+

(37)

J ip

( A c - k2c Xp)’dr - - ( V p - p V c ) 2 d r 5 0. (38) A€ The stationary solutions of the GS equation (34) satisfy the relation

F = --

Vp-pVc=Q,

(39)

which is similar to a condition of hydrostatic equilibrium. Since p = p ( p ) , this relation can be integrated to give p = p ( c ) through

This is equivalent to

zyxw zyx

274

This relation can also be obtained from Eqs. (35) and (22). Therefore, we recover the fact that, in the steady state, p = p ( c ) is a monotonically increasing function of c. We also note the identity

zyxw zyxwvuts

Finally, we note that the relation (40) can also be obtained by extremizing the free energy (37) at fixed mass writing 6F-adM = 0. More precisely, we have the important result: a steady solution of the generalized Smoluchowski equation (34)-(4) is linearly dynamically stable iff it is a (local) minimum of the free energy F [ p ,c] at fixed mass M [ p ]= M . This corresponds to the minimization problem (30). The generalized Smoluchowski equation (34) can also be obtained formally from the damped Euler equation^:^ dP

-

dt

zyx

+ v .( p u ) = 0 ,

dU

1

- + (u . V ) u = - - v p at P


- 0 ) for 52,

+aQ12.

4. Illustrations

(22)

The phase plane and the displacement response will be presented for illustration. The normal vector field is very important to determine the grazing and sliding bifurcations, and the normal vector €ields versus displacement will be presented, and the normal vector field time-history will be given to observe the switchability of the switching dynamical systems. The closed-form solutions in Appendix will be used for numerical computation of motions for such a switching dynamical system. Consider a periodic motion with mapping structure pZlozl for 52 = 2.0, as shown in Fig. 5. In Fig. 5(a), the trajectory in phase plane shows the switching of the motion on the boundary. The switching boundary is depicted by the dotted

294

zy zy zy zy zyxwvu

line. The switching points are labeled by the circular symbols, and the starting point is marked by a large filled circle. The mapping structure is pZlozl = Pzo 4 o p, o Pz04.The sliding motion along the switching boundary is shaded for further discussion. To show the switchability of motion on the discontinuous boundary, the normal vector field plays an important role. The normal vector field distribution along the displacement is illustrated in Fig. 5(b). The motion switchable on the separation boundary requires the normal vector fields in vicinity of switching point on the boundary to have the same sign, and such a switchable condition is given in Eq. (20). For the sliding motion on the boundary, the two normal vector fields in vicinity of switching points have opposite sign, as in Eq. (19). Once a motion flow arrives to the boundary, only if Eq. (19) is satisfied, the sliding motion along the switching boundary in phase plane can be observed. However, when one of the two sets of inequality in Eq. (21) is satisfied, the sliding motion will disappear and the flow will get into the corresponding domains. In Fig. 5(b), the solid curves give the real normal vector field and the dashed curves show the imaginary normal vector field, which can be referred to Luo [8,9]. After the motion is switched, the previous vector field is still used to control the flow as an imaginary flow. From the initial condition, both the normal vector fields in the two domains are positive ( G‘o”’ > 0 and G‘032’ > 0 ), the motion flow should be in domain 9,, which is labeled. Once the motion flow comes back to the switching boundary, both of the two normal vector fields for the switching point are negative (i.e., G‘oxl’< 0 and G‘032’ < 0 ), so it implies that the motion flow will be switched into domain Q Z . When the motion flow in domain 51, returns back to the switching boundary, both of the two normal vector fields possess the opposite sign, i.e., G‘O”’ < 0 and G‘O”’ > 0 . When G‘”” = 0 and G‘o*2’ > 0 , the sliding motion on the boundary disappear, and the motion flow enters into domain 51,. When the motion flow returns again to the switching boundary, the normal vector fields ( G‘O”’ < 0 and G‘”’ < 0 ) are observed. Thus, the motion flow switches into the domain QZ. Finally the motion flow returns to the initial point from domain 51, . The periodic motion ~ i t h , P 2 ~ 0is2formed. ~ The displacement and normal vector field time-histories are also presented in Fig. 5(c) and (d). The responses in each domain are labeled by the mapping and the normal vector fields in each domain are labeled by G‘os”and G(0.2’. The sliding responses are shaded. The time-history of the normal vector fields also shows the analytical conditions for the switchability of the motion flow on the switching boundary, which will not be discussed. The response between the two dashed lines is for a period.

zyxwvut

295

15

zyxwv zyxw zyx zyxw zyxw zyx

-c

200

v

%

0

2 n

A

-g 5

o

y.

::

?8

-15

-200

2 -30

-2.0

0.0

(4

2.0

-400

4.0

Displacement x

0.0

-2.0

2.0

4.0

Displacement x

(b)

400

-

3 -c -

200

m

y.

5

f 2 00

20

0

-200

zyxwvuts zyx zyxwvuts

40

60

80

-400 00

20

40

60

80

Fig. 5. Periodic motion of P,,,,, : (a) phase plane, (b) normal vector field versus displacement, (c) displacement and (d) normal vector field time-history. Initial condition is to = 0.6478 , x,, = -1.4848 and yo =-1.1517. ( r , = O S , k, =50, r, =1.0, k, =150, gl=gz= 1, =150, sz=2.0, @ = O , a = b = l , c = - 3 ) .

zyxw

Consider another periodic motion relative to mapping Pz121with 2excitation periods. The same parameters are used again except for SZ = 17.5. Based such parameters, the initial condition (i.e., to = 0.0282, xo = -1.0243 and yo = -1.9098) are used in Fig.6. The trajectory in phase plane and normal vector field distribution along displacement are illustrated in Fig.6(a) and (b). For this periodic motion in Fig.6(a), it is observed that no sliding motion on the switching boundary exists. Therefore, both the normal vector fields in two domains for all switching points on the switching boundary are with the same sign. In other words, the motion flow switching from SZ, to SZ2 or from SZ2 to requires the normal vector fields are negative or positive, which are

zyx

296

observed in Fig.6(b). In a similar fashion, the displacement and normal vector field time-histories are illustrated in Fig.6 (c) and (d). For each two-excitation periods, the complete periodic cycle is observed.

-480 -1.6

-30 -1.6

0.0

-0.8

08

1.6

24

Displacement x

(a>

3P

'

I

0.0

zyxw

08

I

1.6

2.4

Displacement x

(b) s t

- 2 0 1I " ' " ' '

,

zyxw zyxwvutsr zyxwvut I

-0.8

200

O

2 2

~200

Bf

-400

zyxwvutsrqp zyxwvutsr ' I' '

'

zyx zyxwvutsr "

on

'

05

'

"

' '

'

10

"

15

00

05

10

15

q2,),

Fig. 6 . Periodic motion of : (a) phase plane, (b) normal vector field versus displacement, (c) displacement and (d) normal vector field time-hstory. Initial condition is to = 0.0282 ,xo = -1.0243 and yo=-1.9098. ( r ; = O . 5 , k , = 5 0 , r,=I.O, k2=150, g , = l , gz=I, Qo=lSO, Q=17.5, @ = O , a = b = l , ~ = - 3 ) .

5. Conclusions

The switchability of a flow from one domain into another one in the periodically driven, discontinuous dynamical system is investigated through the boundary. The normal components of the vector fields for a flow switching on the separation boundary is introduced. The switchability conditions of a motion flow on the discontinuous boundary are developed, and the sliding and grazing conditions to the separation boundary are presented as well. The normal vector fields are illustrated to demonstrate the analytical criteria. This investigation will help one better understand the sliding mode control.

Appendix

zy

zy 297

With the initial condition(xi,ii,ti), solution for Eq. (3) in two regions 51,

298

zy zyxw zy zyxw zyxw

zyxwvutsrq ulp'

=

JX

c,Ca) ( x i ,ii,t i )= xi - A(")cos Qti - B(")sin Qti - da), 1 Cp) (xi,X i , t i )= -[Xi - (d,A(") + B(")Q)cos Qt, Ulp'

-( d for Case 111(i.e.,

a B(")

- A(")Q)sin Qti + d, [ x i - C ( a ) ) ] .

4 = ca >,

References

1. Filippov, A.F., 1964, "Differential equations with discontinuous right-hand side", American Mathematical Society Translations, Series 2 , 42, pp. 199231. 2. Filippov, A.F., 1988, Differential Equations with Discontinuous Righthand Sides, Dordrecht: Kluwer Academic Publishers. 3. Aizerman, M. A., Pyatnitskii, E.S., 1974, "Foundation of a theory of discontinuous systems. 1," Automatic and Remote Control, 35, pp. 10661079. 4. Aizerman, M. A., Pyatnitskii, E.S., 1974, "Foundation of a theory of Discontinuous Systems. 2," Automatic and Remote Control, 35, pp. 12411262. 5. Utkin, V. I., 1976 ,"Variable structure systems with sliding modes," ZEEE Transactions on Automatic Control, AC-22.pp. 212-222. 6. DeCarlo, R.A., Zak, S.H., Matthews, G.P., 1988, "Variable Structure

zyxw

zyx zyxwvut 299

7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17.

control of nonlinear multivariable systems: A tutorial,” Proceedings of the IEEE, 76, pp. 212-232. Renzi, E., Angelis, M. D., 2005, “Optimal semi-active control and nonlinear dynamics response of variable stiffness structures,” Journal of Vibration and Control, 11(10), pp.1253-1289. Luo, A.C.J., 2005, “A theory for non-smooth dynamical systems on connectable domains,” Communication in Nonlinear Science and Numerical Simulation, 10, pp.1-55. Luo, A.C.J., 2005, “Imaginary, sink and source flows in the vicinity of the separatrix of non-smooth dynamic system,” Journal of Sound and Vibration, 285, pp.443-456. Luo, A.C.J., 2006, Singularity and Dynamics on Discontinuous Vector Fields, Elsevier: Amsterdam. Menon, S . and Luo, A.C.J. 2005, “A global period-1 motion of a periodically forced, piecewise linear system”, International Journal of Bifurcation and Chaos, 15, pp. 1945-1957. Luo, A.C.J., 2005, “The mapping dynamics of periodic motions for a threepiecewise linear system under a periodic excitation”, Journal of Sound and Vibration, 283,723-748. Luo, A.C.J. and Chen, L.D., 2005, “Periodic motion and grazing in a harmonically forced, piecewise, linear oscillator with impacts”, Chaos, Solitons and Fractals, 24, pp. 567-578. Luo, A.C.J. and Gegg, B.C. 2005 “On the mechanism of stick and non-stick periodic motion in a forced oscillator with dry-friction,’’ ASME Journal of Vibration and Acoustics, 128, pp.97- 105. Luo, A.C.J. and Gegg, B.C., 2006, “Stick and non-stick periodic motions in a periodically forced oscillator with dry-friction,’’ Journal of Sound and Vibration, 291, pp.132-168. Luo, A.C.J. and Gegg, B.C., 2006, “Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction,” ASME Journal of Computational and Nonlinear Dynamics, 1, pp.212-220. Luo, A.C.J. and Gegg, B.C., 2006, “Dynamics of a periodically forced oscillator with dry friction on a sinusoidally time-varying traveling surface,” International Journal of Bifurcation and Chaos, 16, pp.35393566.

300

zyxwvutsr THE FORMATION OF SPIRAL ARMS AND RINGS IN BARRED GALAXIES

zyx zyxw

M. ROMERO-GOMEZ' and E. ATHANASSOULA Laboratoire d'Astrophysique de Marseille, Obseruatoire Astronomique de Marseille Provence, 2 Place Le Verrier 13248 Marseille, France *E-mail: merce.romerogometQoamp.f r

J.J. MASDEMONT

I. E.E. C €d Dep. Mat. Aplicada I, Universitat Politkcnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain

c. GARC~A-GOMEZ D. E.I. M., Universitat Rovira i Virgili, Av. Paasos Catalans 26, 43007 Tarragona, Spain We propose a new theory to explain the formation of spiral arms and of all types of outer rings in barred galaxies. We have extended and applied the technique used in celestial mechanics to compute transfer orbits. Thus, our theory is based on the chaotic orbital motion driven by the invariant manifolds associated to the periodic orbits around the hyperbolic equilibrium points. In particular, spiral arms and outer rings are related t o the presence of heteroclinic or homoclinic orbits. Thus, R1 rings are associated to the presence of heteroclinic orbits, while RlRz rings are associated to the presence of homoclinic orbits. Spiral arms and Rz rings, however, appear when there exist neither heteroclinic nor homoclinic orbits. We examine the parameter space of three realistic, yet simple, barred galaxy models and discuss the formation of the different morphologies according to the properties of the galaxy model. The different morphologies arise from differences in the dynamical parameters of the galaxy. Keywords: galactic dynamics - invariant manifolds - spiral structure - ring structure

1. Introduction Bars are very common features in disk galaxies. According to Eskridge et al. [l]in the near infrared 56% of the galaxies are strongly barred and

zyxwv 301

6% are weakly barred. A large fraction of barred galaxies show either spiral arms emanating from the ends of the bar or spirals that end up forming outer rings (Elmegreen & Elmegreen [2]; Sandage & Bedke [3]). Spiral arms are believed to be density waves (Lindblad [4]). Toomre [5], finds that the spiral arms are density waves that propagate outwards towards the principal Lindblad resonances, where they damp. So other mechanisms for replenishment are needed (see for example Lindblad [6]; Toomre [5,9];Toomre & Toomre [7]; Sanders & Huntley [8]; Athanassoula [lo] for more details). Rings have been studied by Schwarz [ll-131. The author studies the response of a gaseous disk galaxy to a bar-like perturbation. He relates the rings with the position of the principal Lindblad resonances. There are different types of outer rings and they can be classified according to the relative orientation of the principal axes of the inner and outer rings (Buta [14]). If the two axes are perpendicular, the outer ring has an eight-shape and it is called R1 ring. If they are parallel, it is called R2 ring. There are galaxies where both types of rings are present, in which case the outer ring is simply called R1 R2 ring. Our approach is from the dynamical systems point of view. We first note that both spiral arms and (inner and outer) rings emanate from, or are linked to, the ends of the bar, where the unstable equilibrium points of a rotating system are located. We also note that, so far, no common theory for the formation of both features has been presented. We therefore study in detail the neighbourhood of the unstable points and we find that spiral arms and rings are flux tubes driven by the invariant manifolds associated to the periodic orbits around the unstable equilibrium points. This paper is organised as follows. In Sec. 2, we give the characteristics of each component of the model and the potential used to describe it. In Sec. 3, we give the equations of motion and we study the neighbourhood of the equilibrium points. In particular, we give definitions of the Lyapunov periodic orbits, the invariant manifolds associated to them, and of the homoclinic and heteroclinic orbits. In Sec. 4, we present our results and in Sec. 5, we briefly summarise.

zyxwvu

2. Description of the model

We use a model introduced in Athanassoula [15] that consists of the superposition of an axisymmetric and a bar-like component. The axisymmetric component is the superposition of a disc and a spheroid. The disc is modelled as a Kuzmin-Toomre disc (Kuzmin [16]; Toomre [17]) of surface

302

zyxwvu zyxw zyxwv zy

density C ( T ) (see also left panel of Fig. 1):

-3/2

C ( r ) = =2TTd (l+;)

,

where the parameters v d and rd set the scales of the velocities and radii of the disc, respectively. The spheroid is modelled using a spherical density distribution, p ( r ) (Eq. 2), characteristic for spheroids. In the middle panel of Fig.1, we plot the isodensity curves for this density function: p(r) = P b

(

1f

3-3/2,

(2)

where Pb and rb determine the central density and scale-length of the spheroid. Bars are non-axisymmetric features with high ellipticities. We will use three different bar models. In the first one the bar potential is described by a Ferrers ellipsoid (Ferrers [ I S ] )whose density distribution is:

zyxwv zyxwvu +

where m2 = x 2 / a 2 y2/b2. The values of a and b determine the shape of the bar, a being the length of the semi-major axis, which is placed along the x coordinate axis, and b being the length of the semi-minor axis. The parameter n measures the degree of concentration of the bar and po represents the bar central density. In the right panel of Fig. 1, we plot the density function along the semi-major and semi-minor axes of the Ferrers ellipsoid with index n = 2 , and principal axes a = 6 and b = 1.5. We also use two ad-hoc potentials, namely a Dehnen’s bar type (Dehnen [19]) and a Barbanis-Woltjer (BW) bar type (Barbanis & Woltjer [ 2 0 ] )to compare to the results obtained with the Ferrers ellipsoid. The Dehnen’s bar potential has the following expression:

where the parameter (Y is a characteristic length scale and wo is a characteristic circular velocity. The parameter E is related to the bar strength. The BW potential has the expression: %(T,

e) = Z f i ( r 1 - T ) cOs(2e),

(5)

zyxwvu 303

where the parameter the bar strength.

r1

is a characteristic scale length and 2 is related to

zyxwvu zyxw z

Fig. 1. Characteristics of the components. L e f t panel: Density function of the KuzminToomre disc (red solid line) with r d = 0.75 and Vd = 1.5.M i d d l e panel: Isodensity curves for the spherical distribution representing the spheroid with parameters r b = 0.3326and pb = 23552.37. R i g h t panel: Density along the semi-major axis (black solid line) and the semi-minor axis (red dashed line) of a Ferrers bar with n = 2, a = 6, b = 1.5 and po = 0.0193.

a,

zyxwvu zyxw

The bar-like component rotates anti-clockwise with angular velocity = OPz,where 0, is a constant pattern speed a .

3. Equations of motion and dynamics around

L1

and Lz

The equations of motion in a frame rotating with angular speed 0, in vector form are

i: = -V!D-2(aP x i ' ) -a, x

(a, x r),

(6)

where the terms -2aPx i- and -Clp x (52, x r) represent the Coriolis and the centrifugal forces, respectively, CP is the potential and r is the position vector. We define an effective potential C P e ~= CP - :Og (x2 y2), then Eq. (6) becomes i: = -V!D,,ff - 2(OP x i'), and the Jacobi constant is

+

which, being constant in time, can be considered as the energy in the rotating frame. The surface C P e ~= EJ ( E J defined as in Eq. (7)) is called the zero velocity surface, and its intersection with the z = 0 plane gives the zero velocity curve. All regions in which @,,a > EJ are forbidden to a star with this energy, and are thus called forbidden regions. For our calculations we place ourselves in a frame of reference corotating with the bar, and the bar semi-major axis is located along the 2 axis. In this aBold letters denote vector notation. The vector z is a unit vector.

304

z zyxwvut zyxwv

rotating frame we have five equilibrium points, which, due to the similarity with the Restricted Three Body Problem, are also called Lagrangian points (see left panel of Fig. 2). The points located symmetrically along the 5 axis, namely L1 and Lz, are linearly unstable. The ones located on the origin of coordinates, namely L B ,and along the y axis, namely L4 and Lg, are linearly stable. The zero velocity curve defines two different regions, namely, an exterior region and an interior one that contains the bar. The interior and exterior regions are connected via the equilibrium points (see middle panel of Fig. 2). Around the equilibrium points there exist families of periodic orbits, e.g. around the central equilibrium point the well-known 5 1 family of periodic orbits that is responsible for the bar structure. The dynamics around the unstable equilibrium points is described in detail in Romero-G6mez et al. [21]; here we give only a brief summary. Around each unstable equilibrium point there exists a family of periodic orbits, known as the family of Lyapunov orbits (Lyapunov [22]). For a given energy level, two stable and two unstable sets of asymptotic orbits emanate from the corresponding periodic orbit, and they are known as the stable and the unstable invariant manifolds, respectively. The stable invariant manifold is the set of orbits that tends to the periodic orbit asymptotically. In the same way, the unstable invariant manifold is the set of orbits that departs asymptotically from the periodic orbit (i.e. orbits that tend to the Lyapunov orbits when the time tends to minus infinity), as seen in the right panel of Fig. 2. Since the invariant manifolds extend well beyond the neighbourhood of the equilibrium points, they can be responsible for global structures. In Romero-G6mez e t al. [23], we give a detailed description of the role invariant manifolds play in global structures and, in particular, in the transfer of matter. Simply speaking, the transfer of matter is characterised by the presence of homoclinic, heteroclinic, and transit orbits. Homoclinic orbits correspond to asymptotic trajectories that depart from the unstable Lyapunov periodic orbit y around Li and return asymptotically to it (see Fig. 3a). Heteroclinic orbits are asymptotic trajectories that depart from the periodic orbit y around Li and asymptotically approach the corresponding Lyapunov periodic orbit with the same energy around the Lagrangian point a t the opposite end of the bar L j , i # j (see Fig. 3b). There also exist trajectories that spiral out from the region of the unstable periodic orbit, and we refer to them as transit orbits (see Fig. 3c). These three types of orbits are chaotic orbits since they fill part of the chaotic sea when we plot the Poincar6 surface of section (e.g. the section (x,k)near L1).

z

z zyxwvutsrqpo 305

zyxwvutsrq LD

12

< z > L1

zyxwvutsrqp A 0

zyxwv zyxw zyxwvu

ro I

L5

-5

0

5

Fig. 2. Dynamics around the L1 and Lz equilibrium points. Left panel: Position of the equilibrium points and outline of the bar. Middle panel: Zero velocity curves and Lyapunov periodic orbits around L1 and L z . Right panel: Unstable (in red) and stable (in green) invariant manifolds associated to the periodic orbit around L1. In grey, we plot the forbidden region. From Romero-Gbmez et al. 2006, Astronomy & Astrophysics, 453, 39, EDP Sciences.

-10

zyxwvutsrqponm zyx -5

5

0

X

10

10

-5

0

5

X

10

LO

5

0

5

LO

X

Fig. 3. Homoclinic (a),heteroclinic (b) and transit ( c ) orbits (black thick lines) in the configuration space. In red lines, we plot the unstable invariant manifolds associated to the periodic orbits, while in green we plot the corresponding stable invariant manifolds. In dashed lines, we give the outline of the bar and, in (b) and ( c ) , we plot the zero velocity curves in dot-dashed lines. From Rornero-Gcjmez et al. 2007, Astronomy and Astrophysics, 472, 63, EDP Sciences.

4. Results

Here we describe the main results obtained when we vary the parameters of the models introduced in Sec. 2. One of our goals is to check separately the influence of each of the main free parameters. In order to do so, we make families of models in which only one of the free parameters is varied, while the others are kept fixed. Our results show that only the bar pattern speed and the bar strength have an influence on the shape of the invari-

306

zyxwvutsr z

ant manifolds, and thus, on the morphology of the galaxy (Romero-G6mez et al. [23]). Our results also show that the morphologies obtained do not depend on the type of bar potential we use, but on the presence of homoclinic or heteroclinic orbits. If heteroclinic orbits exist, then the ring of the galaxy is classified as rR1 (see Fig. 4a). The inner branches of the invariant manifolds associated to y1 and 7 2 outline an inner ring that encircles the bar and is elongated along it. The outer branches of the same invariant manifolds form an outer ring whose principal axis is perpendicular to the bar major axis. If

zyxwv zyxwv

0

zyxwvutsrqponmlkj zyxwvut zy 0 -1

i

3

in

in-

h 0

' '

' '

I

'

1-

zyxwvu "

I

"

"

I

"

~

"

h 0 -

m

in

I

I

0

0

-

3

3

-

I

1

c

-7 -80

-

rR1RZ ring s t r u c t u r e

-10

0

10

-

' ' ' " 1 , ' -

d

80

-7 -80

-

Barred spiral s t r u c t u r e

zyxw

-10

0

10

a0

Fig. 4. Rings and spiral arms structures. We plot the invariant manifolds for different models. (a) rR1 ring structure. (b) rR2 ring structure. ( c ) RlRz ring structure. (d) Barred spiral galaxy. From Romero-G6mez et al. 2007, Astronomy and Astrophysics, 472, 63, EDP Sciences.

307

the model does not have either heteroclinic or homoclinic orbits and only transit orbits are present, the barred galaxy will present two spiral arms emanating from the ends of the bar. The outer branches of the unstable invariant manifolds will spiral out from the ends of the bar and they will not return to its vicinity (see Fig. 4d). If the outer branches of the unstable invariant manifolds intersect in configuration space with each other, then they form the characteristic shape of R2 rings (see Fig. 4b). That is, the trajectories outline an outer ring whose principal axis is parallel to the bar major axis. The last possibility is if only homoclinic orbits exist. In this case, the inner branches of the invariant manifolds for an inner ring, while the outer branches outline both types of outer rings, thus the barred galaxy presents an R1 R2 ring morphology (see Fig. 4c).

zyxw

5. Summary

To summarise, our results show that invariant manifolds describe well the loci of the different types of rings and spiral arms. They are formed by a bundle of trajectories linked to the unstable regions around the L1/L2 equilibrium points. The study of the influence of one model parameter on the shape of the invariant manifolds in the outer parts of the galaxy reveals that only the pattern speed and the bar strength affect the galaxy morphology. The study also shows that all the different ring types and spirals can be obtained when we vary the model parameters. We have compared our results with some observational data. Regarding the photometry, the density profiles across radial cuts in rings and spiral arms agree with the ones obtained from observations. The velocities along the ring also show that these are only a small perturbation of the circular velocity. Acknowledgements

MRG acknowledges a “Becario MAE-AECI” . References

zyxwvu zyx

1. P.B. Eskridge, J.A. F’rogel, R.W. Podge, A.C. Quillen, R.L. Davies, D.L. DePoy, M.L. Houdashelt, L.E. Kuchinski, S.V. Ramirez, K. Sellgren, D.M. Terndrup, G.P. Tiede, AJ, 119, 536 (2000). 2. D.M. Elmegreen, B.G. Elmegreen, MNRAS, 201,1021 (1982). 3. A. Sandage, J. Bedke, “The Carnegie Atlas of Galaxies”, Carnegie Inst, Washington (1994).

308

4. 5. 6. 7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

zyxwvut zyxw

B. Lindblad, Stockholms Observatorium Ann., Vol. 22,No. 5 (1963). A. Toomre, ApJ, 158,899 (1969). P.O. Lindblad, Stockholms Observatorium Ann., Vol. 21,No. 4 (1960). A. Toomre, J. Toomre, ApJ, 178,623 (1972). R.H. Sanders, J.M. Huntley, ApJ, 209,53 (1976). A. Toomre, “The structure and evolution of normal galaxies”, eds. S.M. Fall and D. Lynden-Ball, Proc. of the Advanced Study Institute, Cambridge, pp. 111-136 (1981). E. Athanassoula, Phys. Rep., 114,319 (1984). M.P. Schwarz, ApJ, 247,77 (1981). M.P. Schwarz, MNRAS, 209,93 (1984). M.P. Schwarz, MNRAS, 212,677 (1985). R. Buta, ApJS, 96,39 (1995). E. Athanassoula, MNRAS, 259,328 (1992). G. Kuzmin, Astron. Zh., 33,27 (1956). A. Toomre, ApJS, 138,385 (1963). N.M. Ferrers, Q.J. Pure Appl. Math., 14,1 (1877). W. Dehnen, AJ, 119,800 (2000). B. Barbanis, L. Woltjer, ApJ, 150,461 (1967). M. Romero-Gbmez, J.J. Masdemont, E. Athanassoula, C. Garcia-Gbmez, A & A , 453,39 (2006). A. Lyapunov, Ann. Math. Studies, 17 (1949). M. Romero-G6mez, E. Athanassoula, J.J. Masdemont, C. Garcia-G6mez, A & A , 472,63 (2007).

309

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LEVY WALKS FOR ENERGETIC ELECTRONS DETECTED BY THE ULYSSES SPACECRAFT AT 5 AU S. PERRI' and G. ZIMBARDO Department of Physics, University of Calabria, Rende, I-87'036, Italy *E-mail: [email protected]

zyx zyxwv

We consider the propagation of energetic particles, accelerated by interplan-

etary shock waves, upstream of the shock. In connection with superdiffusive transport and L&y random walks, we consider the relevant non-gaussian p r o p agator, which has power law tails with slope 2 < p < 3. We show that in the case of superdiffusive transport, the time profile of particles accelerated at a traveling planar shock is a power law with slope y = p - 2 , rather than an exponential decay as expected in the case of normal diffusion. We analyze a dataset of interplanetary shocks in the solar wind detected by the Ulysses spacecraft between July 1992 and November 1993.We find that the time profiles of energetic electrons correspond to power laws, with slopes y N 0.60-0.98, implying superdiffusive transport. On the other hand, the propagation of protons seems to be diffusive even when a non Gaussian statistics is involved.

zyxw zyx

Keywords: anomalous transport; energetic particles; interplanetary shocks.

1. Introduction One of the most far reaching consequences of chaos is that anomalous, non diffusive transport regimes are possible for systems which exhibit non Gaussian statistics and long range correlations. This is the case of particle motion in periodic or quasiperiodic potentials, where chaotic trajectories may alternate long flights, or jumps, to dynamic trapping,lI2 as well as of turbulent fluids, where long range correlations may be found in the velocity of particle^.^ Very similar phenomena are found in plasma physics, and here we show how anomalous transport is useful in describing the propagation of energetic particles in space plasmas. Energetic particles observed in the heliosphere are accelerated at solar impulsive events, as solar flares and coronal mass ejections (CMEs), and at interplanetary shocks waves, as those associated to corotating in-

310

zyx

teraction regions (CIRs), which are compressive zones in the solar wind environment formed when the fast wind (KwN 800 km/s) encounters the slow one (Vsw N 400 km/s); this gives rise to a pair of collisionless shocks, one moving forward and the other one in the reverse d i r e ~ t i o n Ener.~~~ getic particles propagate in the interplanetary space interacting with magnetic turbulence, which causes pitch angle ~ c a t t e r i n g The . ~ ~ ~propagation and acceleration properties of ions seem to be well described by the diffusive shock acceleration (DSA) mechanism,'-1° which takes into account the scattering of particles by magnetic irregularities, located upstream and downstream of the shock this mechanism causes a particle Brownian-like motion during the acceleration process. However, there are experimental evidences of different transport r e g i m e ~ for ,~~ instance ~~ the transport of solar energetic particles (SEPs) varies from diffusive to scatter-free. 1 2 3 1 3 In addition, anomalous transport regimes, characterized by a mean square displacement which grows as (Ax2(t)) 0: t a , both slower ( a < 1) and faster ( a > 1) with respect to normal diffusion ( a = l),have been observed in various system^.'>'^)^^ The superdiffusive regime, i.e., a > 1, can be characterized by L6vy random walks whose jump lengths distribution exhibits At this point an important question is whether, under power law appropriate conditions, anomalous transport of energetic particles can be observed in the heliosphere. For this aim, it is possible to look at the particle spatial distributions, which can be sampled by a spacecraft moving in the solar wind. If normal diffusion works, we expect an exponential decay for the time profiles of energetic particles upstream of an interplanetary shock.1°>17However spacecraft data show that there is a large variety of different behaviours.ls In this work we derive the expressions of particle fluxes by using the propagator formalism both under the hypothesis of normal diffusion and in the case of superdiffusive transport; finally we show that many fluxes of electrons accelerated a t CIR shocks exhibit power law tail corresponding t o a superdiffusive transport, while proton time profiles are often well fitted by an exponential decay.

zyxw zyxwv

2. Deriving energetic particle profiles by the propagator formalism

A spacecraft in the solar wind measures particles accelerated at various times and positions, and the actual time profile reflects the propagation properties from the source to the observer, as well as the source evolution. We consider the propagation of particles, accelerated by interplanetary

zyxwv zyxwvu 311

shock waves, away from the shock. For the present analysis, we consider a large scale planar shock: this assumption is reasonably well satisfied by interplanetary shocks (e.g. Ref. 17), which may be due either to fast CMEs or to CIRs, thanks to their large size (compared to the relevant transport scales). Accordingly, we consider a steady state, one dimensional shock model. We suppose that the energetic particle fluxes measured by a spacecraft at (x,t ) are the superposition of the energetic particles accelerated at the shock moving according to x’ = Kht‘ (in other words, at t’ = 0 the shock will be at the origin of the coordinate system), with Kh assumed to be constant. To fix the ideas, we consider that the observer is at x = 0, upstream of the shock, which is coming from x = -m; then, t < 0 for the relevant time interval. The particle omnidirectional distribution function f(x,E , t ) at the observer will be expressed in terms of the distribution function f & ( dE, , t’) of particles accelerated at the shock as f(X, E , t ) =

S

P ( Z - Z’,

t

- t’)fsh(d, E , t’)dddt’

(1)

zy

with f s h ( d , E , t ’ ) = fo(E)6(x’ - Kht’), where fo(E) represents the distribution function of particles of energy E emitted by the shock. Here, P ( x - x’,t - t’) is the probability of finding a particle at position x at time t , if it was injected at x’ and t’. Also, x - x‘ is the distance upstream of the shock (the source of energetic particles), t - t’ > 0 , and D is the Gaussian diffusion coefficient. This form, P ( x - x’,t - t’),emphasizes the space-time translational invariance of the propagator. In the case of normal diffusion, particles accelerated at the shock are spread in space according to the Gaussian propagator (e.g., Ref. 14,19)

P ( x - x’,t - t’) =

1

(x - x’)2

& G q c - q e x p [-4D(t - t’,1 .

(2)

Using this expression of the propagator we obtain

(3) Exploiting the 6 function, considering that the observer is at the origin of the coordinate system x = 0 and by introducing the variable T = t - t‘ we obtain

zyxwv zyxw

312

It is easy to show that the integral I ( t ) is finite, since the integrand goes to zero for r 0, and decays exponentially for r -+ 00. Application of the Laplace transform given by Eq. (2.12) of Ref. 19,

taking into account that t are left with

zyx

< 0, gives I ( t ) = Vs;’ exp(-Kiltl/2D).

Then we

zy

which coincides with the exponential decay obtained by Ref. 10,17 for the energetic particle distribution function upstream of the shock, starting from a diffusion equation including the convective term I/sha f /ax. Conversely, in the case of superdiffusive propagation, transport can be described in the framework of continuous time random walks. For Lkvy random walks, a jump probability $(r, t ) = Alrl-Pb(t - T / U ) of making a jump of length r in a time t can be adopted.l4?l6The power law behaviour of $ ( r , t ) reflects the fact that very long jumps of length T have small but non-negligible probability, contrary to the case of Gaussian random walk. For p < 3, the probability $ ( r , t ) has diverging second order moment, which corresponds to an infinite value of the mean free path; however, this does not imply an infinitely fast transport because long jumps require long times, as implied by the space-time coupling expressed by b(t - T / u ) . In general, the propagator can be obtained in the Fourier-Laplace space and its explicit inversion is only possible in limiting cases: close to the source, i.e., for Ix - 2’1 > kh’2(t-t’)’/(,-1) the propagator has a power law behaviour described

x’I

by

P ( x - X I , t - t’) = b

t - t’ (x - X ’ ) P

where b and p are constants with respect to position and time, but they may depend on particle velocity and on the relevant transport process;16 the propagator in Eq. (8) goes to zero for x - x’ > v(t - t’), with u the

zyxw zyx zy 313

particle velocity. For 2 < p < 3 superdiffusion with ( x 2 ( t ) )= 2 D 3 is obtained for large t , with a = 4 - p,16,20while for 3 < p < 4 transport is diffusive, but the propagator has non Gaussian, power law tails as above. l6 We assume to be a t large distance from the shock, that is in the tails of the probability distribution, and we determine the energetic particle profile by using the expression in Eq. (1) and the non Gaussian propagator in Eq. (8). A straightforward calculation, reported in Ref. 21,22, yields

zyxw

where we have assumed that the shock starts at t o = --oo and that the observer is at x = 0; then far off the shock the time profile of the accelerated particles is a power law decay with slope y = p-2. Accordingly, an energetic particle profile with 0 < y < 1 implies superdifisive transport with a = 4 - p = 2 - y,while 1 < y < 2 implies a non Gaussian propagator like that in Eq. (8) and a long-tailed distribution for jump lengths, but a diffusive transport with a mean square deviation growing linearly in time. We leave the derivation of the power-law particle distribution function by means of a transport equation including fractional derivatives, which describe anomalous diffusion, for future work.

3. Data analysis We analyze a dataset consisting of repeated shock crossings observed by the Ulysses spacecraft in 1992-1993, a period of low influence of the solar impulsive events due t o the decline in solar activity. From July 1992, Ulysses detected a long series of forward-reverse shock pairs associated with CIRs23124and its heliocentric distance was more than 5 AU; this implyes that the shock can be considered planar with a good approximation. We study fluxes of electrons and protons accelerated both a t the forward and at the reverse shock of the CIRS obtained from the CDAWeb service of the National Space Science Data Center (cdaweb.gsfc.nasa.gov). The data analysis is performed by considering particle time profiles at some distance from the shock front because close to it the propagator can be Gaussian-like even for the case of LBvy walks, (see Eq. (7)). In our analysis we study various shock events, during which Ulysses was a t a heliocentric distance N 4.5-5.0 AU and at a latitude E 25"-30" S. As an example, we show in Figure 1 the event of January 22, 1993. From top to bottom, panels show one hour averages for the plasma radial velocity and the plasma temperature from SWOOPS (PI D. McComas), the

zyxwv

314

zyxwvuts

120ow 14 Jon 1993

1s:ww 18 Jan 1993

zowoo 22 Jan 1993

zyxwv

0ow:w 26 Jan 1993

MWW

30 Jan 1993

Fig. 1. CIR shock event of January 22, 1993, as observed by Ulysses spacecraft. Fkom top to bottom: solar wind radial velocity, solar wind temperature, proton fluxes and electron fluxes. The lower two panels are in semi-log scale. Energies as indicated (from Ref. 21).

semi-log plots of proton fluxes measured by LEFS 60 of HI-SCALE (PI L. Lanzerotti) and those of electrons measured by LEFS 60. We consider the particle time profiles upstream of the reverse shock. In this event, and also in all considered cases, particle fluxes vary slightly more than one order of magnitude over about 200 hours. We notice that several flux fluctuations, with time scale of 20-30 hours, can be distinguished in the time profiles, as well as in the following events. These irregularities are due to the low frequency magnetic t ~ r b u l e n c e which , ~ ~ has a correlation length X 2 35 x lo6 km a t 1 AU (3-4 hours in time scales).26 In addition, turbulence affects the magnetic connection between the spacecraft and shock, causing temporal changes in the energetic particle profiles with the corresponding time scales.27 In Figure 2 we show a comparison between the electron fluxes and the

315 100:

-'-.'-..:

'

'

""'

'

'

' '

""'

'

zyxwvutsrq zyxwvu ........548-781 keV

m

0.11 1

........

. . . . . . . . 10 100 A t (hours) I

I

.

1

1

10 A t (hours)

100

Fig. 2. Comparison between electron fluxes (left panel) and proton fluxes (right panel) of the event indicated in Figure 1. In order to make clear power law decays plots have been displayed in log-log scale. Energies as indicated (from Ref. 21,22).

zyxwv zyx zyxwv zyxw

proton ones upstream of the reverse shock of January 22, 1993; solid lines indicate the best fits. These plots are in log-log scale to better appreciate the power law decay; the time difference At has been calculated as It - tshl, where t is the upstream observation time and t& is the shock crossing time. Since the energetic particle flux J is related to f(x,E , t), we assume a power law decay J = A(At)-y. Calculating the reduced chi-square values for the fits, we obtain for both electron and proton time profiles that power law fits the data better than an exponential decay, J = Kexp(-At/.r), expected for a Gaussian transport (see Figure 2). Note that the power law decay is obtained over more than one decade in electron flux, and over almost 200 hours in time, so that the variations due to the turbulence do not affect the fit. In this event for electrons fluxes we find values of y = 0.81-0.98, implying p = y 2 = 2.81-2.98, i.e., superdiffusion with (Ax2) N t4-p = t1.02-t1.1g,while for protons y = 2.0-2.33 and p = 4.0-4.3. As we argued in Section 2, values of p greater than 3 lead to a diffusive behaviour, even if, as in this case, the statistics is non Gaussian. Other two events, i.e., May 10, 1993 and January 10, 1992, exhibit power law decays in electron time profiles with y E 0.60-0.85, which leads to a mean square displacement (Ax') 2 t1.2-t1.4.Looking a t the event of September 12,1992, associated to a CIR forward shock, in Ref. 22 it has been highlighted that the proton time profile, far from the shock front, displays an exponential decay, typical of a diffusive transport. Finally, we stress that in this kind of analysis we have neglected some effects as solar wind convection and adiabatic deceleration.10>28

+

316

zyxwvuts

4. Conclusions

zyxwv zyxwv zyxwv

In this work, by analyzing electron Ulysses data at 5 AU, we have highlighted that the energetic electron fluxes are well fitted by a power law decay with a slope y E 0.60-0.98 over a period of 100-200 hours, i.e., superdiffusion, while proton fluxes exhibit either an exponential decay and a power law profile but, in the latter case, the exponent y is greater than 1, so the transport can be considered normal. It is interesting to notice that superdiffusive regimes have been observed both in fluids3 and in pla~rnas.~’ In those cases, superdiffusion is related to the existence of series of vortices or magnetic islands, which may exhibit a gerarchy of scale lengths, typical of turbulence, which induce a power law distribution of jump lengths. On the other hand, in the case under consideration the superdiffusive transport is mostly parallel to the background magnetic field, so that the magnetic islands, commonly found in the presence of magnetic turbulence in the plane perpendicular to B0,30 are not so relevant to the creation of a power law distribution of jump lengths. We argue that in this case superdiffusion is due to the weak wave electron interaction, which is caused by small Larmor radii, so that pitch angle diffusion, which drives particles to reverse their direction of propagation, is not very effective. The weak interaction leads to long correlations in the electron parallel velocity, which results in a superdiffusive transport. Further investigations are needed in order to understand the statistical details of such a process. However, we argue that, at least for electrons, shock acceleration mechanisms should be reformulated either in terms of non Gaussian probability distributions or in terms of fractional Fokker-Planck equation^.^?^^ These results promise to have application to models of cosmic ray propagation in space, as well as to the spreading of energetic particles throughout the heliosphere.

Acknowledgments We acknowledge D. J. McComas of Southwest Research Institute for the use of the SWOOPS Ion Measurements data of Ulysses, L. J. Lanzerotti of Bell Laboratories for the use of HI-SCALE LEFS 60 data and M. Lancaster and C. Tranquille of the Ulysses Data System.

References

zy

1. G. M. Zaslavsky, R.Z. Sagdeev, D.K. Chaikovsky, A.A. Chernikov, JETP, 68, 995 (1989). 2. G. M. Zaslavsky, Phys. Reports, 371,461 (2002).

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3. T. H. Solomon, E. R. Weeks and H. L. Swinney, Phys. Rev. Lett., 71,3975 (1993). 4. A. J. Hundhausen, Introduction to Space Physics (eds. M.G. Kivelson and C.T. Russell, Cambridge, 1995). 5. D. V. Reames, Space Sci. Rev., 90,413 (1999). 6. J. R. Jokipii, Astrophys. J., 146,480 (1966). 7. J. Giacalone and J. R Jokipii, Astrophys. J., 520,204 (1999). 8. A. R. Bell, MNRAS, 182,147 (1978). 9. R. D. Blandford and J. P. Ostriker, Astrophys. J. Lett., 221,L29 (1978). 10. L. A. Fisk and M. A. Lee, Astrophys. J., 237,620 (1980). 11. J. E. Mazur, G. M. Mason, J. R. Dwyer, J. Giacalone, J. R. Jokipii and E. C. Stone, Astrophys. J. Lett., 532,L79 (2000). 12. W. Droge, Astrophys. J., 589,1027 (2003). 13. R. P. Lin, Adv. Space Res., 35,1857 (2005). 14. R. Metzler and J. Klafter, Phys. reports, 339,1 (2000). 15. S. Perri, F. Lepreti, V. Carbone and A. Vulpiani, Europhys. Lett., 78,40003 (2007). 16. G. Zumofen and J. Klafter, Phys. Rev. E, 47,851 (t993). 17. M. A. Lee, J . Geophys. Res., 88, 6109 (1983). 18. D. Lario et al., in Proc. Solar Wind 11-Soho 16, (Whistler, Canada, 2005). 19. G. M. Webb, G. P. Zank, E. Kh. Kaghashvili and J. A. le Roux, Astrophys. J., 651,211 (2006). 20. T. J. Geisel, J. Nierwetberg and A. Zacherl, Phys. Rev. Lett., 54,616 (1985). 21. S. Perri and G. Zimbardo, Astrophys. J. Lett., in press (2007). 22. S. Perri and G. Zimbardo, J. Geophys. Res., submitted (2007). 23. S. J. Bame, B. E. Goldstein, J. T. Gosling, J. W. Harvey, D. J. McComas, M. Neugebauer and J. L. Phillips, Geophys. Res. Lett., 20,2323 (1993). 24. A. Balogh, J. A. Gonzalez-Esparza, R. J. Forsyth, M. E. Burton, B. E. Goldstein, E. J. Smith and s. J. Bame, Space Sci. Rev., 72,171 (1995). 25. M. Neugebauer, J. Giacalone, E. Chollet and D. Lario, J . Geophys. Res., 111,A12107 (2006). 26. W. H. Matthaeus and M. L. Goldstein, Phys. Rev. Lett., 57,495 (1986). 27. G. Zimbardo, P. Pommois and P. Veltri, J. Geophys. Res., 109, A02113 (2004). 28. D. Ruffolo, Astrophys. J., 442,861 (1995). 29. S. Ratynskaia et al., Phys. Rev. Lett., 96,105010 (2006). 30. G. Zimbardo, Plasma Phys. Control. Fusion, 47,B755 (2005).

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318

zyxwvutsr

zy zyxwv zyx

WAVE CHAOS AND GHOST ORBITS IN AN UNDERWATER SOUND CHANNEL

D. V. MAKAROV’, L. E. KON’KOV, E. V. SOSEDKO and M. YU. ULEYSKY Laboratory of Nonlinear Dynamical Systems, V.I.I1 ‘ichev Pacific Oceanological Institute of the Far Eastern Branch of Russian Academy of Sciences, Vladivostok, R w s i a *E-mail: [email protected] http://dynalab.poi. d v 0 . r ~ Sound wave propagation in a weakly-inhomogeneous acoustic waveguide in the deep ocean is studied. Vertical oscillations of the inhomogeneity result in strong chaos of rays propagating with small angles with respect to the channel axis. Increasing of amplitude of the inhomogeneity results in multiplication of short periodic orbits, and their phase space distribution becomes irregular. However, the Floquet modes calculated with signal frequency of 100 Hz reveal well-ordered peaks within the chaotic area. We link occurrence of these peaks with periodic orbits of resonance 1:8, which survive as ghost orbits.

Keywords: wave chaos, ray chaos, underwater sound channel, scattering on resonance, ghost orbits, scar.

1. Introduction The semiclassical approximation is often used as the first step for describing wave or quantum dynamics. In this way one first calculates classical trajectories and then imposes wave-based corrections, associated with diffraction and interference. The well-known advantages of semiclassical technique are simplicity (as compared with full-wave calculations) and the possibility to give an intuitive “physical” interpretation of the processes observed. On another front, one must always keep in mind that the semiclassical approximation is nothing but the short-wavelength asymptotics for original wave equations and its applicability has severe restrictions. One necessary restriction is that wavelength must be much smaller than the lengthscale of refractive index (or potential) variability. Another one is concerned with caustics, where Maslov method must be appealed.’ Additional problems in ray-wave (or quantum-classical) correspondence occur when classical ray dynamics

z zyxwvu 319

0 0.5 1

4

1.5

zyxwvut

bi

2

2.5 3

0

20

40

60

r, km

zyxwv 80

100

Fig. 1. Rays in a range-independent underwater sound channel.

zyxwvu

possesses chaotic properties. Wavefield manifestations of ray chaos are referred to as wave chaos. In this case one introduces the so-called mixing (or log) time T A1 Inko, where A1 is the maximal Lyapunov exponent and ko is the wavenumber, playing the role of the inverse Planck constant. Beyond the mixing time classical paths intersect each other in an irregular manner, and interference cannot be neglected. Nevertheless, semiclassical calculations accounting interference between distinct paths can give accurate predictions a t times beyond the mixing timee2l3Furthermore, it is well realized that significant impact into a structure of wave eigenfunctions is given from classical periodic orbits. In particular, one of the most intriguing effects concerned with periodic orbits is the so-called scarring, i.e. forming of intensity peaks at unstable short periodic orbit^.^?^ The present paper is devoted to long-range sound propagation in the ocean. Increasing of hydrostatic pressure with depth, combined with warming of the upper oceanic layer, results in non-monotonic dependence of the sound speed on depth. According to Snell’s law, there occurs a waveguide confining sound waves within a restricted water volume and preventing their contact with lossy bottom. Since low-frequency sound waves are insignificantly attenuating inside the water column, they are capable to propagate over distances of thousands kilometers. Typical ray trajectories of sound rays in the horizontally homogeneous ocean are depicted in Fig. 1. In recent decades long-range sound propagation has attracted increasing attention in the context of hydroacoustical tomography - monitoring of large-scale variability of the ocean using sound signals. The traditional N

320

scheme of hydro-acoustical tomography relies upon analysis of signal travel times along different eigenrays, connecting source and receiver. However, it is realized that even weak longitudinal sound speed variations, mainly associated with ocean internal waves, are sufficient for Lyapunov instability and chaos of sound rays. Ray chaos leads to exponential proliferation of eigenrays with increasing distance between source and receiver, therefore the solution of the respective inverse problem is strongly degenerate under chaotic conditions. Thus ray chaos poses severe limitations on the possibility of extracting information about environment from the hydro-acoustial data.7 Predictability horizon for ray motion, determined by the reciprocal Lyapunov exponent, depends on the topology of a ray. In realistic models of environment this quantity is of approximately 100 km for near-axial rays and of approximately 300 km for steep ones.’ We have shown that strong instability of near-axial rays arise due to resonant scattering on small-scale vertical oscillations of a sound-speed p e r t u r b a t i ~ n . ~These J ~ oscillations are associated with the so-called internal wave structure. On the other hand, vertical wavelength of almost horizontal low-frequency sound waves is not small, that implies weak sensitivity to small-scale sound-speed structures.l1 Thus it is still not enough clear what mechanism accounts for fuzzy wavefield pattern corresponding to near-axial rays. In this way it seems to be worth to examine how chaotic ray dynamics reveals itself in sound propagation with relatively low frequencies, when one should not expect one-to-one correspondence between wave and ray dynamics. In the present paper we report about a specific conflict of ray and wave descriptions, when a wave pattern reveals features having “classical” nature but not allowable on the classical level. These features relate to the scarring phenomenon. The distinctive property of the case we consider is that scars are supported by the so-called ghost periodic orbits,12 which correspond to complex rays.13 The paper is organized as follows. In the next section we briefly describe properties of ray dynamics. Section 3 contains analysis of the Floquet modes. In conclusion we summarize and discuss the results obtained.

zyxw

2. Classical ray dynamics Consider a two-dimensional waveguide with with the sound speed c presented in the form

zyxwvuts 321

where co is a reference sound speed, A c ( z ) represents the range-independent depth change of the sound speed due to the waveguide, and dc(z, r ) is a small term varying with range r. We start with considering ray dynamics. Ray trajectories obey the Hamiltonian equations

z zyxwv dz dr

dp dr

-=--

dH

-

dH

-= P , aP 1 dAc co dz

1 ddc co dz ’

(3)

zyx zyx zyxw zyxwv dz

where p is tangent of ray grazing angle. In the small-angle approximation the Hamiltonian looks as follows

H

=

p2 Ac(z) -1+-+-+-. 2 co

dc(z, r ) CO

(4)

Ray equations ( 2 ) , (3) and the Hamiltonian (4)admit simple interpretation by means of the principle of optical-mechanical analogy: they describe motion of unit-mass point particle in a potential well A c ( z ) with small time-dependent perturbation 6c(z, r ) . In this fashion T plays the role of time, and p is often called as ray momentum. In the present paper we shall consider a range-periodic waveguide with A c ( z ) and 6c(z, r ) given by the following expressions:

21rz 21rr 6c(z, r ) = ~ c o F ( zsin ) -sin -, A, A,

where co = c ( z = h) = 1535 m/s, y = exp(-ah), h = 4.0 km is depth of the ocean bottom, 1.1 = 1.078, a = 0.5 km-’,b = 0.557, A, = 0.2 km, A, = 5 km. F ( z ) is an envelope function, given by expression

where B = 1 km. The channel axis, the depth where function A c ( z ) takes on the smallest value, is given by the formula 1

z, = -In

a

2

-N 1 km. P+Y

The respective unperturbed sound-speed profile is depicted in Fig. 2.

322

zyxwvuts 0

1

4 2 N"

zyxw zyxw

3

4 1480 1500 1520 1540

c, m / s Fig. 2.

The unperturbed sound-speed profile.

zyxw zyxwv

We use the model of range-periodic sound-speed perturbation. Although horizontal periodicity is a strong idealization, our model is relevant for qualitative analysis of the sound propagation in realistic environments. Amplitude of the perturbation term E is a small constant. In this paper we shall consider two cases: E = 0.0005 and E = 0.005. The sound-speed perturbation (6) can represented as a superposition of two waves, propagating upwards and downwards, respectively. This yields 1 ddc

--

co d z

2B

[ g)

zyxwv

~ e - ____ -

~

(1

~

-

/

~

(cosa- - COS@+) 1

1

- k,z (sin@- - sin@+)

,

(9)

where we denoted @* = k,z f k,r, k , = 27~/X,, k, = 27r/X,. Since k , is a large parameter, phases vary rapidly along unperturbed ray paths, except for resonant regions, where

or

d@dr

-= k,p - k,

N

0.

zy zyxwvutsrqp 323

0

1

E

A 2

6 3

4

a)

zyxwvutsrq zyx zyxwvut zyxwv I

-0 2

-0 I

0

0.1

$3

0.2

b)

P

Fig. 3.

._-

-0.2

Poincarh map. (a)

E

-0.1

0

0.1

0.2

I

0.3

P

= 0.0005, (b) E = 0.005.

It is easy to show that these resonances mainly affect near-axial rays, making them strongly chaotic (seeg~l0 for details). This is well demonstrated in Poincar6 sections (see Fig. 3). There occurs a chaotic layer in the part of phase space, corresponding to near-axial rays. The layer is bounded by invariant curves, and steeper rays, not being influenced by resonances, perform stable dynamics.

zy

3. Wave dynamics

In the present section we study how chaotic layer, corresponding to nearaxial rays, reveals itself in structure of the Floquet modes, which are calculated at the signal frequency of 100 Hz. The Floquet modes u, are the eigenfunctions of the shift operator F , defined as F + ( z ,r ) = +(z,?-

+ Ar).

(12)

Phase space representation of the Floquet modes can be obtained by use of the Husimi distribution function

Here Az is the smoothing scale, which we took of 100 m. Mixing ray dynamics inside the chaotic layer presupposes existence of strongly extended Floquet modes with irregular distribution of Husimi zer o ~Nevertheless . ~ ~ the Floquet modes calculated with E = 0.005 reveal almost regular pattern. Although few modes cover wide area in phase space,

324

1

-0.3 -0.2 -0.1

0

0.1

0.2

Fig. 4.

4 -0.3 -0.2 -0.1

zyx

0.3

P An extended Floquet mode.

zyx zyxwvu 7--

0

P

011

-0.2 -0.1

0

0:1

0:2

0.3

P

Fig. 5 . Floquet modes with chainlike topology.

their Husimi zeros are located along certain curves. Since that these modes could be classified rather as weakly irregular than strongly chaotic. One of such modes is illustrated in Fig. 4. We want to focus attention on those Floquet modes which have the specific chainlike topology of peaks, as it is demonstrated in Fig. 5. All the peaks belong to the chaotic sea of the classical system. On another front, a comparison with Fig. 3(b) yields that the peaks in Fig. 5a are located at the elliptic orbits of KAM resonance 1:8 and the peaks in Fig. 5b are placed at the hyperbolic ones. This implies that the respective Floquet modes are localized at the unstable periodic orbits and relate to cars.^^^'^

. zyxwvu . ). zyxw

0.3

* :

0.2 -

0

'

s"

0

*.

0

0

0:.

.

0.0

0

.

O 0 . .

0 0

0

*.

0

0.0

dbo

0

0

0 0

0

0

0

0.1

:

0.

..tt8

0

.

0 :

0

..*@.

O..O

@-

0

Fig. 6. Phase space locations of periodic orbits in the normalized action-anglevari-

ables. Is is the most accessible value of the action for guided rays propagating without reflections from the lossy bottom. Perturbation strength e is of 0.005. KAM resonance 1:8 corresponds to the line I/Is = 0.2.

However the situation we met is more complicated because increasing perturbation amplitude strongly alters phase space distribution of periodic orbits, especially in the range of low values of action, as it is illustrated in Fig. 6. According to this figure, peaks on Husimi plots don't supported by certain periodic orbits. This infers that the ordered peaks are localized at the so-called ghost orbits,12 which correspond to complex ray paths. Occurrence of prominent complex paths may be considered as some manifestation of tunneling through narrow classical barriers, which appear due to rapid depth oscillations of the sound-speed perturbation.

zyx zyxwvut

4. Conclusion

In the present paper we have examined interplay between ray and wave behavior in the model of a range-periodic underwater sound channel. We have shown that small-scale vertical oscillations of the perturbation result in strong chaos of rays propagating with small angles with respect to the channel axis. Increasing the perturbation amplitude results in multiplication of periodic orbits, and their phase space distribution becomes strongly irregular. On the other hand, Husimi plots for the Floquet modes reveal a chain of well-ordered peaks. Phase space locations and number of the peaks indicates on unambiguous link to KAM resonance 1:8. We suppose that periodic orbits of resonance 1:8 survive as the ghost orbits due to tunneling through small-scale sound-speed oscillatghions.

326

This work was supported by t h e projects of the President of t h e Russian Federation, by t h e Program “Mathematical Methods in Nonlinear Dynamics” of t h e Prezidium of t h e Russian Academy of Sciences, a n d by t h e Program for Basic Research of t h e Far Eastern Division of t h e Russian Academy of Sciences. Authors a r e grateful to S.V. Prants, A.I. Neishtadt, A.L. Virovlyansky a n d A.I. Gudimenko for helpful discussions during t h e course of this research.

References

zyxwvu zyxwv zyx

1. V.P. Maslov and M.V. Fedoriuk, Semi-classical Approximation in Quantum Mechanics (Reidel, Boston, 1981). 2. S. Tomsovic and E.J. Heller, Phys. Rev. Lett. 67,664 (1991). 3. S. Tomsovic and E.J. Heller, Phys. Rev. E. 47,282 (1993). 4. E.J. Heller, Phys. Rev. Lett. 53,1515 (1984). 5. E.B. Bogomolny, Physica D, 31 169 (1988). 6. W. Munk and C. Wunsch, Deep-sea Res. 26A, 123 (1979). 7. F.D. Tappert and Xin Tang, J. Acoust. SOC.Am. 99, 185 (1996). 8. F.J. Beron-Vera, M.G. Brown, J.A. Colosi, S. Tomsovic, A.L. Virovlyansky, M.A. Wolfson, and G.M. Zaslavsky, J . Acoust. SOC.Am. 114,1226 (2003). 9. D.V. Makarov and M.Yu. Uleysky, Acoust. Phys. 53,495 (2007). 10. L.E. Kon’kov, D.V. Makarov, E.V. Sosedko, and M.Yu. Uleysky, Phys. Rev. E. 76,056212 (2007). 11. K.C. Hegewisch, N.R. Cerruti, and S. Tomsovic J . Acoust. SOC.Am. 117, 1582 (2005). 12. M. Kus, F. Haake, and D. Delande Phys. Rev. Lett. 71,2167 (1993). 13. Yu.A. Kravtsov and Yu.1. Orlov, Caustics, Catastrophes and Wave Fields (Springer-Verlag, Berlin, 1999). 14. P. Leboeuf and A. Voros, J . Phys. A 23,1765 (1990). 15. 1.P Smirnov, A.L. Virovlyansky and G.M. Zaslavsky, Chaos 14,317 (2004). 16. I.P. Smirnov, A.L. Virovlyansky, M. Edelman and G.M. Zaslavsky, Phys. Rev. E. 72,026206 (2005).

327

DISPLACEMENT EFFECTS ON FERMI ACCELERATION IN RANDOMIZED DRIVEN BILLIARDS A.K. KARLIS*, P.K. PAPACHRISTOU, F.K. DIAKONOSt Department of Physics, University of Athens, GR-15771 Athens, Greece E-mail: * [email protected] t fdiakono Qphys.uoa. gr

zyxw zyx zyxw

V. CONSTANTOUDIS

Institute of Microelectronics, NCSR Demokritos, P. 0. Box 60228, Attiki, Greece

P. SCHMELCHER

Physikalisches Institut, Universitat Heidelberg, Philosophenweg 12, 69120 Heidelberg, Germany Theoretische Chemie, Im Neuenheimer Feld 229, Universitat Heidelberg, 69120 Heidelberg, Germany Fermi acceleration of an ensemble of non-interacting particles evolving in a stochastic Fermi-Ulam model (FUM) and the phase randomized harmonically driven periodic Lorentz gas is investigated. As shown in [A. K. Karlis, P. K. P a pachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. Lett. 97, 194102 (ZOOS)], the static wall approximation, which ignores scatterer displacement upon collision, leads to a substantial underestimation of the mean energy gain per collision. Herein we clarify the mechanism leading to the increased acceleration, through the investigation of a randomized FUM, comprising one fixed and one moving wall oscillating according to a sawtooth time-law. Moreover, it is shown that the presence of a particular asymmetry of the driving function leads to a different particle energy gain compared to the one observed when a symmetric driving law is considered. Furthermore, the impact of scatterer displacement upon collision on the acceleration of particles in the case of a harmonically driven phase randomized Lorentz gas is investigated in terms of a Markov process in momentum space. Generalizing the recently introduced hopping wall approximation (HWA) for application in the Lorentz gas, the time-evolving distribution of particle speeds is obtained. The analysis reveals that the underestimation made by the static approximation in both the Lorentz gas and the FUM, assuming a harmonic driving, is the same. Keywords: Fermi acceleration; Lorentz gas; Diffusion.

328

1. Introduction

zyxw zy

Fermi acceleration was originally proposed in 1949l for the explanation of the high-energy cosmic ray particles. The basic idea was that cosmic ray particles would, on the average, gain energy by scattering off timedependent magnetic inhomogeneities -for a review see Ref. 2. Ever since, Fermi acceleration has been investigated in the context of a variety of systems, pertinent to different areas of physics, including a s t r o p h y ~ i c splasma ,~ physic^,^ atom optics5 and has even been used for the interpretation of experimental results in atomic physics.6 Conceptually, the simplest dynamical system which allows for the investigation of Fermi-acceleration is the Fermi-Ulam model (FUM).7 FUM comprises two infinitely heavy walls, one fixed and one oscillating, and an ensemble of non-interacting particles bouncing between them. FUM and its variants have been extensively studied both theoretically (see Ref. 8 and references therein) and experiment all^.^-^^ Despite the simplicity of the model, the equations defining the dynamics are, in general, of implicit form with respect to the collision time, which hinders the analytical treatment and greatly complicates numerical simulations. For these reasons, a simplification originally introduced by Lieberman et a1.,12 known as the static wall approximation, has become over the time the standard approach for studying the FUM.13-17 The SWA simplifies the process in that the displacement of the moving wall is ignored. However, the time-dependence in the momentum exchange between particle and wall on the instant of collisions is retained, as if the wall were oscillating. More recently, similar simplifying approximations were employed for the investigation of higher-dimensional, spatially extended billiards, such as a periodic Lorentz gas,18 consisting of circular scatterers oscillating harmonically, with their equilibrium positions placed a t the nodes of a square 1 a t t i ~ e . I ~ However, as has been recently shown by Karlis et a1.,20 the SWA leads to a considerable underestimation of the energy gain of the particles evolving in a stochastic FUM. Moreover, in Ref. 20 with the introduction of the so-called hopping wall approximation (HWA) it was made clear that the movement of the oscillating wall in the configuration space affects deeply the diffusion in momentum space. In spite of the externally imposed stochasticity, small additional fluctuations in the time of free flight, caused by the displacement of the wall, act systematically leading to the increased acceleration compared to that predicted by the SWA. In Sec. 2, the physical mechanism a t work, leading t o the increased acceleration due to the displacement of the moving wall, is exemplified through

zyxwv

329

the investigation of a stochastic FUM with one fixed and one moving wall oscillating according to a sawtooth time-law. In addition, it is shown, as indicated in Ref. 20, that in the presence of a particular asymmetry of the sawtooth driving function the increase in acceleration is different from that found considering a symmetric time-law, for all finite number of collisions. However, when the same asymmetric driving is applied on the two-moving wall variant of the FUM, the increase in acceleration due to the displacement of the wall upon collision is once more constant and coincides with that obtained on the assumption of a symmetric force function. These findings open up the prospect of designing specific devices combining driving laws with different symmetries in order to achieve a desired acceleration behavior. In order to determine the effect of scatterer displacement upon collision on particle acceleration in the case of higher-dimensional billiards, the driven phase-randomized periodic Lorentz gas is investigated in Sec. 3. Generalizing the recently introduced hopping wall approximation (HWA) for application in the Lorentz gas, the time-evolving distribution of particle speeds is obtained. The analysis reveals that the increase in acceleration due to scatterer displacement on collision takes place also in higher-dimensional driven billiards and moreover that the increase coincides with that observed in the stochastic FUM assuming harmonic driving. Thus, it can be inferred that the increase of particle energy gain upon collision with an oscillating scatterer is common to many driven dynamical systems and features in any billiard allowing for Fermi acceleration to develop.

2. Fermi-Ulam model with a sawtooth wall driving function In order to clarify the physical mechanism which accounts for the effect of the wall displacement on the acceleration of the particles, a FUM comprising one fixed and one oscillating wall driven according to a sawtooth time-law is investigated. The specific choice for the driving function is justified due to its simplicity, which helps elucidate the important physical aspects of the mechanism in a more intuitive and straightforward manner. The phase of the oscillating wall is shifted randomly (according to a uniform distribution) after each collision with the fixed wall. The stochastic component in the oscillation law of the wall simulates the influence of a thermal environment on wall motion and leads to Fermi a c ~ e l e r a t i o n . ' ~ -The ~~>~~

330

following class of time-periodic laws for the moving wall are considered:

f i^

^e[o,a)

Xi(t) = xo,i+ { -&%%&

^ € [a,b]

I jh^ 1A 2A

u(t)= k

1-6 T

(1)

^e(6,i] QpE[O,a) @yE[u,b] T

te

(2)

Vw' A i

where T is the period of the oscillation, A the amplitude, XO,(L,R) the equilibrium position of the left or right moving wall and r\ a random number uniformly distributed in [0, 2TT). Let us assume that the oscillating wall is on the left. According to the SWA, which treats the oscillating wall as fixed, the time of collision t is given by the intersection of a particle trajectory with the line corresponding to the equilibrium position of the wall, as shown in Fig. l(a). On the other hand, the true collision time t', as predicted by the exact map,22 is the intersection of the line representing the particle trajectory with the piecewise linear function defined by Eq. (1). Furthermore, in Fig. l(a) we observe that if the particle trajectory lies in zone 2 —zones are marked with double arrows in Fig. l(a) and further are delimited by dashed lines— then due to the displacement of the wall toward the right, the actual collision time —obtained using the exact map— compared to that estimated by the SWA (lying in zone 2) is smaller (shifted in zone 1), on which particle and wall have opposite velocities (head-on collision). Moreover, in zone 4, again due to the displacement of the wall, the collision time is shifted toward greater values (in zone 5) in comparison to the that predicted by the SWA (lying in zone 4), when again, particle and wall velocities have opposite directions. Thus, in both occasions, the SWA would render a head-on (energy increasing) collision to a head-tail (energy decreasing) collision. On the other hand, the estimation of the change of particle energy upon collision for trajectories confined between the borderlines of the zones 1, 3 and 5 is the same using either the exact model or the SWA, as the collision times obtained using either of the models correspond to the same wall velocity. Consequently, the SWA leads to the underestimation of the acceleration rate of particles. Our attention now shifts to the dependence of the increase of particle energy gain caused by wall displacement on the specific characteristics (symmetries) of the wall driving function. The symmetry u (m^ + j — tj = —u (m-j + j + t) , (TO = 0,1,2,...), characterizing the harmonic driving

z zyx zyxwv zyxw 331

studied in Ref. 20, in the context of a stochastic FUM, revealed that wall displacement doubles the mean particle energy gain. Let us now investigate the effect on the increase of particle acceleration of the breaking of this particular symmetry. In order to quantify the relative efficiency of the mechanism leading to the increased acceleration between setups with different driving force laws, the ratio Rh ( n ) =

( (V,") -

5 (6Viz)eroct (V,')) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML ~

((V,")-(V,"))s,,

i=l

2(6V:)swa is introduced. The

i=l

specific definition of Rh(n),compared to that given in Ref. 20, is more convenient for numerical calculations as it converges much more rapidly in terms of ensemble averaging. For the harmonically driven FUM (with randomized phase of oscillation) the correction factor Rh(n) can be proved" (within the leading order of (+)) that it is independent of n and equals 2. As already mentioned the ensemble of particle trajectories can be divided into two sets: The first set consists of trajectories for which the acceleration process is identical to that of the exact model. The second set consists of trajectories for which the acceleration in the SWA is underestimated. Since the phase is uniformly distributed the size of the zones determines also the statistical weight of each zone. Thus, denoting the statistical weight of the zones with pi, (i = 1 . ..5) and the particle velocity with V the average energy gain over the phase upon collision is ((6V:)) = C;=lpibV:,i. The ensemble average can be extracted by integrating ((6V:)) over the P D F of the particle velocities (see Ref. 22) to obtain for n >> 1:

where the plus (minus) sign corresponds to a FUM with an oscillating wall on the left (right). Thus, a FUM with a moving wall on the right leads to a slower acceleration rate compared to its counterpart. This is illustrated in Fig. l ( b ) , where numerical results for R h , L ( n )obtained for a = 5 x and b = 0.67 are shown along with the analytical result of Eq. (25) of Ref. 22. As demonstrated in this figure, for the particular choice of the parameter values, the function R h , ~ ( ndeviates ) from the value 2 even for relatively large n. Obviously, for the case of a two-moving wall FUM the factor Rh is given by &(n) = R h z ~ ( n )2+ R h s ~ ( n )= 2. consequently, for a two-moving wall FUM the correction factor for the increased acceleration due to the wall displacement on collision, assuming that both walls follow the dynamics of Eq. (l),is independent of n and equal to that obtained for the case

zyxwvu

332

zyxwv zyx zyxwvu

of harmonic driving. In conclusion, Rh(n) for the FUM setups with one moving wall depends on the number of collisions n and tends to the value Rh (00) = 2 , in a manner specified by the particular oscillation law, which is characterized here by the parameters a , b. On the other hand, when both walls of the FUM are allowed to move, then unless a specific choice for the dynamics of each of the walls is made, Rh(n) is rendered independent of n and equals that obtained when a harmonic driving is considered.

Particle trajectory

a tlT

zyxw b

1

t

l.g 1.8'

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of collisions n

1.6

1.8

I

2

lo5

zyxwvu

Fig. 1. (a) The oscillation law for the FUM (with one oscillating wall). The borderlines of the zones discussed in the text are delimited by the dashed lines of slope X = The zones are also marked by double arrows. (b) Numerical results ( x ) as well as an& lytical results (see Eq. (25), in Ref. (22)), for R h , ~ ( nin) the original FUM, employing and b = 0.67. sawtooth driving for uo = 0.01, a = 5 x

w.

3. Time-dependent Lorentz gas In the theory of dynamical systems Lorentz gas18 acts as a paradigm allowing us to address fundamental issues of statistical mechanics, for instance, ergodicity and and transport processes, such as diffusion in

zyx 333

configuration pace.^^^^^ A generalization of the original periodic Lorentz gas model has recently been introducedlg allowing the velocity of the scatterers to oscillate radially on a square lattice, i.e. static approximation. Due to the inherent strong chaotic dynamics of the static Lorentz gas, owing to the convex geometry of the scatterers, one intuitively expects13 that the introduction of time-dependence induces Fermi acceleration, resulting in unbounded growth of the velocity of the tracer particle. This is in contrast to the FUM, where, on the supposition of smooth periodic force functions, the particle energy remains bounded, due to the presence of a set of spanning KAM curves in the phase space8?l2and only in the presence of external stochasticity does Fermi acceleration become feasible. The acceleration problem can be treated as a Markov process in momentum space. Therefore, the evolution of the probability distribution function (PDF) of the magnitude of particle velocities p(lV1,n) can be determined by the Fokker-Planck equation. In Ref. 19 the study of a square periodic Lorentz gas consisting of “breathing” circular scatterers, i.e. with oscillating boundary of the scatterer in the radial direction, conducted by means of the static approximation, concluded that p(lVI , n ) is a sum of spreading Gaussians. Furthermore, general arguments presented in Ref. 28, where a random time-dependent Lorentz gas has been investigated, also suggest that p(lVl , n) is a Gaussian. However, the numerical results shown in Fig. 2, which correspond to two snapshots for n = 5 x lo4 and n = 5 x lo5, obtained through the iteration of the exact map,22 point to the direction of p(lVl , n) being a Maxwell-Boltzmann like distribution. As mentioned above, p(JV1,n) can be analytically obtained through the solution of the Fokker-Planck equation (FPE)

zyx zy

In Eq. (3), P is the probability of a particle possessing the velocity IVI if it had the velocity IVI - AlVl, A n collisions earlier. Assuming that An = 1, B(V) = (S(V1)and D(V) = ((SlV1)2),where (SlVl) is the mean increment of the magnitude of the particle velocity during one mapping period, i.e. in the course of one collision. To do so the transport coefficients must be calculated taking into account the displacement of the scatterers. However, the implicit form of the exact map describing the dynamics of the system prohibit an analytical calculation. For this reason, the hopping wall approximation (HWA) originally introduced for the stochastic FUM2’ is generalized for application in the Lorentz gas. The key simplifying assumption of the hopping approximation is that

334

zyxwvut zyx zyxwvu .. -

0.6

> 0.4

v

Q

0.2

n

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

IVI

0.4

I

t

n

c..

- 0.3 0.2 -

>

W

Q 0.1

0

0

zyxwvu

zyx zyxwvuts zyxw 2

4

6

8

10

12

14

Fig. 2. Numerically computed P D F for the magnitude of the particle velocity using an ensemble of lo4 trajectories and following the exact dynamicszz for: (a) n = 5 x lo4 and In each case the analytical result provided (b) n = 5 x lo5 with E = 2 215 and llVoll = by Eq. ( 3 ) is also shown (solid line). Velocities are measured in units of ww (w=2.15, angular frequency w = 1)

g.

zyxwvu

a scatterer on the instant of the n t h collision occupies the position it held on the (n - 1)th collision, provided that the time of free flight between two successive collisions is small compared to the period of oscillation, i.e. high particle velocities. Thus, within the framework of this approximation, the deterministic component of the phase of the scatterer's position a t the nth collision is taken to be equal to the deterministic phase component of the scatterer's velocity at the (n - I ) t h collision. This leads to an explicit mapping" with regard to the instant of collision t,, which permits the analytical estimation of the transport coefficients of the FPE. The specific values of the coefficients" together with reflective boundary conditions at V = 0 lead to the following solution of the Fokker-Planck equation:

zyxwvu zyx zyxw zy zyx z zyxwv zyxwv 335

where c is the dimensionless ratio of the amplitude of oscillation over the spacing between the scatters centers a t equilibrium w. In Fig. 2 the analytical result for the PDF, Eq. (4),is also plotted for the sake of comparison. Clearly, the analytics provides an accurate description for the PDF p( IVI, n) for n >> 1. From Eq. (4) if follows that for n >> 1 the ensemble mean square velocity is V," = 2e2n &. Therefore, the mean change of particle energy is (6V,"),,,,t = (V,") - (V,"-l) = 2c2. If the static approximation is applied,22 one finds (6V,")static = c2. Therefore, Rh ( n ) =

+

2(6y2)ezact/(6y2),t,ti, 2 , independent of the number of collisions n n

=

i=l

i=l

and equal to that obtained in the case of a stochastic FUM with symmetric wall driving.

4. Concluding remarks

The basic findings of the present work are: a

Rh(n), which quantifies the increase in acceleration due to the displacement of the moving scatterer, was shown that on the assumption of a driving time law featuring the symmetry -u (m; $ -t) = --u (m; $ t ) , (m = 0 , 1 , 2 , . . .), is independent of the number of collisions n, regardless of the specific billiard, being a FUM or a Lorentz gas. On the other hand, if the driving time-law acting on the moving wall of the FUM -with one fixed and one moving wall- is asymmetric, then Rh(n) is different from 2 for any finite number of collisions n and becomes equal to 2 only asymptotically. The PDF of particle velocities in the driven Lorentz gas resembles a Maxwell-Boltzmann distribution. This is in contrast to the result obtained by the application of the standard SWA, i.e. by not taking into account the displacement of the scatterer, where the corresponding PDFs are found spreading Gaussians.

+ +

+

As a final remark, we note that the understanding of the dependence of particle acceleration behavior on the symmetries of the driving law, helps open up the prospect of designing specific devices combining driving laws with different symmetries in order to achieve a desired acceleration behavior.

336

Acknowledgment T h e project is co-funded by the European Social Fund a n d National Resource (EPEAEK 11) PYTHAGORAS.

References 1. 2. 3. 4. 5.

zyxwv zy zyxwvut zyxw

E. Fermi, Phys. Rev. 75,1169 (1949). R. Blandford, D. Eichler, Phys. Rep. 154,1 (1987). A. Veltri and V. Carbone, Phys. Rev. Lett. 92 143901 (2004). A. V. Milovanov and L. M. Zelenyi, Phys. Rev. E 64, 052101 (2001). F. Saif, I. Bialynicki-Birula, M. Fortunato, W. P. Schleich, Phys. Rev. A 5 8 ,

4779 (1998). 6. G. Lanzano et al, Phys. Rev. Lett. 83,4518 (1999). 7. S. Ulam, in Proceedings of the Fourth Berkley Symposium on Mathematics, Statistics, and Probability, California U.P., Berkeley, 1961, Vol. 3,p. 315. 8. A. J . Lichtenberg, M. A. Lieberman, Regular and Chaotic Dynamics, Applied Mathematical Sciences 38, Springer Verlag, New York, 1992. 9. Z. J. Kowalik, M. Franaszek and P. Pieranski, Phys. Rev. A 37,4016 (1988). 10. S. Warr et al, Physica A 231,551 (1996). 11. S. Celaschi and R. L. Zimmerman, Phys. Lett. A 120,447 (1987). 12. M. A. Lieberman and A. J. Lichtenberg, Phys. Rev. A 5,1852 (1972). 13. E. D. Leonel and P. V. E. McClintock, 3. Phys. A: Math. Gen. 38, 823 (2005). 14. A. J. Lichtenberg, M. A. Lieberman and R. H. Cohen, Physica D 1, 291 (1980). 15. E. D. Leonel, P. V. E. McClintock and J. K. d a Silva, Phys. Rev. Lett. 93, 014101 (2004). 16. E. D. Leonel and P. V. E. McClintock, Phys. Rev. E 73,066223 (2006). 17. D. G. Ladeira and J. J. Leal d a Silva, J. Phys. A: Math. Theor. 40, 11467 (2007). 18. H. A. Lorentz, Proc. Roy. Acad., Amsterdam, 438, 585, 684, (1905). 19. A. Yu. Loskutov, A. B. Ryabov, and L. G. Akinshin, J. Exp. Theor. Phys. 89,966 (1999); J. Phys. A: Math. Gen. 33 7973 (2000). 20. A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. Lett. 97,194102 (2006). 21. G. M. Zaslavskii and B. V. Chirikov, Sov. Phys. Dokl. 9,989 (1965). 22. A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher, Phys. Rev. E, 016214 (2007). 23. Ya. G . Sinai, Russ. Math. Surveys 27,21 (1972); 25 137 (1970). 24. L. A. Bunimovich, Comm. Math. Phys. 65,295 (1979). 25. L. A. Bunimovich and Ya. G. Sinai, Commun. Math. Phys. 78, 247 (1980); 78,479 (1980). 26. H. v. Beijeren, Rev. Mod. Phys. 54,195 (1982). 27. C. P. Dettmann, in Encyclopaedia of Mathematical Sciences, edited by D. Szasz (Springer, Berlin, ZOOO), Vol. 101, Chap. The Lorentz gas: A paradigm for nonequilibrium stationary states, p.315. 28. F. Bouchet, F. Cecconi and A. Vulpiani, Phys. Rev. Lett. 92,040601 (2004).

337

zyxw zyxwvu zyxw zy

MEMORY REGENERATION PHENOMENON IN FRACTIONAL DEPOLARIZATION OF DIELECTRICS V. V. UCHAIKIN* and D. V. UCHAIKIN

Department of Theoretical Physics, Ulyanovsk State University, Ulyanovsk, 432 970, Russia * E-mail: uchaikinQsv.uven.ru

The depolarization of dielectrics with polar molecules is considered as diffusion of points over a spherical surface. The normal (Gaussian) diffusion leads to the Debye relaxation, while the subdiffusion regime expressed in terms of fractional differential equation leads to a non-Debye relaxation law of the inverse power type. When the order a becomes 1, the relaxation takes the exponential form, but when a is close to 1 (but is not strictly equal to it) the relaxation follows firstly exponential law and then inverse power law. Because of the property of fractional derivatives, the end part of the process depends on its prehistory. This phenomenon is interpreted as regeneration of memory.

Keywords: Depolarization; Memory; Relaxation; equation.

Fractional

differential

zyxw

1. Introduction

There exist numerous evidences that the dielectric response of a wide variety of dielectrically active materials manifests deviation from the Debye

relaxation law

f ( t ) = foe-+,

t 2 0,

> 0.

(1) Rich experimental data collections had been gathered and analyzed in Jonscher's books,1,2 proceeding^,^-^ reviews6 and other editions. It is no need t o think, that these observations are quite new: the first of them was made more than hundred years ago7lg and is known now as the Curie-von Schwei7

dler law:

f ( t )0: t-", t

00,

ck

> 0,

(2) Such kind of relaxation is observed in many other natural processes: luminescence decay, chemical reaction processes, mechanical creep and viscoelasticity, trapping charges in p - n junctions and so on. It is so widespread --f

338

zyxw

way of approaching to an equilibrium state that Jonscher named it the “Universal Relaxation Law” .2 The non-Debye systems manifest a new, very important quality - the “memory”. This means that the real law of relaxation depends not only on its value at the switch-off moment t = 0 as Eq (1) predicts but on the whole previous history of the process f ( t ) , t < 0. The rise of the memory has been noted by a few authors of theoretical and experimental works (one of them gave his articleg an expressive title: “Dead matter has memory!”). We present here a new effect connected t o the memory phenomenon: when a being close to 1, the memory about a previous history arises not immediately after time t=O but after a some interval (O,t,) during which the relaxation approximately follows the exponential law independently of the prehistory t < 0.

zyxw

2. Diffusion model of the relaxation

Let us recall the classical image of a polar dielectric as a set of polar molecules freely moving in a boiled neutral liquid representing the influence of heat motion. Each molecule possess its own randomly oriented dipole moment plus a non-random component P caused by an external electrical field. This component is much smaller than the basic one, but namely it produces the polarization, because the mean value of the basic component equals 0. After switching off the external field, the P becomes accessible t o action of the heat bath, changes randomly its direction and the mean value of its projection on the initial direction ( P Z ( t )decreases ) with time. As a result, the polarization of the volume V containing N molecules decreases as f ( t ) = N(Pz(t )) . Two assumptions lie in the base of the model: 1) independent behavior of different molecules and 2) the Markovian character of motion of each of them. They allow us to reduce the relaxation process to a random motion of a single molecule in the direction space or, in other words, t o Brownian motion of a point on the sphere of the unit radius. The probability initially concentrated a t the upper pole of the sphere ( z = 1) gradually spreads over it and eventually fills it uniformly. The relaxation function being proportional to the mean z-coordinate of the distribution goes to 0 as t 4 00, its behavior determines a relaxation law. In terms of spherical coordinates 29, cp, the angular probability density of the point p(29, cp, t ) obeys the diffusion equation

zyxwvuts zy zyxwvu 339

where K is a positive diffusivity in the direction space, and Do,, denotes the angular part of Laplacian. On multiplying both sides of the equation by fo cos 6 sin 6d6dy and integrating over the sphere we arrive at the following equation

dfo = dt

-T-lf(t)

for the seeking function

f ( t )= fo(cos 6 ( t ) )= fo

//

+ foS(t)

(4)

p ( 6 , (p, t )cos 29 sin 6d6dy.

The solution of the equation is exactly the Debye relaxation function Eq (1).What can be improved in this model t o get a solution with the property

(2)? Discussing why relaxation in solids is differ from relaxations in gases, Jonscher fairly notes that the dipoles can not be considered as independent anymore, molecules may unite in clusters. It is naturally to suppose that the clumsy clusters hinder to each other in motion, that they have to wait until they have a space for rotation decreasing their potential energy. A dipole stays almost motionless as if were in a trap (in the direction space). After some random (perhaps very long) time TI it makes a jump to some other direction being close to the previous one. Following the procedure described in" (see alsoll) one can prove that the relaxation process will be of the Debye type in long-time region for any distribution p(t) with a finite mean time ( T ) .However, in the case when 00

zyx

( T )= / t p ( t ) d t = cq 0

one can get another kind of asymptotic behaviour of the relaxation. Namely, assuming that

Prob(T

> t ) cx tKa-', t -+ co, a E (0,1),

we obtain a solution having the property (2). The correspondent diffusion (more exactly subdiffusion) process is described by the time-fractional subdiffusion equation generalizing Eq (3)

where

zy zyxw zyxwv zy zyx zy

340

and -mDF is the Riemann-Liouville fractional differential operator: t

The corresponding generalization of relaxation equation (4) takes the form of an ordinary fractional equation:

-oD,af(t) = -7-

--a

f(t) + for(l

tT“ - a )7

t>0,

O 0, -a < t < 00, O < < I.

(7) 1 the scale factor r does not mean a mean relaxation (Y

3. Numerical calculations

On the assumptions formulated above the solution of Eq (7) is represented in the form

/ t

f(t)=

-a

G(t - t’)F(t’)dt’,

341

where Green's function is expressed through two-parametric Mittag-Leffler functions

via relation13 G(t) = ta-lEa,a(-(t/r)a). Remember some simple cases of this function:

El,2(.) =

E2,2(Z) =

zyx zyxwvutsr

e" - 1 -1

Z

zyxw zyx zy

sinh(&)

&

*

For further details, see.l3?l4A more complicated Green function was used in.15 We compute the solution f ( t ) = U(t) of (7) for the special case F(t)=

{

0, t I -8, E, - e < t I o , 0, t > 0 ,

where 0 < 8 < a. A little bit conditionally, this case can be interpreted as a process of charging of a dielectric capacity by constant electrical field E during time 8 till t = 0, variable V(t) represents a voltage on the capacity as a function of time during charging (t < 0) and following discharging a real dielectric. Using Eqs (9)-(11) in (8) we get

Interchanging summation and integration and using the gamma function property yields

342

According to (9)

zyxwvu zyxw

and finally we have for V ( t ;a , 6')

= U ( t ) / U ( O ) ,t > 0:

6')/71") v(t;a , e) = (t + 6')"-&,a+l(-[(t 6'" +E","+ 1 (-

-

l+(t)taE","+1(-(t/7)")

(6'/7)")

(12) Each value of parameter a E ( 0 , l ) produces a family of curves respective to different durations of charging 6' and only one value of a produces the only curve independently of 8, this is Q = 1. Really, using special formula (11) we get from (12):

In the last case the relaxation follows the Debye law and does not depend on 6': the memory is absent (Fig.1, left panel). However, computing (12) for Q = 0.8 and different charging durations 6' we obtain different relaxation curves. They are represented on the right panel of Fig.1 for charging durations 6' = 1.0 (the upper line), 0.6, and 0.3 (the lower line). It is clearly seen that the process at t > 0 depends on its prehistory at t < 0: we observe the memory effect. This results are in agreement with experimental data discussed in the a r t i ~ l e . ~ An amazing behavior of the solution is observed when Q is very close to 1 but not equal it: in this case the polarization falls according to Debye exponential law for a long time but after some crossover moment we observe splitting with respect to different values of 6' and transition to non-Debye power laws (Fig.2). It seems as if the memory returns to the system after some interval of time. Such behavior may be called the regeneration memory

effect.

zyxwv

4. Where come long waiting times from?

No doubt the key-assumption of this subdiffusional approach is an inverse power type distribution of random waiting (trapping) time T

p T ( t ) 0: t--l

, t --t

00, Q

E

(0,l).

For most physicist, it would be more easy to agree with an exponential distribution, and quantum mechanics confirms this: quasi-classical calculations really show the exponential distribution of trapping time for a particle

zy zyx zyxwvut 343

2

Fig. 1. Charging-discharging of a dielectric capacity. The left panel: a = 1; the right panel: a = 0.8 (0 = 1(1), 0.6(2), 0.3(3))

!2

which can leave a potential hole through an energy barrier of hight tunneling: pT(tlE) = X(€)e-X(')t.

zy

zy

Fig. 2. Transition Debye relaxation to non-Debye one at large times ( a = 0.99, 2, e = qi), ioo(z), iooo(3))

T

E

=

by

(13)

Here the jump rate X has the Arrhenius form

A(€) = vexp(-PE)

(14)

with P being an inverse proportional to the absolute temperature. The presence of hard acting environment in dense matters may shake this point of view, but where come the power law from? What is its physical cause? One could hope that a certain answer will be found from a strict theory taking into account all essential physical details of the process, but a direct

344

zyxwvut zyxwvu zyx zyxwvu

many-body calculation is extremely difficult to perform. Now, there exists a few approximation models demonstrating origin of inverse type distribution on some assumptions replacing strict proofs. We refer here to one of them called the random activation energy model (RAEM).16*19This model is based on relations (13)-(14) under additional assumption that the activation energy E is an exponentially distributed random variable: +(E)

= a!Pe-"PE.

In this case

A survey of reactions in condensed media with power type reaction rates connected with thermally Assisted Tunneling can be found in. l7 5. Summary

zy z

The following statements are worth to be stressed in the conclusion. 1. The idea of trapping of dipole moments carriers in solid dielectrics relaxation dates back to l?ro1ich18 2. The inverse power type relaxation is observed in experiments.2 3. The inverse power type asymptotical behavior of a waiting-hopping

process on a sphere is possible only under condition p ~ ( t cx) t-"-' , t - + 00, a! E (0,1).10 4. The inverse power distribution of trapping time can be considered as a main cause for asymptotical behavior of relaxation to be described by fractional equation. This seems to be a more natural explanation then assumptions about existing fractal structures or even about of fractal time itself.

345

5. Finally, t h e regeneration memory effect can be considered as a n indication that one and t h e same material can reveal Debye and non-Debye relaxation in different time domains. Acknowledgment T h e authors thank t h e Russian Foundation for Basic Research (grant No. 07-01-00517).

References

zyxwvut

1. A. K. Jonscher, Dielectric Relaxation in Solids, (Chelsea Dielectric Press,

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14.

15. 16. 17. 18. 19.

London, 1983). A. K. Jonscher, Universal Relaxation Law, (Chelsea Dielectric Press, London, 1996). Non-Debye Relaxation in Condensed Matter, T. V. Ramakrishnan, M. Ray Lakshmi (Eds), (World Scientific, Singapore, 1987). Relaxation in Complex System and Related Topics, I. A. Campbell, C. Giovannella (Eds), (Plenum Press, New York,1990). Complex Behaviour of Glassy Systems, M. Rubi, C Perez-Vicente (Eds), (Springer Verlag, Heidelberg, 1997). W. T. Coffey, J . Molecular Liquids 114,5 (2004). M. J. Curie , Ann. d e chimie et de physique ser. 6, 18,203 (1889). E. R. von Schweidler, Ann. der Physdk 24,711, (1907). S. Westerlund, Physica Scripta 43,174 (1991). V. V. Uchaikin, J . Exper. Theor. Physics 88,1155 (1999). V. V. Uchaikin, V. M. Zolotarev, Chance and Stability. Stable Distributions and their Applications (VSP, Utrecht, the Netherlands, 1999). V. Volterra, Theory of functionals and of integral and integro-differential equations (Dover Publications Inc., 1959). I. Podlubny, Fractional Differential Equations (Academic Press, New York London, 1999). R. Gorenflo, F. Mainardi, in: Fractals and Fractional Calculus in Continuum Mechanics, Carpinteri A, Mainardi F (eds) (Springer Verlag, Vienna - New York. 1997) 223. V. V. Uchaikin, Intern. J . Theor. Phys. 42 121 (2003). B. K. P. Scaife, Principles of Dielectrics (Oxford Univ. Press, London, 1998). A. Plonka, J Phys Chem B 104,3804 (2000). H. Frohlich. Theory of Dielectrics, (Clarendon Press, Oxford, 1958). M. 0. Vlad, Physica A184,303(1992).

zyxw

346

zyxwvutsr NODAL PATTERN ANALYSIS FOR CONDUCTIVITY OF QUANTUM RING IN MAGNETIC FIELD

zyx

MITSUYOSHI TOMIYA', SHOICHI SAKAMOTO, MASAKI NISHIKAWA and YOSHIFUMI OHMACHI Department of Materials and Life Science, Seikei University, 3-3-1 Misashino-shi, Tokyo 180-8633, Japan 'E-mail: tomiyaOst.seikei.ac.jp

The electron transport inside two-dimensional nanostructure is numerically studied, by the nodal pattern analysis of the wave functions. Especially the doubly connected quantum ring(QR) structure provides us interesting features under an external electro-magnetic field. Recent experimental techniques in nanostructures is now sophisticated enough, for example, to control the electron number in a two-dimensional system. Then one of the main interests on electron transports in nanodevices is to know characteristics of a magnetoresistance. El-om the nodal pattern analysis, it is found that every state cannot contribute to the electron transport, when the overall shape of the device is not near integrable. We also find the wavefunctions that stick to the inner wall of the QR under a weak magnetic field. Classically such strength of magnetic field makes the cyclotron radius near the radius of the inner hole of the device. It implies that there must be the noble relation between the ring size of the inner hole and the cyclotron radius of an electron motion and the electron transport is dependent on the relation. Consequently, the resistance is affected by the drive of an ac electro-magnetic field which is capable of exciting the electron to the higher eigenstate.

1. Introduction

It is known that production of the heterojunctions is getting increasingly promising and great flexibility. Lots of these efforts lead us from submicron t o more smaller sized structures. Then theoretical study for systems of single electron or a few electrons in two dimensions1>2has now urgent needs. On the other hand, the two-dimensional electron gas(2DEG) is still be considered as many electrons in the pm sized semiconductor devices. The pm sized devices already has shown many exotic features, especially under the electro-magnetic field, such as the Aharonov-Bohm(AB) effect,

347

zyxwvuts zyxw zyx zyx

Fig. 1. Schematic illustration of the device shapes. The symbols p and q represent the labels of the lead parts. The size of their rectangle frame is 47.9nm x 14.3nm. (a)nanowire model, (b)strait model, (c)quantum ring(QR) model. At dark parts, the potential is put very large value. The shape of a circle model is the same as the QR model without the inner hole in the middle.

the Altshuler-Aronov-Spivak (AAS) effect the quantum Hall effect, the Shubnikov-de Haas(SdH) oscillation and so on. All these properties are purely quantum mechanical phenomena under external magnetic field.3>4 Moreover, recently the experimental results under the microwave(MW) drive also have attracted considerable attenti~n.~-lO Recently the noble feature of the magnetoresistance of the 2DEG in the pm sized devices has been found by Prof. K. Fhjii's gr0up.l' Their specimen has a hole in the middle of the structure and we shall call it a quantum ring(QR)(fig.l). Thus the structure is doubly connected and makes the physical properties more sensitive especially against the magnetic field. The strong static magnetic field also induces the SdH oscillation in the specimen, however, they uniquely have the dip of the magnetoresistance in the intermediate strength range of the static magnetic field which is perpendicular to a two-dimensional structure, under a radiation of MW. The similar result has been reported in the case of simply connected and pm sized dot ~ a s e . ~ The - ~magnetoresistance has the oscillatory property in the smaller strength of a magnetic field than the range where the SdH oscillation is observed. It is also the oscillatory property of the magnetoresitance, if the condition

zyxwv

is satisfied. Here j corresponds to node of the oscillatory property, w is the frequency of the maicrowave and w c = e H / m c is the cyclotron frequency. Prof. Fujii's group has found the dip instead of the oscillation with respect to the strength of the magnetic field." It can be very easily expected that quantum mechanical study must be inevitable for much smaller systems, which is our main interest. Thus, to investigate the more precise properties of nano-sized devices, we should see the behavior of a single electron more closely inside the devices. Fortunately the progress of computers has been astonishing and very

348

zyxwv

intensive calculation that was beyond the reality a decade ago is now available. Quantum mechanical calculation consumes the computational resource tremendously more than classical calculation in general, however, even the direct study of the nodal pattern of the wavefunctions has been made possible lately.12 2. Simulation method

For the computational simulation, we have made the model of the device structure in the 900~270(=243,000)mesh points(fig.1). The shape of a nanowire is just a rectangle of this size. We put the lattice constant a=la.u.(=O.O529nm). Then, our model size of the wire turns out to be 47.9nm x 14.3nm. In this work, a.u. represents the Hartree atomic unit, i.e. me = e2 = fi = 1. We also take a strait, a QR and a circle shaped model in consideration(fig.1). The circle model has the same size as the QR model and no hole inside. They are all molded in the original rectangle area of the size of 900x270 meshes. Outside the area its potential V is given very large value numerically instead of the infinity. The conductance of the models g is evaluated by the Landauer formula13

zyxwv zyx

zyxwvut zyxw

where Tqpis the transmission function from the lead p to the lead q. The transmission function is derived from the matrix formed Green's function. Due to the discretization of the structure of the devices. By virtue of Data's m e t h ~ d the , ~ retarded Green's function can involve the contribution of the leads up to some degree as

G R = [ E I - 'H - CR]-',

(3)

where 'H is the Hamiltonian, and P

is the contribution from the connection between the leads and the device model. Here pi represents the point in the lead which is adjacent to point i in the device. Then we have

S,R(Pi,Pj)= 4

- l

zyxw

c

xm(Pi)e~~(ik,a)X,(pj),

(5)

m

and t = fi2/2ma2, Xm is the m-th eigenfunction of the transverse mode in the lead, k, = J2m(E - &)/ti and + I , ,is the energy of xm. Finally the

zy zyxwvu zyxwv zyxw zy 349

Fisher-Lee relation makes the evaluation of the transmission function pos~ i b l e . ~we > lget ~ the transmission function from the matrix Green function:

Tqp= t$"GRI',GA],

where the formula

and the relation urn = hkm/m is used. We can also calculate the wavefunctions inside the device. Thus the nodal patterns of the wavefunctions can be examined in detail, solving the Schrodinger equation . a@

2-

at

= 7-N,

and the Hamiltonian

'H= 1 2 (-iv+q)2+V.

(9)

We choose the vector potential A in the Landau gauge A = (0, xH,0) to have the static magnetic field H = (O,O, H ) . Note that the Hartree atomic unit is applied in this work. Outside the device models, we put V = lO3Oo that is almost as large as possible for the double precision variable. If it is necessary to simulate the situation of actual experimental condition when the longitudinal resistance is detected, then the scalar potential for the longitudinal electric field is added in the potential V. 3. Nodal pattern analysis

By the simulation on the mesh, the conductance can be enumerated from eq.(2) and eq.(6). We can check that the conductance or the transmission function is clearly quantized in idealistic cases such as the wire and the straight cases. The shape of the nanowire model and, at least, the narrow strait part of the strait model(fig.1) are just rectangular, and integrable in the classical mechanical point of view. On the contrary, the conductance of the circle dot model oscillates violently. The vibration pattern looks partly similar between the circle and the QR models. The sharp staircase disappears even without a magnetic field and represents just the actual upper limit of the conductance(fig.2). Thus it is the effect of the shape of the dots which have the wider part in the middle of the strait part. It

350

0

0.2

0.4

0.6

energy [eV] Fig. 2. Plots of transmission function(eq.(6)) of the nano-device: the strait, the circle and the QR model.

implies that the electron is refracted by the dot in the middle of the device, and it depends on the shape of the dot. In the case of the nanowire and the strait, they have clear nodal patterns that is typical property in integrable systems. They can be divided into the series that are defined by the number of the nodes in the transverse mode. It originates the staircase shape of the conductance(fig.3a,b). It steps up just at the energy where the base state of the new series appears. The base state has one more nodes in the transverse mode than the previous series and only one node in the longitudinal direction. On the contrary, the QR can not define such clear series of the wave functions. In QR, sometimes the wave functions have the regular-look nodal patterns in strait parts that bridge the leads and the dot in the middle. They also often have the irregular and even occasionally localized patterns(fig.3~). Of course, in general, they would not contribute to the conductivity much. Especially the localized wavefunctions, e.g., the lower left of f i g . 3 are ~ robust

351

Fig. 3. Nodal patterns of the wavefunction inside the nano-device. The square of the wavefunctions of (a)nanowire, (b)strait and (c)QR models are shown.

even after connecting the leads to the device12 and they would not be able to contribute to the conductivity. The circle is itself known integrable, though, the circle model has similar non-regular patterns, due to the chaotic property of whole shape of the device including the leads. Therefore the nodal pattern should have the key role in the physical properties of the devices. A static electro-magnetic field drastically changes the nodal patterns of the wavefunctions. Especially the pattern of the circle and QR models are closely investigated here(fig.4). We often can find the

zyxwv

Fig. 4. Nodal patterns of the wavefunctions inside the nano-devices with and without static electro-magnetic field. (a-1) Circle without EM field, (a-2) Circle with static EM field(E=O.OOla.u., H/c=O.OOJa.u.), (b-1) Circle without EM field, (b-2) Circle with static EM field(E=O.OOla.u., H/c=O.OOJa.u.). All sequences of the wavefunctions are from the 19th t o the 24th state.

352

zyxwvutsr

zyxwv zyxwvu z

Fig. 4.

(Continued)

regular circular shaped wavefunctions both in the circle and QR under B certain strength bf the magnetic field. Most of them seem to be attached to the outside wall. We also can discover the wavefunctions that stick to the inner waII of the QR. Moreover, we check that their appearance is very robust against the shape of the inner walls. They are similar to the edge states in the context of the Hall effect. Therefore we call the former states as the inner edge states and the latter states as the outer edge states.

353

zy

4. Discussions

The dip in the magnetoresistance is a recent discovery of Prof. Fujii’s group. It seems to be mainly determined by they size of the radius of the inner hole. The strength of the magnetic field at the dip makes the cyclotron radius closed to the radius of the inner wall. On the other hand the averaged size of circular electron motion measured by the experiments of the AB effect tell us that the radius should be a bit larger. It means that the method detects only the electrons which really contribute to the conductivity and the measured radius of the circle that is just in the middle of the QR. Under the MW drive, some another quantum mechanical effect detects the size of the inner hole. It is not dependent on the frequency of MW, as far as it occurs. The existence of the inner edge states can explain this property of the dip under the MW drive. They can only detect the size of the inner hole, because the probability density of electrons concentrates in the close neighborhood of the inner wall. Thus this kind of wavefunctions do not contribute to the conductivity in general. Under the MW drive, the interaction between these wavefunction and the usual conducting wavefunction must be activated. The interaction would usually be scattering, and then the conductance would become smaller under the MW. The activation of the interaction with the inner edge state makes the longitudinal conductance, then the reistance also becomes proportionally smaller. The relation between two-dimensional longitudinal conductance oxxand resistance pij (2, j = z, y) leads

zy zyxwvu zyxwvutsrq zyxwvuts rxx =

P I X

PZX

+ P&

‘

Then the change of the longitudinal resistance would be proportional to the change of the longitudinal conductance

under the condition pxx > 3, which describe arrays of conservative nonlinear oscillators.

Acknowledgments T. Manos was partially supported by the “Karatheodory” graduate student fellowship No B395 of the University of Patras, the program “Pythagoras 11” and the Marie Curie fellowship No HPMT-CT-2001-00338. Ch. Skokos was supported by the Marie Curie Intra-European Fellowship No MEIFCT-2006-025678. The first author (T. M.) would also like to express his gratitude t o the Institut de M6canique Celeste et de Calcul des Ephkmerides (IMCCE) of the Observatoire de Paris for its excellent hospitality during his visit in June 2006, when part of this work was completed. References 1. Ch. Skokos, T. Bountis and Ch. Antonopoulos, Physica D, 231, 30, (2007). 2. Ch. Skokos, J . Phys. A : Math. Gen., 34, 10029, (2001). 3. Ch. Skokos, Ch. Antonopoulos, T. Bountis and M. Vrahatis, Prog. Theor. Phys. Suppl., 150, 439, (2003).

364

zyxwvut zyx zy

4. Ch. Skokos, Ch. Antonopoulos, T. Bountis and M. Vrahatis, J . Phys. A , 37, 6269, (2004). 5. H. Kantz and P. Grassberger, J. Phys. A : Math. Gen, 21 L127, (1988). 6. T. Bountis and Ch. Skokos, Nucl. Instr. Meth. Phys. Res. A, 561,173, (2006). 7. H. Christodoulidi and T. Bountis, ROMAI Journal, 2, 2, (2007). 8. M. A. Lieberman and A. J. Lictenberg, Regular and Chaotic Dynamics, Springer-Verlag, (1992).

I