Long-Range Interactions, Stochasticity and Fractional Dynamics: Dedicated to George M. Zaslavsky (1935-2008) 7040291886, 9787040291889

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NONLINEAR PHYSICAL SCIENCE

NONLINEAR PHYSICAL SCIENCE Nonlinear Physical Science focuses on recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physic al science and nonlinear mathem atics with engineering applications . Topics of interest in Nonl inear Physical Science include but are not limited to: - New finding s and discoveries in nonlinear physics and mathematics - Nonlinearity, complexity and mathematical structures in nonlinear phy sics - Nonlinear phenomena and observ ations in nature and engineering - Computational methods and theories in complex systems - Lie group analysis, new theories and principles in mathematical modeling - Stabil ity, bifurc ation, chaos and fractals in physical science and engineering - Nonlinear chemical and biological physics - Discontinu ity, synchronization and natural complexity in the physic al sciences

SERIES EDITORS Albert C.J . Luo

Nail H. Ibragimov

Department of Mechao ical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1 805, USA Email: aluo @siue.edu

Department of Mathematics and Science Blekinge Institute of Technolog y S-371 79 Karlskrona, Sweden Email : [email protected]

INTERNATIONAL ADVISORY BOARD Ping Ao , University of Washington . USA; Email: aoping@ u.was hington.edu Jan Awrejcewicz, The Technic al University of Lodz, Poland ; Email: awrejeew@ p.lodz.pl Eugene Bcnilov, University of Limerick. Ireland ; Email: Eugene.Benilov@ ul.ie Eshel Ben-Jacob, Tel Aviv University, Israel; Email: cshclts' tnmar.tau.ac.il Maurice Courbage, Univcrsitc Paris 7, France; Email: mauriee.eourb [email protected] Marian Gid ea , Northe astern Illinois University, USA; Email: mgidea @neiu.edu James A. Glazier, Indiana University. US A; Email: glazicr @indiana.edu Shijun Liao, Shanghai Jiaotong University. China; Email: [email protected] Jose Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto , Portugal; Email: jtm @dec.i sep.ipp .pt Nikolai A, Magnitskll, Russian Academy of Sciences. Russia; Email : nmag @isa.ru Josep J. Masdemont, Univcrsitat Poliiccnica de Cata lunya (UPC). Spain ; Email: jos [email protected] Dmitry E. Pelinovsky, McMaster University. Canada; Email: dmpeli @math .memaster.ea Sergey Prants, Vl.Il'i chcv Pacific Occanological Institut e of the Russian Academy of Sciences, Russia; Email: prants @poLdvo.ru Vlctor I. Shrira, Keele University, UK; Email : [email protected]. uk Jian Qiao Sun, University of California. USA; Email: jq sun @uemereed .edu Abdul-Majid \Vazwaz, Saint Xavier University, USA; Email: wazwaz@sx u.edu Pei Yu, The University of Western Ontario . Canada; Email: pyu @uwo.ea

Albert C.l. Luo Valentin Afraimovich

Long-range Interactions, Stochasticity and Fractional Dynamics Dedicated to George M. Zaslavsky (1935-2008)

With 114 figures

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HIGHER EDUCATION PRESS

~ Springer

Edito rs

Albert C.J . Luo Department of Mechanical and Industrial Engineering Southern Illinoi s University Edwardsville Edwardsville, IL 62026-1805 , USA E-mail : aluo @siue.edu

Valent in Afraimovich IICO-UASLP, Av. Karakorum 1470 Loma s 4a Seccion , San Luis Potos i SLP 782 10, Mexico E-mail : [email protected]

ISSN 1867-8440

e-ISSN 1867-8459

Nonline ar Physical Science ISBN 978-7-04-029188-9 Higher Educat ion Press, Beijing ISBN 978-3-642-12342-9

e-ISBN 978-3-642-12343-6 (eBook)

Springer Heidelberg Dordrecht London New York Library of Congre ss Control Number: 2010924294

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To the memory of George M. Zaslavsky

Preface

George M. Zaslavsky was born in Odessa , Ukraine in 1935 in a family of an artillery officer. He received education at the University of Odessa and moved in 1957 to Novosibirsk, Russia. In 1965, George joined the Institute of Nuclear Physics where he became interested in nonlinear problems of accelerator and plasma physics . Roald Sagdeev and Boris Chirikov were those persons who formed his interest in the theory of dynamical chaos . In 1968 George introduced a separatrix map that became one of the major tools in theoretical study of Hamiltonian chaos. The work "Stochastical instability of nonlinear oscillations" by G. Zaslavsky and B. Chirikov, published in Physics Uspekhi in 1971, was the first review paper "opened the eyes" of many physicists to power of the theory of dynamical systems and modern ergodic theory. It was realized that very complicated behavior is possible in dynamical systems with only a few degrees of freedom . This complexity cannot be adequately described in terms of individual trajectories and requires statistical methods . Typical Hamiltonian systems are not integrable but chaotic , and this chaos is not homogeneous. At the same values of the control parameters, there coexist regions in the phase space with regular and chaotic motion . The results obtained in the 1960s were summarized in the book "Statistical Irreversibility in Nonlinear Systems" (Nauka , Moscow, 1970). The end of the 1960s was a hard time for George . He was forced to leave the Institute of Nuclear Physics in Novosibirsk for signing letters in defense of some Soviet dissidents. George got a position at the Institute of Physics in Krasnoyarsk , not far away from Novosibirsk. There he founded a laboratory of the theory of nonlinear processes which exists up to now. In Krasnoyarsk George became interested in the theory of quantum chaos . The first rigorous theory of quantum resonance was developed in 1977 in collaboration with his co-workers. They introduced the important notion of quantum break time (the Ehrenfest time) after which quantum evolution begins to deviate from a semiclassical one. The results obtained in Krasnoyarsk were summarized in the book "Chaos in Dynamical Systems " (Nauka, Moscow and Harwood, Amsterdam, 1985). In 1984, R. Sagdeev invited George to the Institute of Space Research in Moscow. There he has worked on the theory of degenerate and almost degenerate Hamilto-

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nian systems, anomalous chaotic transport, plasma physics, and theory of chaos in waveguides . The book "Nonlinear Physics : from the Pendulum to Turbulence and Chaos" (Nauka, Moscow and Harwood, New York, 1988), written with R. Sagdeev, is now a classical textbook for everybody who studies chaos theory. When studying interaction of a charged particle with a wave packet, George with colleagues from the Institute discovered that stochastic layers of different separatrices in degenerated Hamiltonian systems may merge producing a stochastic web. Unlike the famous Arnold diffusion in non-degenerated Hamiltonian systems, that appears only if the number of degrees of freedom exceeds 2, diffusion in the Zaslavsky webs is possible at one and half degrees of freedom . This diffusion is rather universal phenomenon and its speed is much greater than that of Arnold diffusion . Beautiful symmetries of the Zaslavsky webs and their properties in different branches of physics have been described in the book "Weak chaos and Quasi-Regular Structures" (Nauka , Moscow, 1991 and Cambridge University Press, Cambridge, 1991) coauthored with R. Sagdeev, D. Usikov, and A. Chernikov . In 1991, George emigrated to the USA and became a Professor of Physics and Mathematics at Physical Department of the New York University and at the Courant Institute of Mathematical Sciences. The last 17 years of his life he devoted to principal problems of Hamiltonian chaos connected with anomalous kinetics and fractional dynamics, foundations of statistical mechanics, chaotic advection, quantum chaos , and long-range propagation of acoustic waves in the ocean . In his New York period George published two important books on the Hamiltonian chaos: "Physics of Chaos in Hamiltonian Systems " (Imperial College Press, London, 1998) and "Hamiltonian chaos and Fractional Dynamics" (Oxford University Press , NY, 2005) . His last book "Ray and wave chaos in ocean acoustics : chaos in waveguides " (World Scientific Press, Singapore, 2010) , written with D. Makarov , S. Prants , and A. Virovlynsky, reviews original results on chaos with acoustic waves in the underwater sound channel. George was a very creative scientist and a very good teacher whose former students and collaborators are working now in America, Europe and Asia. He authored and coauthored 9 books and more than 300 papers in journals. Many of his works are widely cited . George worked hard all his life. He loved music, theater, literature and was an expert in good vines and food . Only a few people knew that he loved to paint. In the last years he has spent every summer in Provence , France, working, writing books and papers and painting in water colors. The album with his water colors was issued in 2009 in Moscow. George Zaslavsky was one of the key persons in the theory of dynamical chaos and made many important contributions to a variety of other subjects . His books and papers influenced very much in advancing modern nonlinear science . Sergey Prants Albert C.J. Luo Valentin Afraimovich March, 2010

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Contents

1 Fractional Zaslavsky and Henon Discrete Maps Vasily E. Tarasov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1.1 1.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Fract ional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 1.2.1 Fractional Riemann-Liouville derivative s . . . . . . . . . . . . . . . . . .. 3 1.2.2 Fractional Caputo derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 1.2.3 Fraction al Liouville derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Interpretation of equations with fraction al derivatives . . . . . . . . 6 1.2.5 Discrete maps with memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 8 1.3 Fractional Zaslavsky maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9 1.3.1 Discrete Chirikov and Zaslavsky maps . . . . . . . . . . . . . . . . . . . .. 9 1.3.2 Fractional univers al and Zaslavsky map . . . . . . . . . . . . . . . . . . . 10 1.3.3 Kicked damped rotator map .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.4 Fractional Zaslav sky map from fractional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Fract ional Henon map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Henon map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.4.2 Fractional Henon map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IS 1.5 Fractional derivative in the kicked term and Zaslavsky map . . . . . . . . 16 1.5.1 Fractional equation and discrete map . . . . . . . . . . . . . . . . . . . . . . 17 1.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Fract ional derivative in the kicked damped term and generalizations of Zaslavsky and Henon maps. . . . . . . . . . . . . . . . . . . . 21 1.6.1 Fractional equation and discrete map . . . . . . . . . . . . . . . . . . . . . . 21 1.6.2 Fractional Zaslavsky and Henon maps . . . . . . . . . . . . . . . . . . .. 24 1.7 Conclusion. . . . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . .. 25 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Self-similarity, Stochasticity and Fractionality Vladimir V Uchaikin

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Contents

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1.1 Ten years ago . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Two kinds of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Dynamic self-similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.1.4 Stochastic self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Self-similarity and stationarity . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2 From Brownian motion to Levy motion. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Self-similar Brownian motion in nonstationary nonhomogeneou s environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Stable laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Discrete time Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Continuous time Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2.6 Fractional equations for continuous time Levy motion . . . . . . 2.3 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Differential Brownian motion process . . . . . . . . . . . . . . . . . . . . . 2.3.2 Integral Brownian motion process . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Fractional Brownian motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Fractional Gaussian noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.3.5 Barne s and Allan model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Fractional Levy motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fractional Poisson motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Renewal processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4.2 Self-similar renewal processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Three form s of fractal dust generator . . . . . . . . . . . . . . . . . . . . .. 2.4.4 nth arrival time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Fractional Poisson distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Fractional compound Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Compound Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Levy-Poisson motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Fractional compound Poisson motion . . . . . . . . . . . . . . . . . . . .. 2.5.4 A link between solutions 2.5.5 Fractional generali zation of the Levy motion . . . . . . . . . . . . . . Ackno wledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix. Fractional operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Reference s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Long-range Interactions and Diluted Networks Antonia Ciani, Duccio Fanelli and Stefano Ruffo 3.1 Long-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1.1 Lack of additivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Equilibrium anomalies: Ensemble inequivalence, negative specific heat and temperature jumps . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Non-equ ilibrium dynamical propertie s. . . . . . . . . . . . . . . . . . . . 3.1.4 Quasi Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 28 29 30 31 32 32 35 40 45 50 51 54 55 56 58 60 61 62 64 64 65 66 68 68 70 70 71 73 74 75 77 77 79

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3.1.5 Physical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.1.6 General remarks and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 3.2 Hamiltonian Mean Field model: equilibr ium and out-ofequilibrium features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 3.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.2.2 Equilibrium statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2.3 On the emergence of Quasi Stationary States: Nonequilibrium dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.3 Introducing dilution in the Hamiltonian Mean Field model . . . . . . . 120 3.3.1 Hamiltonian Mean Field model on a diluted network . . . . .. 120 3.3.2 On equilibrium solution of diluted Hamiltonian Mean Field 121 3.3.3 On Quasi Stationary States in presence of dilution . . . . . . . . . 123 3.3.4 Phase transition 129 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 130 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4

Metastability and Transients in Brain Dynamics: Problems and Rigorous Results Valentin S. Afraim ovich. Mehmet K. Muezzinoglu and Mikhail f. Rabinovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.1 Introduction: what we discuss and why now 4.1.1 Dynamical modeling of cognition . . . . . . . . . . . . . . . . . . . . . .. 4.1.2 Brain imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Dynamics of emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2 Mental modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.1 State space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.2 Functional networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.3 Emotion-cognition tandem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.4 Dynamical model of consciousness . . . . . . . . . . . . . . . . . . . . . . 4.3 Competition-robustness and sensitivity. . . . . . . . . . . . . . . . . . . . . . .. 4.3.1 Transients versus attractors in brain . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Cognitive variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.3 Emotional variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.4 Metastability and dynamical principles . . . . . . . . . . . . . . . . . .. 4.3.5 Winnerless competition-structural stability of transients . . 4.3.6 Examples: competitive dynamics in sensory systems . . . . .. 4.3.7 Stable heteroclinic channels 4.4 Basic ecological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.1 The Lotka-Volterra system . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.2 Stress and hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Mood and cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.4 Intermittent heteroclinic channel . . . . . . . . . . . . . . . . . . . . . . .. 4.5 Conclusion . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . .

133 134 134 135 136 137 137 137 140 142 144 145 146 147 148 148 ISO lSI 153 153 ISS 157 160 161

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Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 167 169

5 Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions Konstantin A. Gorshkov, Lev A. Ostrovsky and Yury A. Stepanyants . . . . 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stable soliton lattices and a hierarchy of envelope solitons . . . . . . . . 5.3 Chains of solitons within the framework of the Gardner model . . .. 5.4 Unstable soliton lattices and stochastisation . . . . . . . . . . . . . . . . . . . . . 5.5 Soliton stochastisation and strong wave turbulence in a resonator with external sinusoidal pumping . . . . . . . . . . . . . . . . . . . .. 5.6 Chains of two-dimensional solitons in positive-dispersion media 5.7 Conclusion Few words in memory of George M. Zaslavsky . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 179 188 193 202 204 212 212 214

6 What is Control of Turbulence in Crossed Fields'!-Don't Even Think of Eliminating All Vortexes! Dimitri Volchenkov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stochastic theory of turbulence in crossed fields : vortexes of all sizes die out, but one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 The method of renormalization group . . . . . . . . . . . . . . . . . . .. 6.2.2 Phenomenology of fully developed isotropic turbulence . .. 6.2.3 Quantum field theory formulation of stochastic Navier-Stokes turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Analytical properties of Feynman diagrams . . . . . . . . . . . . . .. 6.2.5 Ultraviolet renormalization and RG-equations . . . . . . . . . . . . 6.2.6 What do the RG representations sum ? . . . . . . . . . . . . . . . . . . . 6.2.7 Stochastic magnetic hydrodynamics . . . . . . . . . . . . . . . . . . . .. 6.2.8 Renormalization group in magnetic hydrodynamics . . . . . .. 6.2.9 Critical dimensions in magnetic hydrodynamics. . . . . . . . .. 6.2.10 Critical dimensions of composite operators in magnetic hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.2.11 Operators of the canonical dimension d = 2. . . . . . . . . . . . .. 6.2.12 Vector operators of the canonical dimension d = 3. . . . . . .. 6.2.13 Instability in magnetic hydrodynamics. . . . . . . . . . . . . . . . .. 6.2.14 Long life to eddies of a preferable size. . . . . . . . . . . . . . . . .. 6.3 In search oflost stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Phenomenology of long-range turbulent transport in the scrape-off layer (SOL) of thermonuclear reactors . . . . . . . . . .

219 220 221 221 224 226 229 229 233 233 235 238 240 241 242 243 244 249 249

Cont ents

xv

6.3.2

Stochastic models of turbulent transport in cross-field systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.3.3 Iterative solutions in crossed fields . . . . . . . . . . . . . . . . . . . . . .. 6.3.4 Functional integral formulation of cross-field turbulent transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Large-scale instability of iterative solutions . . . . . . . . . . . . . . . 6.3.6 Turbulence stabilization by the poloidal electric drift . . . . . . 6.3.7 Qualitative discrete time model of anomalous transport in the SOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.4 Conclusion . . . .. . . . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . . . .. . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

252 256 259 262 266 267 272 272

Entropy and Transport in Billiards M. Courbag e and S.M. Saberi Fathi 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7.2 Entropy . .. . . . . . ... . . . . . ... . . . . ... . . . . . ... . . . . ... . . . . . ... . . . . 7.2.1 Entropy in the Lorentz gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Some dynamical properties of the barrier billiard model. . . 7.3 Transport. 7.3.1 Transport in Lorentz gas 7.3.2 Transport in the barrier billiard . . . . . . . . . . . . . . . . . . . . . . . . .. 7.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Index

277 277 278 283 290 297 299 301 306 307 310

Contributors

Valentin S. Afraimovich Instituto de Investigacion en Comun icacion Optica, Universidad Autonoma de San Luis Potosi, Karakorum 1470, Lomas 4a 78220 , San Luis Potosi, S.L.P., Mexico, e-mail: [email protected] Antonia Ciani Dipartimento di Fisica, Universita di Firenze, and INFN, Via Sansone I, 50019 Sesto Eno (Firenze), Italy, e-mail: [email protected] M. Courbage Laboratoire Matiere et Systemes Complexe s (MSC), UMR 7057 CNRS et Universite Paris 7- Denis Diderot, Case 7056, Batiment Condorcet, 10, rue Alice Domon et Lonie Duquet, 75205 Paris Cedex 13, France, e-ma il: courbage @ccr.jussieu.fr S.M. Saberi Fathi Department of Physics, University of Wisconsin-M ilwaukee , 1900 E. Kenwood Blvd., Milwaukee, WI 53211 , USA, e-mail : saberi @uwm.edu Duccio Fanelli Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze, Italy, e-mail : duccio .fanelli @unifi.it K.A. Gorshkov lAP RAS, Nizhny Novgorod , Russia, e-mail : Gorshkov @hydro.appl. sci-nnov.ru Mehmet K. Muezzinoglu BioCircuit Institute, University of Californ ia, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0328, USA, e-mail : kerem .muezzinoglu @ gmail.com L.A. Ostrovsky ZeITech/NOAA ETL, Boulder, USA and lAP RAS, Nizhny Novgorod, Russia, e-mail : Lev.A.Ostrovsky @noaa.gov M.I Rabinovich BioCircuit Institute, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0328, USA, e-mail : mrabinovich @ucsd.edu

Contributors

xvii

Stefano Ruffo Dipartimento di Energetica and CSDC, Universita di Firenze, and INFN, via S. Marta, 3, 50139 Firenze , Italy, e-mail: [email protected] Y.A. Stepanyants: Department of Mathematics and Computing, USQ, Toowoomba, Australia, e-mail : [email protected] Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics , Moscow State University, Moscow 119991, Russia, e-mail : tarasov @theory.sinp.msu.ru Vladimir V. Uchaikin Ulyanovsk State University, Ulyanovsk, Russia, e-mail : [email protected] Dimitri Volchenkov The Center of Excellence Cognitive Interaction Technology (ClTEC) , University of Bielefeld , Postfach 100131, D-3350 I, Bielefeld, Germany, e-mail : [email protected]

Chapter 1

Fractional Zaslavsky and Henon Discrete Maps Vasily E. Tarasov

Abstract This paper is devoted to the memory of Professor George M. Zaslavsky passed away on November 25, 2008 . In the field of discrete maps, George M. Zaslavsky introduced a dissipative standard map which is called now the Zaslavsky map. G. Zaslavsky initialized many fundamental concepts and ideas in the fractional dynamics and kinetics . In this chapter, starting from kicked damped equa tions with derivatives of non-integer orders we derive a fractional generalization of discrete maps. These fractional maps are generalizations of the Zaslavsky map and the Henon map. The main property of the fractional differential equations and the correspondent fractional maps is a long-term memory and dissipation. The memory is realized by the fact that their present state evolution depends on all past states with special forms of weights .

1.1 Introduction There are a number of distinct areas of mechanics and physics where the basic problems can be reduced to the study of simple discrete maps. Discrete maps have been used for the study of dynamical problems, possibly as a substitute of differential equations (Sagdeev et aI., 1988; Zaslavsky, 2005; Chirikov, 1979; Schuster, 1988; Collet and Eckman, 1980). They lead to a much simpler formalism, which is particularly useful in computer simulations. In this chapter, we consider discrete maps that can be used to study the evolution described by fractional differential equations (Samko et aI., 1993; Podlubny, 1999; Kilbas et aI., 2006). The treatment of nonlinear dynamics in terms of discrete maps is a very important step in understanding the qualitative behavior of continuous systems described by differential equations. The derivatives of non-integer orders (Samko et aI., 1993) are Vastly E. Iarasov Skobeltsyn Institute of Nuclear Physics, Moscow State Univer sity, Moscow 119991, Russia, e-mail: [email protected]

2

Vasily E. Tarasov

a natural generalization of the ordinary differentiation of integer order. Note that the continuous limit of discrete systems with power-law long-range interactions gives differential equations with derivatives of non-integer orders with respect to coordi nates (Tarasov and Zaslavsky, 2006; Tarasov, 2006). Fractional differentiation with respect to time is characterized by long-term memory effects that correspond to intrinsic dissipative processes in the physical systems. The memory effects to discrete maps mean that their present state evolution depends on all past states. The discrete maps with memory are considered in the papers (Fulinski and Kleczkowski, 1987; Fick et aI., 1991; Giona, 1991; Hartwich and Fick, 1993; Gallas, 1993; Stanislavsky, 2006; Tarasov and Zaslavsky, 2008 ; Tarasov, 2009; Edelman and Tarasov, 2009) . The interesting question is a connection of fractional equations of motion and the discrete maps with memory. This derivation is realized for universal and standard maps in (Tarasov and Zaslavsky, 2008; Tarasov, 2009) . It is important to derive discrete maps with memory from equations of motion with fractional derivatives. It was shown (Zaslavsky et al., 2006) that perturbed by a periodic force, the nonlinear system with fractional derivative exhibits a new type of chaotic motion called the fractional chaotic attractor. The fractional discrete maps (Tarasov and Zaslavsky, 2008; Tarasov, 2009) can be used to study a new type of attractors that are called pseudochaotic (Zaslavsky et aI., 2006). In this chapter, fractional equations of motion for kicked systems with dissipation are considered . Correspondent discrete maps are derived . The fractional generalizations of the Zaslavsky map and the Henon map are suggested . In Sect. 1.2, we give a brief review of fractional derivatives to fix notation and provide a convenient reference . In Sect. 1.3, the fractional generalizations of the Zaslavsky map are suggested . A brief review of well-known discrete maps is considered to fix notations and provide convenient references . In Sect. lA, the fractional generalizations of the Henon map are considered. The differential equations with derivatives of non-integer orders with respect to time are used to derive general izations of the discrete maps. In Sect. 1.5, a fractional generalization of differential equation in which we use a fractional derivative of the order 0 ::; f3 < I in the kicked term, i.e. the term of a periodic sequence of delta-function type pulses (kicks). The other generalization is suggested in (Tarasov and Zaslavsky, 2008). The discrete map that corresponds to the suggested fractional equation of order 0 ::; f3 < I is derived. This map can be considered as a generalization of universal map for the case 0 < f3 < I. In Sect. 1.6, a fractional generalization of differential equation for a kicked damped rotator is suggested . In this generalization, we use a fractional derivative in the kicked damped term, i.e. the term of a periodic sequence of deltafunction type pulses (kicks) . The other generalization is also suggested in (Tarasov and Zaslavsky, 2008) . The discrete map that corresponds to the suggested fractional differential equation is derived. Finally, a short conclusion is given in Sect. 1.7.

3

I Fractional Zaslavsky and Henan Discrete Maps

1.2 Fractional derivatives In this section a brief introduction to fractional derivatives are suggested. Fractional calculus is a theory of integrals and derivatives of any arbitrary order. It has a long history from 1695, when the derivative of order a = 1/2 has been described by Gottfried Leibniz. The fractional differentiation and fractional integration goes back to many mathematicians such as Leibn iz, Liouville, Grunwald, Letnikov, Riemann, Abel, Riesz, Weyl. The integrals and derivatives of non-integer order, and the fractional integro-differential equations have found many applications in recent studies in theoretical physics, mechanics and applied mathematics. There exists the remarkably comprehensive encyclopedic-type monograph by Samko, Kilbus and Marichev, which was published in Russian in 1987 and in English in 1993. The works devoted substantially to fractional differential equations are the book by Miller and Ross (1993), and the book by Podlubny (1999) . In 2006 Kilbas , Srivastava and Trujillo published a very important and remarkable book, where one can find a modem encyclopedic, detailed and rigorous theory of fractional differential equations. The first book devoted exclusively to the fractional dynamics and application of fractional calculus to chaos is the book by Zaslavsky published in 2005 . Let us give a brief review of fractional derivatives to fix notation and provide a convenient reference.

1.2.1 Fractional Riemann-Liouville derivatives Let [a,b] be a finite interval on the real axis lit The fractional Riemann-Liouville derivatives Dg+ and D~_ of order a > 0 are defined (Kilbas et aI., 2006) by (D~+j)(x)

=

D';(i~+a)(x)

1

r(n-a)

D11

1 x

x a

j(z)dz (x-z)a -11+1

(x > a),

(DLj)(x) = (-I )"D~(i~=a)(x) (-1)" D11 r(n - a) x

1· (z-j(z)dz x)a-11+1 b

x

where n = [aj + 1 and [aj means the integral part of tive of order n. In particular, when a = n E N, then (D~+j)(x)

(x 0

L 8jY(tj + r) = L 8je-Htje-H'X(e'etj) ~ L 8je-HtjX(etj) = L 8jY(tj) "

"

j=l

j=l

and if {Y (r),

-00

I

"

"

j=l

j=l

< t < oo} is stationary, then for any positive tl , ... ,tIl

and a

" " " L" 8jX(atj) = L 8ja HtfY(lna + lnrj) ~I L 8ja HtfY(lntj) = L 8ja HX(tj) .

j=l

j=l

j=l

j=l

32

Vladimir V. Uchaikin

Notice, however, that every stationary process is O-ss in the mean.

2.2 From Brownian motion to Levy motion Brownian motion in stationary homogeneous environment and self-similar diffusion with a time- and space-dependent diffusion coefficient are described . Levy-motion is derived as a self-similar generalization of Brownian motion . As a result, we come to fractional differential equations for Levy-stable probability density functions .

2.2.1 Brownian motion The most known self-similar L-process is Brownian motion (Bm) that is the Lprocess with transition pdf

p(x, t)

=

V

I ;;=;-:=

2nt(J

{x - -2 (J t 2

exp

2 } .

It can also be determined as follows . Definition of Bm, The random process {X (r), t ~ O} is called (standard) Brow-

nian motion (Bm), if I) X (0) = 0 almost certainly; 2) {X (r}, t ~ O} is a process with independent increments;

3) X(t+r) -X(t)~rI/2G at any t and r where G is a normally distributed random variable with the variance 2:

Recall that a random process is called Gaussian process if, for any n ~ I and t",t,,_I , ... ,tl, the density p"(x,,,t,,; ... ;XI ,tI) is an n-dimensional normal (Gaussian) distribution. Consequently, the 8m is a homogeneous Gaussian process with independent increments, starting from the origin and having no drift. Let us list first properties of Bm .

1. Finite-dimensional densities It is joint pdf's of B(td , ... ,B(t,,) for 0

< tl < ... < tIl

are given by the products

f(x" ,t,,;... ;x2,t2;xI ,t I) = f(x",t"lx,, -I ,t,,- I) ' " f(X2 ,t2IxI ,tI )f(XI ,tI)

= f(x

ll -

Xll -I

,til -

til -I)'"

exp{ -(I / 2(JJ )[(xll

f(X2 -

XI

.t: - tI)j(Xl ,tI)

x ll _d 2/ (tll - tn-d + ...+ (X2 - xd 2/ (t2 - td +xT;td} (JJ(2n)Il/2[(tll - til -I) ' " (t2 - tl)td I/ 2 -

2 Self-similarity, Stochasticity and Fractionality

33

One can directly verify that the final-dimensional distributions of {B(at) , t ~ O} are identical to the finite-dimensional distributions of {a l / 2B(t), t ~ O} and consequently the process is self-similar with H = 1/2.

2. Covariance function As a Gaussian process the Bm-process is completely defined by its mean values

(B(t )) = 0 and covariance function Cov(B(td ,B(t2)). The latter can be computed directly from the definition. Letting 0 < tl < t: and taking into account (B(t )) = 0 we get

CoV(B(tI),B(t2 )) = (B(tI)B(t2)) = (B(tI) [B(tI) + B(t2) - B(tl)]) = (B(tl )B(tI) ) + (B(td [B(t2) - B(tl)]) = (B2(td) + (B(tl))([B(t2) - B(tdD = cr6t" Consequently, correlation of Bm coordinates at an arbitrary pair of times described by the auto-covariance function

t( , t:

is

The function can be rewritten in a form more convenient for algebra:

When time :

tl

= t: = t then the covariance becomes the variance linearly COV(B(t) ,B(t)) = (B2(t )) =

increasing with

cr6ltl .

Since the function Itll + It21-ltl - t21 is non-negative, the correlations between B(tl) and B(t2) are always positive . This results from independence of Bm increments : the more B(tl) the more (in average) B(t2) = B(tl) + [B(t2) - B(tl )], becau se the increment in square brackets does not depend on the first summand.

3. Hitting time Let us denote by Ta the first time of Brownian motion process hits the point ~ a} and conditioning on whether or not Ta ::; t :

a > 0 and compute P(Ta ::; t) by considering the event {B(t) P(B(t) ~ a) = P(B(t) ~ alTa ::; t)P(Ta ::; r) + P(B(t)

~

alTa > t)P(Ta > t).

Taking into account that

P(IB(t)1 ~ alTa ::; t) =

I,

P(B(t)

~

alTa ::; t) = 1/2 ,

we obtain for the cumulative distribution function

1

00

F7;,(t) = P(Ta ::; t) = 2P(B(t) ~ a) =

2~

cry2nt

a

exp( _x2/ 2cr6t )d.x

34

Vladimir V. Uchaikin

=

(fl""

V7r

a/cro lt

exp(

-Z2 / 2)dz

and for the probability density

a;;=;-=exp [-a2/(2(j6t)]t -3/2, a > 0, t > O.

PTa(t) =

(jOY 21t'

This is the Smirnov-Levy stable distribution density . Because of the symmetry of Bm-process this formula can be extended to the a of an arbitrary sign:

PTc,(t)=

la1-::exp[-a2 /(2(j6t)]t -3 /2,lal >0,t >0. (jOY 21t'

The integral of this density converges to I for any a i- 0, this means that the Brownian particle sooner or later hits any point x E (-00,00). The Laplace transform of the Smirnov-Levy density is of the form

pdA) =

la1-:: ["" exp[-At - a2 /(2(j6t)]t - 3/2dt = exp[-(a / (jo)J2I]

(jOY 21t'

Jo

which shows that the case a = 0 is characterized by degenerated D-distribution:

4. Brownian Sample Paths (I) The Brownian particle being at point x = a at time t will with unit probability visit both regions X a and x < a during any small time interval (t, t + h). (2) With a unit probability, the Brownian particle sooner or later hits any point

XE(-OO,oo) . (3) With a unit probability, the Bm-trajectories are continuous. Mathematically, it is expressed in terms of the Lindeberg condition

lim P(IX(t + r) -X(t)1 ~ .1) /-r = 0 for all A > O.

, ~o

Indeed, on substituting here the one-dimensional Bm density,

P(IB(t + -r) - B(t)1

~ .1 )

-r

- - dx 1"" (-2(j2-r = - - 1"" exp (- Z2) - dz V27i-r 2 =

exp I V27i(jO-r 3/ 2 L1

2)

X

I

L1 / (cro vr)

and applying the rule of L' Hospital lim P(IB(t+-r)-B(t)I r H O

~.1) =Iim_l_~ [""

H OV27i dr JL1 /(crovr)

exp

(_Z2) dz 2

35

2 Self-similarity, Stoch asticity and Fractionality

. L1 exp [-L12 / (2O

2J2ir3 / 2

'

we verify the Lindeberg condition.

2.2.2 Self-similar Brownian motion in nonstationary nonhomogeneous environment 2.2.2.1 Three types of the stochastic integral Let us consider the stochastic equation dX (t) dt

= b(X (r), t g (r),

where b(x ,t) is a non-random function of the - 0 0 < x Broeck, 1997), we take the shock model of the noise

~ (t) =


0 Eq. (2.23) takes the form ap(x,t;a , 1) __ Oa ( . ) at Orpx,t ,a ,l, . where oO~ is given by (AS). It describes the one-sided Levy motion in the positive direction of x-axis .

2.3 Fractional Brownian motion In this section , Brownian motion is interpreted as a random function obeying some stochastic differential equation with white random noise in right hand side. Replacing integer order of the derivative by fractional order opens another way to generalization of Brownian and Levy motions . We obtain fractional Brownian and Levy motions . Their main property is the memory.

55

2 Self-similarity, Stochasticity and Fractionality

2.3.1 Differential Brownian motion process As shown above, correlations of Bm coordinates at an arbitrary pair of times tl .t: are described by the covariance function

Consider the differential Bm process(dBm), i.e. the process of Bm increments

dB(t)

= B(t + dr) -

B(t) , dt = const.

Evidently,

{dB(t)} ~ B(dt), and therefore,

(dB(t )) = 0,

O}B = er6dt .

Autocorrelations in dB(t) are described by the covariance function Cov( dB(tl) ,dB(t2)) which can easily be calculated from correspondent expression for Bm:

Cov(dB(t,) ,dB(t2))

= (dB(t l )dB(t2)) =

a2(B(tl a )B(t2)) a dtldt2 t]

2

d 2 a l (t ] - t2 )d d 25:( = -er6a2(ltll+lt21-lt]-t21)d :l :l tl t: = era :l tl t: = era u tl 2

otlot2

at,

ti

)d d t]

ti -

The differential Bm process dB(t) is an example of stochastic differentials dX (z ). Many authors prefer to write dB(t)=~(t)dt

or even

dB(t) = ~ (r) dt and call equations of such kind stochastic equations, and the "functions" ~ (r) random noises. In this special case, when B(t) represents Brownian motion, the noise ~ (z ) is called the white noise. We shall use for it the notation

As follows from above, the dBm process possesses the following properties. I) Its mean value is zero : 2) It is delta-correlated:

3) The white noise is a stationary stochastic process. 4) Its stochastic integral

56

Vladimir V. Uchaikin

B(t) =

t Jo

dB(t') = lim

maxL1t ~O

[,L1B(ti) = 11

t Jo

B(I )(t')dt'

is a Gaussi an random variable :

l

B(I )(t' )dt'

~ G(O, cr6t ).

The latter property can be generalized to integration of any arbitrary integrable function, namely : the integral

is a Gaussian random variable with the mean

and the variance

b crl = ([l f(t )B(I )(t)dtf) b

= l dt'lb dt2f(td f (tz) \ B(I)(tl )B(I)(tZ)) = cr61bfZ (t)dt .

2.3.2 Integral Brownian motion process A stochastic process

B(-I )(t ) = oltB(t)

=l

B(t' )dt'

is called the integral Bm (iBm) . The iBm proce ss is also a Gaussian proce ss. One can easily verify it by repre senting the integral as a limit of approximation sums

and taking into account that any set of linear superpositions of independent norm ally distributed random variables L1B(tk) = B(tk) - B(tk- l), k = 1,2 ,3 , ... , is jointly normal. At the limit, we have

2 Self-similarity, Stochasticity and Fractionality

57

Since {B (- I) (r), t ?: O} is Gaussian process, it follows that its distribution is completely determined by its mean value and covariance function . They are easily computed and have the form :

(B(-l) (t )) = . : B(t')dt') =

l

(B(t' ))dt' = 0;

Note, that the process {B (- I) (r), t ?: O} is not a Markov process , however, the vector process { {B(- I) (r), B(t) },t ?: O} is again a Markov process . It is a jointly Gaussian with zero mean and covariance

The concepts of stochastic integrals and differentials are generalized to operating with arbitrary (in some sense) random functionsX(t) , Y(t) , Z(t) , W(t) :

1=

lb

Y(t)dX(t),

dW(t) = X(t)dt

+ Y(t)dZ(t).

The simplest (after the Bm) example of such equation is

~;t) = -pX(t) + ~(t) . Interpreting X as the velocity of a Brownian particle we can recognize in - pX the Stocks viscous force . The solution of the equation under condition X(O) = XQ

is the Gaussian process with the mean

and variance

58

VarX(t) =

([l e-,u(t-t')~(t')dt'r) a6 l =

Vladimir V. Uchaikin

e- 2,u(t - t' )dt' =

At each t , X (t) has the normal distribution . In the limit t an equilibrium distribution -00

----+

0 and for all t, we observe

Z(at) ~ aHZ(t). 2. Non-stationarity The autocorrelation function is

2.3.6 Fractional Levy motion Further generalization of the way of inserting hereditarity into self-similar processes is based on using stochastic integrals with respect to the random measure

that describes the random increment of the Levy motion process in (t,t + dz) and

X(t + r) - X(t)

=

lH

dL(a)( r')

~ r ll a S(a,!3 ).

Here, the hereditarity is introduced using the function h(t , r), which determines the contribution of a unite measure at time r to the state of the process at time t :

X(t)

=

i:

h(t ,r)dL(a)(r) .

If the function h (z ,r) is invariant with respect to shift in time,

h(t, r) = h(t - r), such a process is referred to as a moving-average process(MA process). The Ornstein-Uhlenbeck-Levy process can serve as an example of MA process:

63

2 Self-similarity, Stochasticity and Fractionality

Constructed on the same principle, process

with 0 < H < I and H -I- 1/ a, is called the fra ctional Levy motion(tLm), since it is obtained from a Levy motion process by fractional-order integration. Note two important properties of the process {Xfj (z )}. First, it is self-similar with the exponent H, i.e., for any a > 0 and tl , ... .t;

Second, its increments are stationary,

Xfj (r) - Xfj (0) ~Xfj(t + r) - Xfj( r) . In the particular case a = 2, H = 1/2, tLm turns to an ordinary Brownian motion , in the case a = 2, H -I- 1/2 we deal with fra ctional Brownian motion. Its mean value is zero, the variance is

and the covariation function is

The case of H

= 1/2 and

C1 /2( 2

tl, t:

) = { (J2mintr} , t2) , if t] and t: are of the same sign , 0, if t] and t: are of opposite signs.

corre sponds to the ordinary Brownian motion. Since the Bm has stationary increments, the sequence

{z, = xf (j + I) - xf (j) ,

j

= ...,- I ,0, I , ... }

is stationary and is called the fractional Gaussian noise . Its auto-covariance function is

Vladimir V. Uchaikin

64

2.4 Fractional Poisson motion The self-similarity condition being applied to renewal processes leads to fractional generalization of the Poisson process . The link between fractional character of the differential equation and fractal kind of random point distribution is discussed.

2.4.1 Renewal processes The above scheme of the anomalous diffusion process is based on the self-similar generalization of Brownian motion . Historically, it was developed in a different way, using asymptotic analysis of jump processes. The groundwork for this approach was laid by Montroll and Weiss (Montroll and Schlesinger, 1984), and none of the review articles on anomalous diffusion has avoided making a reference to their study (see also an excellent review by Montroll and Shlesinger(Samko et al., 1993)). We note here the main milestones on this avenue using the terminology of the renewal theory (Repin and Saichev, 2000) . Being less formal, this way is more ocular and more productive for physical interpretations in different problems. Let T called the waiting time or interarrival time, be a positive random variable with pdf q(t) and TI, T2 ,' " be a sequence of its independent copies . The new sequence

T(n)

=

n

L Tj,

T(O) = 0,

j =1

will be referred to as the renewal timesor arrival times. In physical processes, some transitions from one state of a system to another, collisions of particles, emission or absorbtion of photons, etc, take such a little time that can be considered as instant transitions. The registered transitions of this kind generate in a measuring electric device a correspondent sequence of current pulses of a very short duration. In many cases, they can be considered as zero-duration pulses. We will call these zero-duration phenomena events or jumps. Let N(t) denote a random number of the events in the interval (O ,t]. In this case, the difference N(t2) - N(tl) means the number of events in the interval (tl' t2]' The random process {N(t) ,t :2: O} is said to be a counting process if it satisfies: (a) N(t) :2: 0; (b) N(t) is integer valued ; (c) N(tl) ~ N(t2) if tl < t2. The function N(t) jump-like increasing at each arrival time is called a counting function . Thus, TN(t ) denotes the arrival time of the last event before t while TN(t )+1 is the first arrival time after t. In these terms, N(t) can be determined as a largest value of n for which the nth event occurs before or at time t :

N(t)

= max{n: T"

~

t} .

2 Self-similarity, Stochasticity and Fractionality

65

In other words, the number of events by time t is greater than or equal to n if and only if the nth event occurs before or at time T: N(t) 2n~T,, ~t .

Feller noted that considering renewal processes we deal merely with sums of independent identically distributed random variables, and the only reason for introducing a special term is using such a power analytic tool as the renewal equation. Let us call the mean number of events by time t (N (t) ) the renewalfunction. It can be represented in the form

(N (t )) =

L P(T(n) < t) = 11 >0

l°

q*" (t') dt' ,

q*o(t) = o(t).

The renewal function is a non-decreasing, finite-valued, non-negative and semi additive function :

(N (t + s)) ~ (N(t )) + (N(s)), It obeys the renewalequation

(N (t )) =

l

t,s 2 0,

[I + (N (t - t' ))]q(t' )dt'.

Its interpretation is very clear: the mean number of events within (O ,t) is equal to the contribution of the first event plus the mean number of subsequent events . For the mean frequency of the events

we obtain from here the similar equation:

f(t)

= q(t) +

l

f(t - t')q(t')dt '.

(2.24)

2.4.2 Self-similar renewal processes Let us try to answer the following question: what form should have transition pdf

q(t) for the process N(t) to be v-ss in medium? In other words, we want to find such q(t ) == 1JIy (r) that (2.25) Following B. Mandelbrot (Jumarie, 200 I), we will call such a set of random points on t-axis the fractal dust and the pdf 1JIy(x) the fractal dust generator (fdg) . As

66

Vladimir V. Uch aikin

follows from (2.24), it is linked with the mean fractal dust density Iv (x) via equation

lJfv (t) = Iv (t) -

l

Iv (t - t') lJfv (t')dt ' .

(2.26)

Applying the Laplace transform

yields the expression

~ ( /\.' ) -_ lJfv

lv(?) ~ 1+ Iv(?)

J1 J1+?v '

(2.27)

which for v = I coincides with the corresponding expression for the ordinary Poisson process :

{2'lJf, (t)}(?) =

---.1!:..." J1+/\.

lJf,(t) =J1e- tu .

Wang and wen of (2003) used formula (2.27) for introducing fractional Poisson processes and derived the fractional integral and differential equations for this density

and oD~lJfv(t)

+ J1lJfv(t) = 8(t) .

2.4.3 Three forms offractal dust generator The solution of the above equations was represented in three forms, two of them were obtained in (Wang and Wen, 2003 ;Wang et al., 2006) . First of them is obtained by performing the backward transition

with the use of the geometrical progression formula

and the relation

67

2 Self-similarity, Stochasticity and Fractionality

Th is leads to the first representation of IfIv (r) in terms of two-parameter MittagLeffler function :

(2.28)

In particular,

1fI1 /2(t) =

~-

2IErfc(p

p2 e /1

yTCt

vt),

where Erfc(t) is the complementary error function : Erfc(t) making use of the formula

=

};r j/'" e-

z2

dz. By

one can verify , that the density (2.28) has really the Laplace transform (2.27). The second form is

IfIv(t) = -I t

j 'OO e-xv(pt /x)dx, 0

where

(2.29)

sin(vn)

v(~) = n [~v+~ -v+2cos(vn)]' It allows with easy to find asymptotical expressions for small and large time :

lfI(t) "-'

u" v -I y-t , (v)_v

vp - v- I { ql_v)t ,

t

--+

0,

t--+ oo •

The third form is given by the next Lemma proved in (Laskin, 2003):

Lemma . The complement cumulative distribution function

P(T > t) =

J OO IfIv(t')dt'

can be represented in the form (2.30)

68

Vladimir V. Uchaikin

2.4.4 nth arrival time distribution For the standard Poisson process, the pdf of the nth arrival time is given by Gamma distribution

I/I*"(t) = J1

( II t )" - 1

r-

(n - I)!

T (I1 )

= T, +...+ T" (2.31)

«!" .

According to the central limit theorem

As numerical calculations show, p (l1 ) (r) practically reaches its limit form already by n = 10. In case of the fPp,

ET = 10'' ' I/Iv (t)tdt = 00 and the central limit theorem is not applicable. Applying the generalized limit theorem (Uchaikin and Zolotarev, 1999), we obtain :

2.4.5 Fractional Poisson distribution Now we consider another random variable: the number of events (pulses) N(t) arriving during the period t . According to the theory of renewal processes

PIl(t)=P(N(t)=n)=P

(

11 LTj >t ) -P

J=I

("+1LTj >t) ,

n =0, 1,2, .. .

J=I

and the following system of integral equations for PIl (r) takes place :

After the Laplace transform with respect to time, we obtain from here

The inversion yields :

69

2 Self-simil arity, Stoch asticity and Fractionality

oD~ PIl(t)

t- V

= J1 [PIl-l (z) - PIl(t)] + 1(1 _ v) 0 0, 0 < V :::; I. 11

(2.32)

This is a master equation system for the fractional Poisson processes. When v it becomes the well known system for the standard Poisson process :

dpll(t) = J1 [PIl-l (r) - PIl(t)] + O(t)OIlO' dt

--+

I

(2.33)

System (2.32) produces for the generating function

L U"p"(t) 00

g(u,t) ==

(2.34)

11=0

the following equation:

oD~ g(u,t) When v

--+

t- V

= J1(u - 1)g(u,t) + 1(1 _ v)

(2.35)

I it becomes the well known equation for the standard Poisson process :

dg(u,t) = J1(u - I )g(u,t) + O(t). dt

(2.36)

Comparing (2.32) with (2.33) and (2.35) with (2.36), one can observe that the equa tions for standard processes are generalized to the equations for correspondent fractional processes by means of replacement of the operator d/ dt with oD~ and of right side the term O(t) with t- V / 1( 1- v) . The solution to Eq. (2.35) is of the form

Applying the binomial formula to each term of the sum and interchanging the summations , one can rewrite it as the series

g(u,t) = I~U 00

11

[all (m+n)!(-a)lIl] n! n{;o m!1(v(mk+ n) + I) . 00

(2.37)

Comparing (2.37) with (2.34) yields a"

PIl(t) = n! n{;o 00

(m+n)! (-a)1Il v m! 1((m+n)v+ 1) ' a = J1t , 0< v :::; I.

This distribution becoming the Poisson one when v = I can be considered as its fractional generalization, called fractional Poisson distribution. The correspondent mean value and variance are given by

70

Vladimir V. Uchaikin

iu"

(N(t )) = r(v+ I) and

(J"2(t ) = (N(t )){ l + (N(t ))[2 1-2VvB(v , 1/ 2) - In .

Table 2.3 shows the properties of Fpp compared with those of the Poisson process . Tab le 2.3 Properties of FPP compared with those of the Poisson process Poisson process (v

IJI(I )

u cr!"

P(n,l )

(J1t)" e -J1t n!

(N (I ))

pI

2

pI

(}N(t )

=

Fractiona l Poisson process (v

I)

< I)

plV-1 e; v(_ p IV) 00 (H Il) ! (_ J1 t v) k (J1t V ) " - Il! - L k=O - k! - n V(H Il)+ I )

J1 t V nv+ l ) V

J1 t r( v+1)

{

J1 t

V

[VII(V. I/ 2)

1+ r(V+I ) ~ - I

]}

2.5 Fractional compound Poisson process The last section joins both kinds of models and leads to bi-fractional (with respect to space and time variables) different ial equations. Their fundamental solutions form a new exten sion of Levy stable distributio ns - the class of fraction ally stable distributions.

2.5.1 Compound Poisson process The Poisson process admits a very simple but productive genera lization called compound Poisson process . The idea of this generalization is based on replacing unit jumps at random arrival times by jumps of random length X U), j = 1,2 ,3, ... at the same times. The random variable s are independent of each other and of arrival times. Consequently, instead of random function N(l )

N(t ) =

LI

j =l

for the Poisson process we have

2 Self-similarity, Stochasticity and Fractionality

X(t) =

71

N(t)

E x U)

j =l

for the compound Poisson process . Let N(t) be the Poisson process with the rate f1 and p(x) , - 0 0 < x < 00, denotes the probability density function for X U), then pdf f(x ,t) for X (r) is represented in the form :

f(x ,t) = e- I·U

00 (f1t)j . E - .,-p*J(x) , t > O.

j =O

J.

This density obeys the integro-differential equation

d! = _ f1f(x ,t) + f1 ) '00 p(x - x')f(x' ,t )dx' = 0, ot

(2.38)

-00

with the initial condition

f(x ,O+) = 8(x)

(2.39)

or, equivalently, the equation

it =

-f1f(x,t) + f1

with the condition

i:

p(x-x')f(x' ,t)dx' = 8(x)8(t)

f(x ,O) = 8(x)8(t) .

(2.40)

(2.41)

The solution can be obtained by passing to characteristic functions :

d!~~,t)

= -f1[i - p(k)]!(k,t) = 8(t) ,

!(k,t) = exp{ -f1t [l - p(k)]}, f(x,t) = - 1 2n

Joo exp{ -ikx -

f1t [1 - p(k)}dk .

-00

2.5.2 Levy-Poisson motion Let us call the Levy-Poisson motion such a compound Poisson process which has a stable distribution of random jumps, that is

p(x) = g(x;a, B), It is readily seen than

and consequently,

ji(k) = exp {-Ikl a exp{ -i( Ban/2)sign k}}.

Vladimir V. Uchaikin

72

Therefore, the strict solution of Eq. (2.1) is represented by the series

Substituting here

a = I,

e = I and taking into account that g(x;

I, I) = 8(x-I) ,

we obtain

f(x,t; I , I) =

f:

f:

(j1.?j e- fl t r I8 (r ' x - l ) = (j1~)j e- fl t 8(x - j) . j=O J . j=O J .

After integrating this expression over small interval (n- £,n+£) , n = 0 , 1,2 , ... , 0 < e < I, we arrive at the ordinary Poisson distribution

From the other side, at the asymptotic of large

(j1t)j fl t L _.,_e... "-' 00

j=O J .

1 00

ut dj8(j - j1t)· · ·

0

and we obtain the Levy motion :

f(x ,t ;a ,e) "-'rl"(x,t ;a , e) = p(x,t)

= (j1t)-I /ag((j1t)-I /ax;a , e) .

As shown above, this pdf obeys the equation

op(x,t ;a ,e) Ot with the initial condition

= L( a ,e )p (x.t: a ,e ), t > 0, p(x,O+) = 8(x),

or, equivalently, the equation

op(x,t Ot;a,e) with the initial condition

= L( a , e) p (x .t; a , e) + u5:(x )5:() u t , t ~ 0, p(x,O-) = o.

(2.42)

73

2 Self-similarity, Stochasticity and Fractionality

2.5.3 Fractional compound Poisson motion Let us come back to Eqs. (2.39)-(2.40) describing evolution of a jump-like Markov process started from the origin at t = 0. It can be rewritten in the form

a

at [f (x,t ) - l +(t)f(x,0)] = Kf(x,t) =

°

(2.43)

where

1+(t)={O, t :::;O, I,

and

t

> 0,

f(x,0)=8(x)

Kf(x,t) = -J1f(x,t) + J11: p(x-x')j(x',t)dx'.

Replacing in Eq. (2.43) differential operator a/at by its fractional version aD;, we arrive at the correspondent generalization of fractional Poisson equation

aD; f(x,t) = Kf(x,t) + f(x,O)cI>v(t) where

(2.44)

t- V

cI>v(t) = oD,I+(t) = r(1 - v) Observe that the presence of function cI>v (z ) in the right-hand side of the equation guarantees against violating normalization. Indeed, because

1 : Kf(x ,t)dx = 1 : [-J1f(x,t) + J11: p(x-x')f(x' ,t)dx'] dx = - J11:f(x,t)dx+ J11:p(x)dxl :f(x' ,t)dx' =0, the condition

must be fulfilled. Following the same way as above we can arrive at the time-space bi-fractional differential equations for a model of fractional Levy-motion being alternative to that considered in Sect. 2.3.6. The correspondent master equation is of the form :

aD; p(x,t; a , e, v) = L( a , e)p(x,t; a , e , v) + 8(x)cI>v(t) , t

~

with the initialization condition

p(x,t) = 0, t < 0. This equation generalizes Eq. (2.42) and takes its form when v ----+ I.

0,

(2.45)

74

Vladimir V. Uchaikin

2.5.4 A link between solutions Let us dwell on a link between solutions of first-order and fraction al (v E (0, I )) which allows to avoid a special computational algorithms. One of the first applications of this approach can be find in the authors works (Ucha ikin and Zolotarev, 1999; Uchaikin, 1999). The space variable x doesn't participate in this transformations and will be tempor arily omitted. Consider equation s

and

df = Kf (t ) + 8(t ). dt Recall, that (13)] . f3 Since the inverse LFT of a concave function is always concave, one cannot recover the initial microcanonical entropy, which displays a "convex intruder". Hence s( £) i:s* (e) . Indeed, one gets the concave envelope of the entropy function as reported in the left side of Fig. 3.3 with the dashed-dotted line. Moreover we note that, at 13(, the left and right derivatives of 1/>(13) (given by £1 and £2 respectively) are different. This is the first order phase transition point in the canonical ensemble . The BEG model displays, indeed, phase transitions, both of first and second order, but the transition curves in the phase diagram obtained within the two ensemble are different, as clearly shown in Fig. 3.4. First, the position of the tricritical points, which connect the first order curve to the second order one, is not the same, thus implying that there is a region in which the canonical phase transition is first order, while the microcanonical one is second order. It is precisely in this region that the specific heat is negative. Again this is a general fact that we here learned with reference to a specific application: The region with negative specific heat develops in correspondence of a first order transition in the canonical ensemble. Back to the BEG model, in the microcanonical ensemble, beyond the tricritical point, the temperature experiences a jump at the transition energy. The two lines emerging on the right side from the microcanonical tricritical point (MTP) correspond to the two limiting temperatures, which are reached when approaching the transition energy from below and from above . Let us now turn to clarifying the ,

" ~ '0

(Z")J - I , n

(3.75)

where n is a assumed to be a real number. The central idea con sists in carrying out the computation for all integers n, extending the results for all n, and performin g in the end the limit for n ----+ O. The HMF replicated partition funct ion reads ZIl

=

[Jndeidpiexp-/3H (ei,Pi)]" N

. /3 /3N j nn detdp jexp [-"2 LL(pj)2 - 4N L L (I -cos(et - ej))] . i= 1

=

11

N

a= l i = l

N

a i= 1

L a (b.,,})

(3.76) Averaging over disorder eventu ally yie lds

122

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

x

LP(Jij)exp [f3 N L L

4NL a (io'!' j)

Jij

=

(I-cos(er -

f3

ej))]

I TITI derdpf exp [- 2 LL(Pf) 2] "

N

a= ll = 1

X

N

a 1=1

N

LP(Jij)TIexr[f3 L(I-cos(er-ej))] Jij

(3.77)

4NL a

ie]

Recalling Eq. (3.73) one then obtains (Z")J

=

I fIn X

=

derdpfexp [-

a= ll= l

I

~ LE(pf)2] a 1= 1

[f3

N Y! 2 I I -2- -I- +---exp -TI i* j [ y! N 2-y y! N2- y 4Ny-1 "

f3

N

TITIderdPfexr[ -2

a= l l= 1

1

J

N

LL(Pf)2] a 1=1

f;; [I-~-I-+~-Iy! N2- y y! N2 - y

xexp ["In

x exp [

La (I-cos(e·a-e·a))]]

:;~!I ~ (I - co s( er - en) J]

l

(3.7 8)

To proceed further we series exp and the exponential and the logarithm and retain the lead ing order in N (Z")J

=

I fI nderdpf a= ll= 1

X

exr[L:L I*J a

So (ln Z)J

exp [ -

~ L E(pf)2] a 1=1

;~ (1- cos(er - en] = Z::,.

(3.79)

= InZ ", and one obtains the corresponding Hamiltoni an (3.80)

y:s:

In conclu sion , and as anticipated, for large enough N values and for I < 2 the diluted system is equivalent to Eq.( 3.26) . In other words, and irrespectively from the dilution amount, system (3.71) is expected to relax asymptotically to the equilibr ium configuration, which is eventually attained by the original, full y coupled, analogue.

3 Long-range Interactions and Diluted Networks

123

3.3.3 On Quasi Stationary States in presence of dilution We will here focus on investigating the emergence of Quasi Stationary States for the HMF model on a diluted network . We shall be in particular concerned with testing the robustness of QSS versus the dilution amount. Are the QSS still present when the average number of links per node is progressively reduced? And, in that case, how the duration time scales with the dilution parameter y? These are the questions that we plan to address in the following . Let us start by clarifying the numerical procedure that will be employed in the forthcoming characterization. The equation of motion of the diluted HMF model can be readily obtained and read

(3.81)

Our numerical implementation relies on the same symplectic 4th-order integrator used for the HMF model (Me Lachlan and Atela, 1992). The timestep here selected is dt = 0.5. The disorder is of the quenched type : The configuration of assigned links is fixed for every realization, without being further adjusted during the simulations. The quantities of interest are averaged over several configurations of the underlying network of contacts. To keep contact with the preceding discussion, we will limit our analysis to the water-bag initial condition specified by Eq. (3.42) . We recall again that, for this specific case the initial magnetization and the energy are obtained from the parameters of the water-bag as in Eqs .(3.43) . We begin our discussion by presenting the results relative to an initially homogeneous distribution (rna = 0). We focus on two different choices of the energy, E = 0.58 and e = 0.69, respectively below and above the transition line point f c = 7/12. Consider first the case e = 0.69, which in the fully connected scenario (y = 2) is shown to yield to an almost demagnetized QSS. In Fig. 3.21 we report on the temporal evolutions of the magnetization as recorded in our numerical simulations, for different values of the dilution , and by varying the system size (three choices of N in each panel, respectively N = SOD, 1000, 2000). Several observations can be made, as it follows from a straightforward qualitative inspection of the figures. On the one hand the QSSs do exist even in presence of dilution . The magnetization settles down into an intermediate characteristic plateau which is eventually maintained for long times (notice the logarithmic scale on the x-axis), displaying a sensible dependence on the number N of simulated particles, as we shall be commenting in the following . On the other hand, the value of the magnetization associated to the QSS regime of Fig. 3.21 is shown to decrease when increasing the system size. We hence argue that the QSS is of the homogeneous type, as it is indeed found for the fully connected reference case for the same choice of parameters (s > f c.). As a final comment, we also stress that the asymptotic value of the magnetization is independent on the specific choice of the dilution and compat-

124

An tonia Ciani, Ducc io Fane lli and Stefano Ruffo

y =1.7

0--0

-

S

e

N =500 N =1000 N =2000

0.2

0. 1

00

10

100

1000

10000

t

Y= 1.5

0.4 0--0

0.3

-

N =500 N =2000 N =5000

S 0.2

s

0. 1

Fig. 3.21 Temporal evol ution of the magnetiza tio n m(t ) for different particle numbers N = 500 , 1000, 2000, 5000. Different panels refer to disti nct choices of y. Horizontal solid lines represent the equi lib rium value meq c::: 0.3 . The energy of the system is always set to e = 0 .69 , and mo =O

3 Long-range Interactions and Diluted Networks

125

ible, within statistical fluctuations due to the finite number of realizations, with the equilibrium value calculated for the fully connected case (m eq '" 0.3, solid lines in the figures). This finding confirms in turn the adequacy of the theoretical argument developed in Sect. 3.3.2. In Fig. 3.22 we reorder the simulation outputs so to appreciate how the dilution r affects the QSS lifetime . The more the system is connected the longer the QSSs survive. We emphasize that, as the dilution takes the lowest value here considered (namely r = 1.5), the QSS lifetime gets reduced by two orders of magnitude, with respect to the corresponding fully connected case. While QSS are certainly present when dilution is accounted for, they tend to progressively reduce their duration as r approaches the limiting value I, where they formally fade off.

N =1000 0.4

0.3

S

E

'-'y =1.5 - y= 1.6 - y =1.7 -y=1.8 -fully

0.2

0.1

0\

100 t

10000

N =2000 0.4

0.3

S

E

.-.y = 1.5 - y =1.6 - y = 1.7 - y =1.8 -fully

0.2

0.1

0\

100

t

Fig. 3.22 Temporal evolution of the magnetization m(t) for different values of yand for N (upper panel) and N = 2000 (lower panel). The energy of the system is £ = 0.69.

=

1000

Antonia Ciani, Duccio Fanelli and Stefano Ruffo

126

In order to quantify our observation we turn to measuring the QSS lifetime TQSS via the very same fitting procedure as introduced in Sect. 3.2.3.5. We here recall that the sigmoid profile (3.44) can be numerically superposed to the simulated curves, by properly tuning the free parameters a(N),c(N) and d(N) . Result of the analysis are displayed in Fig. 3.23, where TQSS is plotted as a function of N in log-log scale, for different choices of y (symbols). The data follow a power-law trend , the exponent (slope of the linear profiles) being sensitive to the selected value of y. Starting from this observation, it would be extremely interesting to elaborate on a reasonable ansatz, physically motivated, which is capable of reproducing the scaling observed for y < 2, while converging to the well-known N1.7 solution as the y ----+ 2 limit is performed. We stress again that the 1.7 factor is being suggested to apply when homogeneous QSS are concerned, but our simulations as reported in Sect. 3.2.3.5 seem equally compatible with the more sound 1.5 choice . We introduce again a to label such a controversial exponent, regardless of its specific numerical value. A rather natural proposal would be to replace in the aforementioned relation the global number of degrees of freedom N, with the effective quantity Neff = N (y-l ), which quantifies the average number of links per node. Under this assumption a Na (y-l ) (382) TQsS 0--

2

State

2

3

State

=

~

4

EGHE

4

:x13 ~

4

2

3

BE

4

State

2

4

3

State

3

10 9

8 "" 7 ~6 u 5 Z4 3 2 I

~--'.=!:---=-~=-...J

20

2

5

5

Firing Rate (Hz)

-

-

-

4 10 10 Firing Rate (liz)

20

"4

2

Firing Rate (liz)

20

5

:5

2

Firing Rate (liz)

Fig . 4.7 Neuro ns in the rat's gustatory cortex generate a taste-specific sequential pattern . The top row shows the sequential WLC activity among 10 cortex neurons in response to four taste stim uli. A Hidden Markov Model (HMM) of joint temporal activity (the ticks denoting the action potentials) reveals that the network behavior is best represented by four discrete states in a winnerless competition setting . The dashed horizontal line denotes the threshold, above which (with probabil ity 0.8) the network is considered to be occupy ing the correspond ing state (i.e., the state becomes the winner) . The second row lists the outcome of the four replicates of the previous experiment on the same network and confirms the reproducib ility of the sequential activity: the order of the observed states is the same in each trial. Note, however, that the switching times are irregular. The translation of the four HMM states into firing rates for each stimul us are given on the last row. (Figure adopted from Jones et aI., 2007).

4.3.7 Stable heteroclinic channels The mathematical object that represents a reproducible transient activity is the Stable Heteroclinic Channel (SHe), consisting of saddle sets, their vicinities, and the pieces of trajectories connecting them. A SHC is characterize d by two properties: (i) the conditions on the structural stability of the SHC, and (ii) the relatively long (but finite) passage (or exit) time that the system spends in the vicinity of a saddle in the presence of moderate noise.

Valentin S. Afraimovich et al.

152

, ,

I

1_

,,'.11 '11 ' 111 1

(a)

•

(b)

I

,

_

"

. 1 111 111 11

,

" ; " lilli' 111." : .', ',: " I 1. 1

.,

I

1I ~ 1 1I ~

,I I , II

,,~ :

1"

(c)

(d)

Fig. 4.8 Spatio-temporal representation of the sensory information in the locust olfactory system (antennal lobe): (a) Schematic of an insect antennal lobe sectioned through its equatorial plane; (b) response of 110 neurons of the antennallobe to an odorant that lasted 1.5s; (c) projection of the neural activity on 3D principal component space (black trajectory is the average of 10 different experiments); (d) diversity of heteroc1inic sequences in the same ensemble of neuronal group .

Let us consider a channel that consists of saddles each having one-dimensional unstable manifolds, i.e., a separatrix leading to the next saddle . To obtain the condition on the channel's stability, we must consider elementary phase volume in the neighborhood of each saddle that is compressed along the stable separatrices and stretched along the unstable separatrix . Let us order the eigenvalues of saddle i as

Ai > 0 > Re { Ad}

~ Re {A~} ~

... ~ Re {A,~} .

(4.1)

4 Met astability and Transients in Brain Dynamics

153

The number (4.2) is called the saddle value. If Vi >

I,

(4.3)

then the compression along the stable manifolds dominates the stretching along the unstable manifold, and the saddle is called as a dissipative saddle . If all saddles in the heteroclinic chain are dissipative, then the trajectories in their vicinity cannot escape from the chain, providing stability. In the absence of noise, the state vector approaching to a saddle along a stable manifold is confined to the neighborhood of the saddle indefinitely. The exit from a saddle 's neighborhood is possible only due to a strong enough perturbation . The dependence of the exit time on the noise level was studied in (Kifer, 1981) and (Stone and Holmes , 1990). A local stability analysis around a saddle fixed point results in the relation

.= Ai1 (1) T11f '

t'

In

(4.4)

where '[;i is the mean time spent in the neighborhood of saddle i (provided that initial points belong to the stable manifold), i.e., the life time of a metastable state, and 111 1is the level of noise. In the framework of a specific model , one can derive the inequalities on the model parameters using the conditions (4.2) and (4.3) that guarantee the stability of heteroclinic channel.

4.4 Basic ecological model 4.4.1 The Lotka-Volterra system The competition within and between cognitive and emotional modes can be described by the Generalized Lotka- Voterra (GLV) model (Lotka, 1925), given by (4.5) Here X i ;::: 0 is the i-th competing agent , E is the input that captures all (known) external effects on the competition, '[; is the time-constant, pi's are the increments that represent the resources available to the competitor i to prosper, qJij is the competition matrix , and 11 (z ) is a multiplicative noise perturbing the system . The system (4.5) has many remarkable features, see Appendix 2. Depending on the control parameters' ratio, this model can describe a vast diversity of behaviors. When connections are nearly symmetric, i.e., qJij ;::::: qJji, two or

Valentin S. Afraimovieh et al.

154

more stable states can co-ex ist, yielding a multi-stable behavior - the initial condition determines the final state. When the connections are strongly non-symmetric, a heteroclinic contours or limit cycles in their vicinities can emerge. Dynamical chao s can be observed in this case (Muezzinoglu et al., to appear) . A specific kind of dynamical chaos , where the order of the switching is deterministic, but the life-time of the metastable states is random , can also be observed (Varona et al., 2004) . We think that such reproducibility of metastable states ' order, despite the irregularity in timing, can be interesting for the processing of observed data. For a given model , the values of the control parameters that ensure the stability of the transients can be obtained from inequalities to be derived from (4.1) and (4.2). In (Afraimovich et al., 20 I0) such conditions have been generalized in the case of weakly-interacting subsystems like (4.5). As we already discussed , cognition and emotion are strongly connected. Nevertheless, it is reasonable to suppose that the modes within one family are more strongly connected than the modes between these two families . That means, one can consider that, one family does not "destroy" the dynamics of the other family, but modulates it. In particular, cognition support emotional equilibrium, whereas emotion excites or inhibits cognition. Therefore, it is natural to describe their interaction based on coupled subsystems of type (4.5). Taking also into account the dynamics of resources, we should write a third set of equations, describing the resource modes (i.e., attention, memory, and energy) . It is important to emphasize the special role of attention in this interaction: it selects the sensory cues that are critical for current decision making process . Based on experimental evidence, the dynamics of attention can also be described by a competition among informational entities . For the sake of simplicity, let us consider that these entities are total emotional i3 = L~l BJt) and total cognitive A: = L~, Ai(t) activities . Finally, we write the model in the following form .

"fA ' :tAi(t) = Ai(t) [(}i(I,B,D) ·RA-

E,

Pij(D)Aj(t)] +Ai(t) · T/(t),

(4 .6)

i= 1, ... ,N ,

te-:tBi(t) = Bi(t) [Si(S,A,D) ·RB-

~ ~ij(D)Bj(t)] +Bi(t) · T/(t) ,

(4 .7)

i= 1, ... ,M ,

"fRA' : /A(t) = RA(t) [A: - (RA(t) + tPA(I,D)RB(t))] ,

(4 .8)

"fRA' :/B(t) = RB(t) [i3 - (RB(t) + tPB(S,D)RA(t))] .

(4 .9)

The nonnegative variables Ai and Bi, as described above, correspond to the cognitive and emotional modes , the union of which are denoted by A and B respectively . The proposed model is merely a formul ation of the competition within and among these two sets of modes . Both of these modes are open to the external world through the quantities / and S, denoting the cognitive load and the stressor, respectively, and

4 Met astability and Transients in Brain Dynamics

155

D is the control parameter characterizing the medication. The coupled processes evolve on time scales determined by the parameters 'LA and 'LB. Both processes are open to the brain noise, which appears as the multiplicative perturbation 11 (t) in the equations. The variables RA and RB characterize the resource dynamics, where rf!A and rf!B control the level of competition for resources . The competition within cognitive and emotional modes are regulated by the selfexcitations (J and S, and by the competition matrices p and c, and by the time constants 'LA and 'LB, respectively . Note that the GLV equations modeling the two coupled processes has a rich repertoire of dynamical behaviors. Therefore, the choice of the triples ((J , p, 'LA) and (S, ~ , 'LB) determines not only the quantitative attribute s (i.e., time scales, transient and/or steady-state characteristics), but also the qualitative nature of each behavior. In fact, the cognitive and the emotional brain processes have different qualities : the former is usually characterized as a sequentially ordered brain activity advancing on a regular pace, whereas the latter is a highly variable, fast, and sometimes unpredictable activity. Based on these observations, the suitable operating regime for a (healthy) cognitive process is the stable heteroclinic chain. There is no particular constraint posed at this point on the quality of emotional dynamics ; it can follow also a heteroclinic sequence with a short switching period, a recurrence behavior or a strange attractor. Cognitive and emotional brain processes have different qualities : the former is usually characterized as a sequentially ordered brain activity advancing on a regular pace , whereas the latter is a highly variable, fast, and sometimes unpredictable activity. Based on these observations, the suitable operating regime for a (healthy) cognitive process is the stable heteroclinic chain . There is no particular constraint posed at this point on the quality of emot ional dynamics; it can follow also a heteroclinic sequence with a short switching period , a recurrence behaviour or a strange attractor.

4.4.2 Stress and hysteresis In this example, as adopted from (Rabinovich and Muezzinoglu, 2009), an auxiliary stressor S triggers these emotions, which in turn disrupts an ongoing cognitive sequence. Thus, the simulation demonstrates the feed-forward chain of events : S =} negative emotions =} cognitive disruption. Let us consider N = 5 cognitive modes and M = 5 emotional components. The multiplicative perturbation 11 (r) is a white noise with variance 10- 8 and 10- 3 for the cognitive and the emotional dynamics, respectively, and the time constants are 'LA = 'LB = 20. Without loss of generality, we prescribed the finite heteroclinic sequence of saddles el ----+ e2 ----+ ••• ----+ es for the emotional modes. The mode es is a stable attractor (i.e., without any unstable manifold so that the system is confined to the vicinity

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of es once it enters its domain of attraction). This state marks the terminal cognitive mode, such as the execution of a certain coping strategy, whereas the preceding modes denote the cogn itive tasks that lead to this resulting activity. They could be named, for instance, as perception, appraisal, evaluation, and selection, in their order of appearance in the sequence. The feasible values of Pij that can establish the desired heteroclinic skeleton in the A network constitute a broad continuum in the parameter space. A set of sufficient conditions that determine a part of this region in the form of simple inequalities on o, and Pi) can be found in (Afraimovich et al., 2004b) . Following these conditions, we set Pii = 1.0 for i E {I, ... , 5}, Pi-l ,i = 1.5 for i E {2, ... , 5}, Pi,i+l = 0.5 for i E {1, ... ,4 }, and Pij = Pj -l ,j+2 for j E {2,3,4} and i rf:- j -I ,j,j+ I. In this illustration, the five emotional modes were organized as a heteroclinic sequence, yet as a cyclic one by introducing es ---+ el transition. We note that we do not necessarily name the emotional components individually, but interpret their mean activity as the degree of anxiety, a negative emotional state. In this respect , the precise dynamical quality of the emotional network is not of primary consideration in our design; for the sake of our illustrations, the emotional behavior could have been realized simply as a limit cycle , or as a strange attractor. The ~ij was evaluated as done above for Pij , yet taking into account the es ---+ el transition, which results in ~s ,s = 0.5. We disregard any transient drug effect in the simulations, thus assume that D, thus both matrices P and ~ are fixed. These matrices configure the competition within the cognitive and the emotional modes . The interaction between them are regulated by the choice of the increment functions a and S, as well as through the resource competition (4.8) and (4.9) . All five increments o, in the cognitive process were modeled as I - L=l Bi(t), i.e., inversely proportional to the total (negative) emotional activity. The increments Si for the five emotional modes were considered as independent of the cognitive activity in this example; they were all equal to the externally applied stressor quantity S, which was assumed to be non-negative. The resource competition RA vs. RB is regulated by Eqs. (4.8) and (4.9) with parameters CPA = CPB = 0.3 and random initial conditions. With the selected parameters, the integration of the ordinary differential equa tions were performed by the Milstein approximation. The results shown in Fig. 4.9 were obtained. The figure illustrates the suppression of and delay in the cognition due to the emotional activity, which is induced by an external stressor. An interesting prediction that can be derived from the model is the contrast in the switching regimes of the total activity in the cognitive and the emotional network during the rising and decay periods of S. This can be better observed in Fig. 4.10 .

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4 Metastabi lity and Transients in Brain Dynamics

'tr:xL

.~ iO:~

u

0

0

100

200

300

400

500

600

100

200

300

400

500

600

:E

n:: 700

800

900

1000

700

800

900

1000

u -
ReA(i) > ... -> ReA(i) 3 d

We will use the saddle value Vi

=

-ReAdi)

A(i)

(4.11)

I

The saddle Qi is dissipative if Vi > I . It means that a displacement from the stable manifold of Qi becomes much smaller after going through a vicinity of Qi (Shilnikov et al., 1998) and (Shilnikov et al., 200 I). Definition A2. The heteroclinic sequence T is called the stable heteroclinic sequence (SHS) if (4.12) Vi > I, ... ,N. It was shown in (Afraimovich et al., 2004a,b) thatthe conditions (4.11) and (4.12) imply stability of F , in the sense that every trajectory started at a point in a vicinity of QI remains in a neighborhood of T until it comes to a neighborhood of QN. In fact, the motion along this trajectory can be treated as a sequence of switching between the equilibria Qi, 1,2, ... , N . Of course , the condition r; + C WQi+ 1 indicates the fact that the System (4.10) is not structurally stable and can be only occurred either for exceptional values of parameters or for systems of a special form . As an example of such a system one may consider the generalized Lotka- Volterra model (4.5) (see (Afraimovich et al., 2004a,b». In the space of the generalized Lotka- Volterra models, the occurrence of heteroclinic connections is a structurally stable event.

Stable heteroclinic channel We consider now another system, say,

x = Y(x),

x

E jRd ,

(4 .13)

that also has N equilibria of saddle type QI , Q2, ... , QN with one-dimensional unstable manifold WQi = r; + Ur; - U Qi, and with Vi > I, i = 1, ... , N . Denote by Vi a small open ball of radius e centered at Qi (one may consider, of course, any small

4 Metastability and Transients in Br ain Dynamics

165

neighborhood of Qi) that does not contain invariant sets but Qi. The stable manifold WQi+ ! divides Vi onto two parts: V/ containing a piece of Ij + and another one, V i-.

Assume that Ij + n ir;

I

:f. 0 , i

= 1,2, ... ,N -

I , and denote by Ij ;+I the connected

r;\ Ujf-iVt containing Qi. We assume that Ij ;+1 coincides with the connected component of r;\ Viti containing Qi and that Ij ;+1 n tr = 0 if

component of

j:f. i, i + I . Denote by 0 0 (Ij ; +I ) the 8-neighborhood of Ij ;+1 in jRd.

uI:,1

)UJ=I

v:

Definition A3. Let V(c , 8) = 0 0 (Ij ;+1 We say thatthe System (4 .13) has a stable heteroclinic channel in V (e, 8) if there exists an open set V \ V~ of initial points such that for every Xo E V there exists T > 0 for which the solution x(t ,xo), 0 :::; t :::; T, of (4 .13) satisfies the following conditions: i) x(O,xo) = xo, ii)for each 0 :::; t :::; T, x(t ,xo) E V(c, 8), iii)for each I :::; i :::; N there exists t, < T such that X(ti'XO) E

vt

Thus, if E and 8 are small enough then the motion on the trajectory corresponding to x(t ,xo) again can be treated as a sequence of switching along the pieces Ij ; +1 of unstable separatrices between the saddles Qi, i = I, ... ,N. It follows that the property to possess a SHC is structurally stable: if a System (4 .13) has a SHC then a C l - close to (4 .13) system also has it. We prove this fact here under additional conditions. Denote by Ij t c the inter-

vt

section r i i+1 n It is a segment for which one end point is Qi while the other one , say Pi, belongs to oV/, the boundary. Let W/ foc := WQi n Vi, the piece of the

stable manifold of Qi and Vi(Y) := 0y(W/foJ n V/ ' Y < e, where Oy(B) is the yneighborhood of a set B in jRd. The boundary OVi(y) consists of W ; loc: a (d - 1)dimensional ball , Bi, "parallel to" W ; foc and a "cylinder" homeomorphic to Sd-2 x I where Sd-I is the (d - 2)-dimensional sphere and I is the interval [0, 1]. Denote by Ci( y) this cylinder. The proof of the following lemma is rather standard and can be performed by using a local technique in a neighborhood of a saddle equilibrium (see (Shilnikov et al. , 1998,200 I; Afraimovich and Hsu, 2003)).

Lemma At. There is 0 < Co < I such that for any e < Co and any I :::; i :::; N there exist Ci < E and I < J1i < Vi for which the following statement holds: if fi :::; Ci Xo E Clfi) then (4 .14) where "dist' is the distance in jRd, t' > 0 is the time and x( 1:i,XO) is the point of exit of the solution of (4.13), going through xo,from

vt

rt.;

A segment has two end points: one of which is ~ and another one, say Ri+1 E oVitl ' Fix C < Co.

Lemma A2. There exist members 11 < )1, then : i) there is t;

x, >

> 0 such that x(t; ,xo)

I and v;

E OVitl'

> 0 such that ifxo E

Oy;(~) , 0


0 such that every system x=Y(x)+Z(x)

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4 Met astability and Transients in Brain Dynamics

where IIZllcl < (J also has a SHC in V(s, 8), maybe with a smaller open set U of initial points. The proof of the corollary is based: i) on the fact that the local stable and unstable manifolds of a saddle point for an original and a perturbed system are CI-close to each other; ii) on the theorem of smooth dependence of a solution of ODE on parameters and iii) on the open nature of all assumptions of Theorem A I . The conditions R, E C, (et+ I) with ei « I looks rather restrictive, generally. Nevertheless, for an open set of perturbations of a system possessing a SHS, they certainlyoccur. Theorem A2. If a System (4.13) has a SHS then there is an open set 'f,f in the Banach space of vector fields with the C r-norm such that the system

i=X(x)+Z(x) has a SHC, for every Z E 'f,f . See the proof in (Rabinovich et al., 2008b). So, an orbit stays in a SHe until it goes out of a vicinity of QN. Then it could go to an "inner part" of the phase space and after some time come again to the same or other SHe. Some numerical simulations show that such a behavior indeed occurs in GLYM. GLYM. It is even more observable for the case of intermittent heteroclinic channels. Qualitatively it is very similar to the behavior of a trajectory of a Hamiltonian system processing sticky sets (Zaslavsky, 2005 ; Afraimovich et al., 2004a) . It is an interesting problem to study the similarity and difference of statistical features for both cases.

Appendix 2 We are going to investigate a transient multispecies competition in the framework of the following form of GLYM: (4.19) Here each ai(t) ?: 0 represents an instantaneous density of the i-th specie 's, ?: 0 is the interaction strength between species i and i . (Ji(E) is the growth rate for species i that depends on the environmental parameter E «(JJ Pii) is the overall carrying capacity of species i in the absence of the other species ; T/i is environmental Pij

noise. The product a, [(Ji(E) + T/i(t )] determines the interaction of the species i with the environment. We will consider a non-symmetrical species interaction,

Pij

-I- Pji .

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A Generalized Lotka- Volterra model in the absence of noise

°

The system (4.19) provided that 11i(t) == is a simplest lattice dynamical system with quadratic inhibitory couplings. It has many remarkable features . Let us list some of them (Afraimovich et aI., 2004b) . (i) It is a dissipative system that is simple to see fixing the ball of dissipation 1l

La] = R 2 , where R is large enough . j

(ii) It has n invariant hyperplanes a, = 0, i = I, ... .n, and many invariant linear subspace formed by the intersections of some of them. (iii) Because of that, it is simple to calculate the eigenvalues of the matrix of the linearized at the equilibrium S, = (0 0 · · · ()j 0 · · · ) system (we set ()j(E) = ()i). They equal to - o, and ()j - Pji()i, j = I, ... , n, j i- i. Thus , if all of them are negative then S, is the stable node, and if ()jl - Pj;i()i > for some ji then S, is the saddle point. (iv) Assume that it is true and consider a restriction of the system onto the invariant plane ~ji = {aj = O}.lt has the form

n

°

f hl ,i

The point (()i, 0) is the equilibrium saddle point , and the system has no other equilibria in the positive quadrant provided that I - Pijl P N i- 0. Therefore, the unstable separatrix has no choice but to go to the equilibrium (0, ()jl) that is the stable node, and the phase portrait in the positive quadrant is very simple : all trajectories except the stable separatrices of the saddle (()i , 0) and the unstable node in the origin go to the node (0, ()jl ) as t ----+ +00.

Stable heteroclinic sequence Selection of saddles. We look for the conditions under which the system (4.19) has a SHS consisting of saddles Sk = (0, ... ,0, ()ik ' 0, ... , 0) linked by heteroclinic trajectories, k = I , ... ,N ~ n. The saddles Sk have the following increments (eigenvalues of the linearized system at Sk): ()j - PPk ()ik' j i- ik, and -()ik · The saddles Sk = (0, ... ,0, ()ik' 0, ... ,0), k = 2, ... ,N are selected in such a way that : there is one positive eigenvalue, and the rest, are negative . Then the following inequalities are verified (4.20) and the other eigenvalues are negative .

4 Metastability and Transients in Brain Dynamic s

169

Heteroclinic connections. To assure that there is a heteroclinic orbit fik-1 ik joining Sk-I and Sb as it was said, the following condition has to be satisfied (4.21) This orbit belongs to the plane Pik-l ik = n j""ik_1,ik{a j = O}, where the point Sk has a I-dim strongl y unstable direction (determ ined by ik+ d. Leading directions. Under the following conditions (4.22) and (4.23) the separatrix fik-1 ik come s to Sk follow ing a leading direction, transversal to the on the plane ~k-l ik (Afraimovich et al., 2004b).

a ik -axis

Dissipativity of saddles. The saddle value (4.24) is defined for every saddle Sk . We assum e that Vik >

I, k= l , oo . ,N.

(4.25)

It means that every saddle Sk is "dissipative" . It was shown in (Afraimovich et al., 2004a,b) that if all saddles have onedimensional unstable manifolds, then under the conditions above, the SHS consi sting of the saddles Sk and joining them separatrices fik - I ik is stable in the following sense: if one choo ses a positive initial cond ition in a small neighborhood of So, the trajectory going through it will follow the sequence {fik - 1i k } , staying in a small vicinity of them until it come s to a neighborhood of the last saddle SN. In anoth er words the system possess a SHC in a vicinity of this SHS.

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Chapter 5

Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions Konstantin A. Gorshkov , Lev A. Ostrovsky and Yury A. Stepanyants

Abstract A brief review of soliton dynamics constituting one-dimensional periodic chains is presented. It is shown that depending on the governing equation, solitons may have either exponential or oscillatory-exponential decaying tails. Under certain conditions, solitons interaction can be considered within the framework of Newtonian equations describing the dynamics of classical particles . Collective behaviour of such particles forming a one-dimensional chain may be simple or complex and even chaotic. Specific features of soliton motions are presented for some popular models of nonlinear waves (Korteweg-de Vries, Toda, Benjamin-Ono, KadomtsevPetviashvili, and others) .

5.1 Introduction One of the key topics of George Zaslavsky's research has been transition from regular motions to chaotic ones and the chaotic behavior of dynamic systems . He developed this approach from relatively simple nonlinear oscillators (starting to work with B.Y. Chirikov) to quantum chaos and ray chaos in acoustics . The brief review below deals with the complex dynamics of "wave particles " - solitons which can form one-d imensional "chains" and two-dimensional "lattices" and chaotic ensemK.A. Gorshkov Institute of Applied physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia, e-mail : Gorshkov @hydro.appl.scin-nov.ru L.A. Ostrov sky ZelTech/NOAA ETL, Boulder, USA and Institute of Applied physics of the Russian Academ y of Sciences, Nizhny Novgorod , Russia, e-mail : Lev.A.Ostrovsky @noaa.gov Y.A. Stepanyants Department of Mathematics and Computmg, faculty of SCIences, University 01 Southern Queensland, Toowoornba, Australia, e-mail : Yury.Stepany ants@usq .edu.au

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bles and still preserve their identity upon interactions. In general this topic is extremely broad . Here we limit ourselves by using an asymptotic approach based on the assumption that interacting solitons are sufficiently strongly separated in space to be overlapped only by their asymptotic "tails" so that the effect of their interaction and the effect of the latter reveals itself at time intervals significantly exceeding the characteristic duration of each soliton . This allows application of the direct perturbation method developed in (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000) . As a result, a number of non-trivial features of soliton ensembles can be assessed. It turns out, for example, that soliton chains in the Kortewegde Vries equation behave as Toda solitons , that the wave fronts-kinks can form a double-chain system, that solitons in resonators can behave chaotically, and solitons in non-integrable systems can exist as "multi-hump" solitons with the peaks distributed both regularly and chaotically. In spite of such a plethora of effects, we, as already mentioned, address only one kind of approach to soliton interaction and do not consider other, also effective approximate methods such as the Green function method (Keener and McLaughlin, 1977) or the perturbation method based on the application of the inverse scattering method to systems close to integrable ones (Karpman and Maslov, 1977). In a number of cases the solutions described here are confirmed by exact solutions and/or numerical calculations. Complex dynamics of nonlinear wave systems can often be described in terms of the interaction of compact coherent structures such as solitons or kinks. The solitonic concept is especially successful when the governing model equations belong to the class of completely integrable systems or are close to such class. In this brief review it will be demonstrated that even in the case of completely integrable equations, e.g., the Korteweg-de Vries (KdV) equation, the behaviour of the soliton ensembles can be rather complex and interesting . In the case of non-integrable equations that are close to integrable ones, complex dynamics can be revealed even within few interacting solitons . For instance, even three oscillatory-tail solitons in a circular resonator can demonstrate very complex and, apparently, even stochastic behaviour. In what follows the KdV equation plays a role of a basic model equation and we will refer to it many times. In the reference frame moving with the speed of long linear perturbations it can be presented in the form

u, + txuu, + f3u xxx = 0,

(5.1)

where ex and f3 are some constant coefficients, which are determined by a specific physical problem . As is well known, Eq. (5.1) possesses periodic and solitary solutions . The former solutions are known as cnoidal waves, whereas the latter ones are known as solitons . Solitons playa fundamental role in the dynamics of localized initial perturbations due to their stability and persistence in their interactions with each other and even more general perturbations. The solitary solution for Eq. (5.1) can be presented in the form

u(x,t) == UKdV(~ =x- Vt) =A sech 2 [(x - Vt) /il]'

(5.2)

5 Dynamics of Soliton Chains

179

where V = aA /3 is the soliton speed and ~ = (12f3/aA) 1/2 is the characteristic soliton width, whereas A is its amplitude (maximum deviation from the zero level). KdV solitons as well as solitary solutions for other nonlinear partial differential equations (PDEs) will be treated as interacting classical particles whose dynamics is determined by Newtonian equations of motion (ODEs) . The interaction potential between solitons is completely determined by the asymptotics of fields of individual solitons. Such a concept was introduced for the first time in (Gorshkov et a1., 1976) and then was developed in many papers [see the reviews (Gorshkov and Ostrovsky, 1981; Kivshar and Malomed, 1989; Ostrovsky and Gorshkov, 2000) and references therein] .

5.2 Stable soliton lattices and a hierarchy of envelope solitons In many cases stationary nonlinear waves in dispersive media can be presented as a periodic sequence of solitary waves. In several cases such representation has been rigorously proven (Toda, 1989; Zaitsev, 1983). In particular, a well-known periodic cnoidal wave solution to the KdV equation can be presented as (Toda, 1983):

cn2 (2Kx ) =

~k2 {(~)2 2K'

f. sech/ [nKK' (x- i)] + ~K + .s: -I} + I 2KK' ,

i= - oo

(5.3)

where cn (x,k) is the elliptic function of the argument x and modulus k, E(k) and K(k) are complete elliptic integrals with the modulus k [see, e.g., (Weisstein, 2003)], K'(k) = K(k'), and k' =~. Important question then arises regarding the stability of nonlinear periodic waves and chains of solitons . To a certain extent, the stability problem is related to the problem of the evolution of an initially perturbed periodic wave. One of the widely adopted approaches to this problem is the application of one of the versions of the averaging method (Whitham, 1965, 1974; Ostrovsky and Pelinovsky, 1972; Karpman and Maslov 1977; Keener and McLaughlin, 1977; Kaup and Newell 1978; Grimshaw, 1979). However, this approach has a drawback because the governing set of equations for the parameters of the perturbed periodic wave (amplitude, frequency, wavenumber) contain in the first approximation hydrodynamic-type nonlinearities which are not balanced by dispersion or dissipation. As a result, nonphysical discontinuities appear for modulated waves; they can be removed , however, in the high-order approximations. Another evident limitation of the averaging method is in its inapplicability to the description of perturbations whose period is comparable with the period of a carrier wave. The situation becomes much simpler when a nonlinear wave can be treated as a sequence of well-separated solitons . In this case a modulated nonlinear wave can be considered as a perturbed chain of particles - solitons interacting with each other. As has been shown in (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000), if soliton velocities (as well as, other parameters such as widths, amplitudes)

180

K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

differ only slightly from some mean value Vo (i.e. Wi - Vol « Vo) , then equations describing coordinates of their centers (maximums) take the form of Newtonian equations for interacting particles : 2----+ d S if3 m yf3 ~ t

(----+ = L ----+ Fy Sj j#i

-

----+)

Si

(5.4)

,

----+

where S i are spatial coordinates of solitons , m yf3 are elements of the "tensor of mass" which is determined via the derivative of the y-component of the soliton's mo----+ ----+ mentum overthe l3-component of the mean soliton velocity Vo : m yf3 = a p y/ a V f3 '

Is

Sil] than their characteristic widths, ISi- Sjl > ~ (the difference between soliton

and F y '" exp [-A,(Vo) j are forces which are determined by the asymptoties of individual solitons . Distances between solitons may be arbitrary but greater widths ~i and ~j is insignificant in this approximation). For the Lagrangian systems a condition of reciprocity takes place: (5.5a) and the force

F y (s)

can be determined via the "potential function" V

(s): (5.5b)

Equations (5.4) adequately describe soliton interactions when their collisions are elastic; in this case the number of solitons before and after the interaction is unchanged, and the nonsolitonic wave field (radiation) is absent. In the onedimensional case, Equations (5.4) are the equations describing a set of particles moving and interacting on a straight line. If the wave field of an individual soliton diminishes sufficiently fast in space (in many cases it decreases exponentially with distance from the center), then one can consider only the interaction of an n-th soliton with its nearest neighbours having numbers n - I and n + I. Equations (5.4) reduce in this case to a chain of coupled nonlinear oscillators (Gorshkov and Papko, 1977): aPd

2S

11

aV dt2 =V SI1- SI1 -1 -V SI1+I- S11 , (

)'(

)

,

(5.6)

where the prime sign stands for the differentiation V' (x) = dV / dx. The simplest solution to this equation is SI1 = nAo which represents an equidistant set of "particles" on a line or a periodic nonlinear wave in terms of original field variables with the spatial period Ao (see, e.g., Eq. (5.3) when k ----+ I). By linearizing Eq. (5.6) around stationary state SI1 = nAo, one can show that the chain is stable if V'(Ao)ap /av > 0, otherwise, when V'(Ao)ap/av < 0, it is unstable. Physical interpretation of this result is straightforward: the chain is stable when solitons repel each other, and unstable, when they attract each other. In the

5 Dynamics of Soliton Chains

181

long-wave limit in the stable case, when the differe nces in the right-hand side of Eq. (5.6) can be replaced by the differential operator Vi (Ao)d 2 / dn2 , the corresponding POE is of a hyperbolic type, while in the unstable case the POE is of an elliptic type (Whitham, 1965, 1974). Meanwhile, due to the discrete nature of Eq. (5.6), the dispersio n is essential in that equation, especially for the small-scale perturbations. The dispersion stabilises the process of singularity creation and provides the existence of smooth solutions and stationary waves. In many cases soliton fields possess exponential asymptotics, so that Vi (S) ----+ aexp( - AS) when S ----+ 00 , where a and A are some constants. In a stable case of repelling solitons with such exponential asymptotics, Equations (5.6) represe nt the well-known model of the Toda chain (Toda, 1989). This model has an exact solution in the form of a periodic stationary wave:

where dn (x,k) is another elliptic function linked with the en-function [see Eq. (5.3)] by the formu la dn2(x ,k) = k2 [cn2 (x, k) - I] + I [see, e.g., (Weisstein, 2003)], and V and k are related by the following dispersion relation:

0.5 u (x)

I

o

(a)

50

100

ISO

200

1.5 ,

< ,

-.

-,

-- -

"

-

-,

'.

"

-

250 x

---

,-

300

"

-

350

, '.

400

450

,

.-- - -

-- -

500

"

"-

u (x)

0.5

(b)

o

100

200

300

400

500 x

600

700

800

900

1000

Fig.5.1 (a) Unperturbed chain of KdV solitons and (b) modulated chain of the same solitons (note that the horizontal scales are different).

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K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

cn 2 (2kK )

2VK(k)

I - cn2 (2kK)

E(k)

+ K(k) =

(5.8)

I.

In the limiting case of an infinite period, solution (5.7) reduces to the solution for solitary wave - the Toda soliton:

n - Vt ) ] , 5,,-5,,-1 =Ao+~ln [ I +f3 2 cosh _2 ( -~-

(5.9)

with the following relationships between the parameters: f3 = sinh (I / ~),v = sinh (1 / A). With respect to the original sequence of solitons, Equations (5.7) and (5.9) can be treated as stationary waves of envelopes. Solutions (5.7)-(5 .9) are illustrated in Fig. 5.1. One of the important properties of the Toda chain model is its complete integrability (Manakov, 1974). An outcome of this property is that the interaction of solitons (in our case - the envelope solitons) conserves their number and individual parameters (only phase shifts appear). Arbitrary localized perturbation (of the appropriate polarity - this will be specified further) evolves into the diverging sequence of solitons described by Eq. (5.9) . Similar properties of repelling chains of solitons were revealed from the exact solution for the KdV equation (Kuznetsov and Mikhailov, 1974). The existence of periodic envelope waves (5.7)-(5.9) provides a way to constructing a wide class of solutions . Indeed, periodic envelope waves (5.7)-(5.9) can be treated again as chains of equidistant envelope solitons (5.9) [see Eq. (5.3)], if the period of the wave is not too short in comparison with the characteristic soliton 's width . Inasmuch as solitons (5.9) possess exponential asymptotics and behave as repelling particles, their sequence can be treated again as the Toda chain. This procedure can be repeated again and again , one can construct multi hierarchical envelope waves of different orders . An envelope wave at each order represents a perturbation of the soliton chain of the previous order. An envelope wave of the N-th order can be presented in the analytical form (Gorshkov and Papko, 1977): ~

(x,t) =

L

(0 )[x- Vot-5j(t)],

(5 .10)

j = - oo

~ [ f32 - 2 nm- 10"tm- 5"1/1+1 (tm+ I ) ] 5"1/1 -5"1/1 -1 = Am-I +~m-I ~ In 1+ mcosh ~ llm - - OO

tm =

[H!E PT

~exp

m-I

-u-

(Am-I )] tm-I, m-I

tn

m= 1,2, · · ·,N.

Here (x,t) is the initial wave field, (0) (~ = x - Vat) is soliton solution for the initial wave field, 10" and Am are the average speed and mean distance between solitons in the m-th order chain, correspondingly, aPT is the coefficient of interaction

5 Dynamics of Soliton Chains

183

for Toda solitons. The quantities 13m and I::!.m can be presented in terms of Vm + dSIlIII + 1 / dt in each order using the relationships presented after Eq. (5.9). Solution (5.10) represents a N-periodic function with aliquant periods in general. Note that for the integrable models there are so-called N-zone exact solutions which represent periodic and conditionally periodic analogs of multisoliton solutions . Such solutions are described in (Zakharov et aI., 1980; Ablowitz and Segur, 1981) for the KdV equation, and in (Dubrovin et aI., 1976) for the Toda equation and some other equations. A detailed comparison of exact solutions against the approximate ones (5.10) has not been done thus far. It is obvious that the solutions (5.10) correspond to only a part of the general exact solutions . Indeed, as follows from the condition of applicability of Eq. (5.6) in each order of the hierarchy, V,1l » dSIlIII + 1 / dr, when all N periods tend to infinity, Equations (5.10) will be reduced to the equation describing multisolitons with close velocities only, whereas the exact solutions in the same limit describe multisolitons with arbitrary velocities. The Toda equation can be generalized for waves of modulation in the chain s consisting of envelope solitons of the initial wave field (Gerdjikov et al., 1996). In this case the real values SIl(t) should be replaced by the complex ones ZIl(t) = SIl(t) + i f{J1l (r), where SIl(t) represent the coordinates of the centers of envelope solitons, and f{J1l (r) are the phases of the carrier wave of the initial wave field. The Toda equation in the complex form preserves the integrability property and the form of the solution with the replacement of real parameters characterizing the solution by their complex counterparts. The twofold increase of the number of parameters results in the richer family of solutions of the Toda equation. Such solutions were investigated in details in (Gerdjikov et al., 1996) for the groups of almost equid istant envelope solitons within the framework of the nonlinear Shrodinger (NLS) equation. For slightly modulated nonlinear waves close to the periodic sequences of solitons, the approximate results described above are more general as they are not related to the integrability of the initial nonlinear equation and provide clear physical interpretation in terms of interacting Newtonian particles . One may say that the equation of Toda chain plays a similar role to that which the NLS equation plays for quasi-harmonic waves. These two equations, to a certain extent, are complementary to each other. Indeed, if the modulated quasi-harmonic wave can be described by the NLS equation and the envelope wave is a periodic cnoidal wave of a large period , it can be presented as an infinite chain of solitons with exponential tails (Toda, 1989; Gerdjikov et al., 1996). Hence , the perturbation of such solitons can be described by the Toda chain equation. On the other hand, if the initial sequence solitons is only slightly modulated in amplitude, the envelope quasi-harmonic wave can be described by the NLS equation. And to complete this scheme, one should mention the case when the envelope of slightly modulated quasi-harmonic wave is described by the NLS equation and represents again quasi-harmonic wave, which in turn can be described by the NLS equation, and so on, and so on. The relationship between the NLS and Toda equations can be schematically illustrated in Fig. 5.2.

184

K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

Where a) Toda Eq.

----+

Toda Eq.

----+

Toda Eq.

----+ •• •

b) Toda Eq. ----+ Toda Eq. ----+ NLS Eq. ----+

•••

Toda Eq. ----+ NLS Eq. ----+

•••

c) NLS Eq. ----+ Toda Eq. ----+ Toda Eq. ----+

•••

NLS Eq. ----+ Toda Eq. ----+

•• •

d) NLS Eq.

•• •

----+

NLS Eq. ----+ NLS Eq. ----+

Fig . 5.2 Schematic presentation of various possibilities for modulated periodic waves: a) pure Toda equation hierarchy ; b) and c) random intermittent hierarchy of Toda and NLS equations when the original wave is describ ed either by Toda equation (case b) or by NLS equation (case c), d) pure NLS equation hierarchy.

It should be noted, however, that in such a hierarchical structure amplitudes of the tallest solitons must still be relatively small to be in consistent with the weak nonlinearity assumption under which the basic equation was derived . At each level of the hierarchical structure, envelope solitons are smaller, wider, and the time of interaction between them increases. One more interesting example of application of this theory is a periodic sequence of slightly modulated Benjamin-Ono (BO) solitons (Ablowitz and Segur, 1981). These solitons posses algebraically, rather than exponentially, decaying tails and are described by the function:

(5.11 ) where the soliton half-width ~ is related to the amplitude, ~ = 4/ A. It is interesting to note that more general stationary solutions to the BO equation in the form of periodic waves can be presented similarly to Eq. (5.3) as an infinite sequence of BO solitons (5.11) (Zaitsev, 1983) (such representation is rather general ; as shown in the cited paper, it is valid for a wide class of "soli tonic" equations, includ ing KdV, BO, Toda equation, Kadomtsev-Petviashviliy equation, and others). Then, the function determining the force exerting on the nearest soliton by its neighbor is (see Eq. (5.6» : U'(S) = (2A /~)(NS? For small soliton deviations S from their equilibrium distances Ao, the forces in the right-hand side of Eq. (5.6) can be approximately presented as

U'(S) ;:;: -3(2A /~)(NAo? [S/Ao-4(S /Ao?] .

(5.12)

Such presentation, which is simply a Taylor's expansion of the force, is rather general ; it may work for solitons with non-exponential asymptotics. The first nonlinear term in the Taylor series (5.12) is usually quadratic, but it can be, in principle, cubic or even of higher-order nonlinearity. Chains of quadratically and cubically interacting particles were studied in the Report by Fermi , Pasta and Ulam (1955) which gave rise to what is known as the FPU-problem - anomalously slow stochas-

5 Dynamics of Soliton Chai ns

185

tizatio n in chains of nonlinear oscillators (see also the paper by (Dauxois, 2008) where the significant contribution of Mary Tsingou to the numerical comp utations on that problem is described). As is well-known (Toda 1989), a discrete chain of quadratically interacting particles can be reduced to the KdV equation in the longwave limit; similarly, a chain of cubically interacting particles can be reduced to the modified KdV (mKdV) equation in the same limit. The n, KdV equation has periodic stationary solutions which can be presented by infinite sequences of solitons with exponential tails on the basis of which the Toda chain hierarchy can be constructed as described above (the same is true for the mKd V equation). From this consideration it follows that in the first row of the scheme presented in Fig. 5.2, the entry equation may be any nonlinear equation possessing soliton solutions with any kind of asymptotic behavior at infinity. To concl ude this section, let us discuss briefly the case of strong modulation of soliton lattices . Notice first that Eq. (5.6) is able to describe the interact ion of solitons not only of almost equal speeds, but solitons whose speeds are significantly different. In the latter case, two solitons may overlap in the process of interaction despite the repulsion force which may act between them - this case is known as the "overtaking" interaction in contrast to the "exchange" interaction taking place for solitons of almost equa l amp litudes and speeds [see, e.g., (Scott et aI., 1973)]. These two cases of soliton interactions are illustrated in Fig. 5.3. u

3

1.0

'I'\ 2 1\ ,\I'

'I I I I ,

(a)

I I I ,

0.5

I ,

I I

,/

0.00

(b)

/

20

u 1.0

I

I

I I

1\

I I , I

\ \

I I I I

\ \

I J

' I

\/

I \ I I

I

I I I

\,

,

-,

50 x

30

3

0.5

0.00

50 x

Fig . 5.3 Exc hange (a) and overtaking (b) interactio n of solitons. In the former case, energy from the tallest soliton transmits into the smallest soliton, which gradually grows, accelera tes and moves ahead. In the latter case, the tallest solito n simply covers the smallest soliton and forms jointly with it a single-crest pulse, which then disi ntegrates then into the same two solitons. (In Fig . 5.3a curves 2 and 3 are shifted back artificially to visualize the interaction process in the chose n space interval.)

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

186

If now there is an infinite sequence of small-amplitude solitons in front of a large-amplitude soliton, then the latter will consecutively interact with each of them separately. As a result of that it will move non-uniformly, decelerating at the rear slope of each small soliton and accelerating at the frontal slope. The mean speed of the large-amplitude soliton can be determined by its own speed in a free space plus a correction to that speed caused by phase shifts arising each time when it passes through the next small-amplitude soliton . Thus, in the case of strong modulation, the motion can be understood as a propagation of a "dislocation" in the soliton lattice, as shown in Fig. 5.4. u 1.0

(a)

0.5

10

20

30

40

50 x

10

20

30

40

50 x

u 1.0

(b)

0.5

Fig. 5.4 Single soliton dislocation nonuniformly moving on the small-amplitude soliton chain . Number I - designates the soliton chain; number 2 - shows large-amplitude soliton .

If instead of one fast soliton there is a sequence of fast solitons , then a particular case of strong periodic modulation of a soliton lattice occurs (Fig. 5.5) . Such cases have been studied by Zakharov (1971) and Zaslavsky (1972). Actually, the dislocations represent to a certain extent a limiting kind of motion originated from the envelope waves considered above in the case of strong modulations of soliton sequences. It should be stressed, however, that the modulation waves in the form of envelope solitons are obtained by neglecting the inelastic effects associated with the radiation . In many cases the inelastic effects are either absent (e.g., when the governing equation is integrable) or are negligibly small. Description of the inelastic effects, when they are essential, requires a separate consideration. In the meantime, in some situations, the influence of inelastic effects is rather obvious. For instance, the radiation leaking from the region of interaction of the solitonic group of the initial wave field (this group forms the envelope solitary wave), leads

5 Dynamics of Soliton Chains

187

u

u

x u 1.0

x u

x

x

Fig. 5.5 Interaction of two soliton chains.

188

K.A . Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

to i) either slow decay of the envelope solitary wave or ii) fast destruction of that wave, depending on the intensity of radiation.

5.3 Chains of solitons within the framework of the Gardner model In this section we consider a nontrivial generalization of stable soliton chains to the case when soliton interactions within the framework of particle-like approximation are described by equations which are different from the Newtonian equations of motion (5.4)-(5.6). This occurs when the basic model equation contains solitary solutions of a more complex structure than the simple one-parametric KdV solitons . One of the typical representatives of such a class of equations is the Gardner equation also known as the combined or extended KdV equation [see, e.g., (Ostrov sky and Stepanyants, 1989,2005; Apel et al., 2007)] . This equation in the dimensionless variables reads (Gorshkov and Soustova, 2001 ; Gorshkov et al., 2004) :

au au a3u at + 6u(1 - au) ax + ax3 = 0,

(5.13)

where a = ± I is the parameter characterising a sign of the cubic nonlinearity. The Gardner equation became popular in recent years as the model equation describing nonlinear wave processes in the case when wave amplitude is not small. Such situation occurs , for instance, for internal waves in oceans, see (Lee and Beardsley, 1974) as the pioneering paper in this field and reviews (Ostrovsky and Stepanyants, 1989, 2005 ; Ape! et al., 2007) for further references. Equation (5.13) with a = I possesses solitary solutions with exponential asymptoties [when a = -I , Equation (5.13) has solitary solutions of different types; among them there is a one-parametric family of solitons both with the exponential and algebraic asymptotics [see, e.g., (Grimshaw et al., 1997) and references therein] :

u(x,t)

= UG(x- Vt) = ~ {tanh [~(x- Vt+~)] -tanh [~(x- Vt-~)]},

(5.14a) where V = k 2,~ = (I 12k) In [(I +k) /(1 - k)] and k is a free parameter varying from o to I. Depending on the value of this parameter, soliton (5.14) may be very close to the KdV soliton when k --+ 0, or to a superposition of separated kink and antikink, which form the so called "fat" or "tabletop" soliton when k --+ I (see Fig. 5.6). For k = I, solution (5.14a) degenerates into a single kink or antikink :

u(x,t)

=Ukink(X-t) = ~ [I ±tanh (x~t)] .

(5.14b)

5 Dynamics of Soliton Chains

189

u

- 20

- 15

- 10

- 15

o

5

10

20 x

15

Fig.5.6 Shape of Gardner solitons (5.14a) for different parameter e = I - k. (from the smallest to the highest solitons e = 0.5 , 0.1, 10- 2 , 10- 4 , 10- 6 , 10- 8). For very small e, one of the soliton slopes reduces to the kink (5.14b), whereas another slope reduces to the antikink. Width of the widest fat soliton is shown by horizontal lines with the arrows.

The peculiarity of tabletop solitons is the logarithmic divergence of their widths j!": Va (x)dx when k --+ I . Because of that, when two tabletop solitons collide , their neighbouring fronts begin to interact , whereas their other slopes are not influenced by such interaction for awhile . This fact is not reflected in the point-particle model (5.4) or (5.6) : the left-hand side of Eq. (5.6), dP / dt, does not adequately describe strong variations of the solitons' widths and momentums for large values of k when solitons cannot be treated as point particles . Details of tabletop soliton interactions can nevertheless be described within the framework of the same asymptotic method (Gorshkov and Ostrovsky, 1981; Ostrovsky and Gorshkov, 2000) if the interaction of tabletop solitons is treated as the interaction of kinks and antikinks which constitute such solitons . Application of that method to the kink-antikink interaction results in the following equation for their coordinates (Gorshkov and Soustova, 200 I; Gorshkov et al., 2004): ~ rv In (I - k) and momentums P =

dSn dt

= -4 [e - (Sn+I -Sn) +e- (Sn-Sn-I )] +D

'

(5.15)

where D is an arbitrary constant. Equations (5.15) are symmetric with respect to coordinates of kinks (even index numbers) and antikinks (odd index numbers) or, in other words, solitons ' fronts and rear slopes. These equations, known as the Kac-Moerbeke system [see, e.g., (Toda, 1989)], describe well both soliton's widths and distances between solitons . The Kac-Moerbeke system is completely integrable and represents the Backlund transformation of the Toda-Iattice equations. From a physical point of view one may say that the evolution of tabletop solitons is adequately described in terms of the positions of their front and rear slopes rather than the coord inate of soliton centres. This reflects the non-synchronous motion of soliton fronts and rear slopes in the external non-uniform field (e.g., in inhomogeneous medium) . In particular, when two solitons approach each other, the front of one of them begins interacting with the rear slope of another, and only later do their other slopes enter into the interaction. We focus here on an infinite series of solitons (soliton chains) leaving aside the details of two-soliton interactions [they are presented in terms of exact solu-

190

K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

tions of the Gardner equation in (Slyunaev and Pelinovski , 1999; Slyunyaev, 200 I; Grimshaw et al., 2002) for both signs of the parameter a in Eq. (5.13)]. Differenti ation of Eqs. (5.15) on t and elimination of derivatives dS,,/dt with the help of the very same Eqs. (5.15) leads to the two independent sets of Toda-chain equationsfor odd and even numbers n : (5.16) Each of these sets describes the evolution of soliton fronts and rears, whereas Eqs. (5.15), playing the role of links between them, are the Backlund transformation for each set. As a result of this, solutions ofEqs. (5.15) can be presented as the solutions of two Toda-chain sets of Eqs. (5.16) , provided they are linked with each other by the Backlund transformation (5.15). In the general case, the constitutive portions of such a composite solution have different structures. In the degenerative case of D =0, the Backlund transformation does not change the solutions structure of both subsets described by Eq. (5.16) and only imposes conditions which relate the parameters of these solutions . Such solutions for the finite chains describe N-soliton interactions of tabletop solitons. Although the character of interaction of tabletop solitons is qualitatively similar to the interaction of particles with the repulsive potential between them, there is no direct mechanical interpretation for the Eqs. (5.15), as was mentioned above. The specifics of the tabletop solitons ' interaction can be easily seen from the example of two-soliton collision when the solutions for the pairs of soliton fronts and rears are the same but shifted in time and space with respect to each other. When two solitons approach each other, the front of one of them interacts with (repeals from) the rear of another one, and then after the corresponding delays front-front, rear-front and rear-rear interactions occur. This picture is in the correspondence with the exact solution of the Gardner equation obtained in (Slyunaev and Pelinovsky, 1999; Grimshaw et aI., 2002) for the case of Eq. (5.13) with a = I. Returning to the description of infinite chains of Gardner solitons, note first that there exist trivial solutions to Eqs. (5.15):

S" = { nA + Vt + 8, nA+ Vt,

for even n, for odd n.

(5.17)

Such solutions correspond to periodic nonlinear waves in the original PDE model (5.13) with the spatial period 2A and the given on-off time ratio 8 which ranges in -A < 8 < A (Fig. 5.7) Substitution of solution (5.17) into Eqs. (5.15) yields the dispersion equation v = -2e- A cosh 8, (5.18) which relates the speed of a periodic wave with the parameters A and 8 . Wide classes of solutions can be constructed for an infinite soliton chain described by Eqs. (5.15). Among these classes there are modulation waves on the background of a periodic sequence of tabletop solitons (5.17). Bearing in mind the

5 Dynamics of Soliton Chains

191 u

o

8

A

8 +4A

8 +2A

8 +6A

x

Fig. 5.7 Sketch of a typical chain of Gardner solitons . Positions of kink and antikink fronts are indicated underneath .

compound character of solutions of Eqs . (5.15) (see above), we conclude that the N soliton solution of one of the subsets of Eqs . (5.16) for soliton fronts or rear slopes corresponds to the (N + 1)-soliton solution of another subset of these equations:

Sn(t) =

S,~N)(t),

for even n (odd n),

{ SI~N+I )(t) ,

for odd n (even n).

(5.19)

Within each subset of Eq. (5.19) asymptotics of solutions S~N\t) and S,~N+l ) (t) when t ----+ 00 represent sequences of Toda-chain solitons ordered in amplitudes and speeds . As the solution S,~N+ I) (r) is obtained by a single application of Backlund transformation to solution S~N) (r), N solitons in each of the subsets are the same in pairs. This means that solution (5.19) describes a collision of N pairs of linked solitons from the different subsets plus one extra soliton in one of these subsets. Collisions between such solitons possess all attributes of soliton interactions in the integrable systems: there is no radiation or energy leakage from solitons, the number of solitons and their parameters are preserved after the interaction, and the solitons acquire only phase shifts in the course of interaction . However, in contrast with the single Toda chain, the set (5.15) allows the existence of two types of localised formations that correspond to the presence of Toda solitons either in only one of the subsets or simultaneously in both of them :

Sn(t) =

where V

Sn(t) =

{

nA + Vt + In {Cosh [A,(n - 2) - f3 t]} , cosh(A,n - f3t)

even n (odd n) ,

nA+Vt ,

oddn (evenn) ,

= -2 cosh (2A.) exp( -A) , 13 = sinh (2A.) exp( -A) , ;

Ito"" -..,

rr>.

'""'

'""'

r:

.5

f\

-200

- 100

o

\

100

200

x

Fig. 5.9 Example of the envelope soliton on the sequence of initial Gardner solitons (the soliton represents a smooth transition from the periodic sequence of table-top solitons in the left to wider soliton in the center and then, back to the same periodic sequence of solitons in the right).

5 Dynamics of Soliton Chains

193

The shift parameter A for an envelope kink is unambiguously related to the on-off time ratio 0 of the periodic chain (5.17) in the limit of 0 ----+ 21. . Envelope solitons exist on the background of any periodic chain (5.15); they are presented by the family (5.21) with the parameter A independent of O. This situation is similar to that occurring for solitons and kinks in the basic model (5.13) , where the unique kink (or antikink) solution (5.14b) exists in parallel with the entire family of solitons (5.14a) depending on the parameter k. The kink (antikink) solution (5.14b) corresponds to envelope kinks related to the excitation of Toda solitons in different subsets (5.20). Envelope solitons (5.21) can be treated as compound formations which are formed by a pair of envelope kink-antikink from different subsets (5.20). With the help of envelope solitons or envelope kinks one can construct an infinite periodic chain . Quasi-sinusoidal perturbations of such a chain can be described in terms of the NLS equation (see Fig. 5.2), whereas large-amplitude perturbations of the chain can be described in terms of second-order envelope solitons or kinks with all the aforementioned properties. Repeating this argument, one can construct a family of solutions in the form of a hierarchy of envelope waves of various orders described by multi-periodic functions of aliquant periods in general. It is important to note that the composite character of two-parametric Gardner solitons is repro duced on each hierarchical level of description; this property is inherited from the basic Gardner model (5.13). Similar solutions in the form of a hierarchy of envelope waves within the models having "simple" one-parametric solitons (such as KdV or sine-Gordon solitons) demonstrate only the same "simple" envelope solitons on either hierarchical level.

5.4 Unstable soliton lattices and stochastization Soliton chain dynamics becomes much richer and multifarious when the solitons are described by non-integrable equations. Here, rather complex phenomena up to stochastisation, may occur even in a progressive wave. In particular, an interesting dynamics is possible in chains consisting of solitons with non-monotonous asymptoties in the form of oscillatory "tails," as shown in Fig. 5.10. 4.5 u

- 15

Fig.5.10 A solitary wave with oscillatory tails.

15

x

194

K.A. Gorshkov , L.A . Ostrovsky and Y.A. Stepanyants

The first example of such solitons was apparently obtained by Kawahara (1972) , who numerically constructed a solitary solution with oscillatory tails for the fifthorder KdV equation: au au a 3u a 5u + u ax + ax3 + Yax5 = 0, (5 .22a)

°

at

where Y > is the dimensionless parameter (when y < 0, Eq. (5.22a) has only solitons with monotonic tails). Equation (5.22a) presented here in the dimensionless form was earlier derived by (Kakutani and Ono , 1969) for magnetosonic waves in plasma [a similar equation was later derived for many other types of waves, among them the gravity-capillary waves on thin liquid films (Hasimoto, 1970; Stepanyants, 2005), electromagnetic waves in nonlinear electric circuits (Gorshkov and Papko, 1977, Nagashima, 1979)]. Similar properties demonstrates an equation describing nonlinear waves in with two types of dispersion: small-scale (KdV-type) and large-scale (waveguide-type) dispersion : (5.22b) Similar to the previous case, here solitons with oscillating tails exist only when y > 0, whereas there are no soliton solutions at all when y < (Leonov, 1981; Galkin and Stepanyants, 1991; Liu and Varlamov, 2004). Equation (5.22b), also presented in a dimensionless form, was derived for the first time by Ostrovsky (1978) for surface and internal waves in a rotating ocean (for these types of waves y < 0). Later a similar equation with both signs of y was derived for other types of waves, among them elastic waves in curved thin rods (Rybak and Skrynnikov, 1990), waves in relaxing media (Vakhnenko, 1999), and oblique magnetosonic waves in rotating plasma (Obregon and Stepanyants, 1998), for further references see, e.g., (Stepanyants, 2006) . The stationary solutions of these equations satisfy the ordinary differential equations : d4 u d2 u u2 (5 .23a) Yd~4 + d~2 -Vu+ 2 =0,

°

d 4u

I d 2u2 d~4 -v d~2 +yu+"2 d~2 =0, d 2u

(5.23b)

where ~ = x - V t , V is the wave speed . (Note that the linear parts of these equations are similar). The asymptotic wave field, which corresponds to a soliton with oscillating tails, can be described by the formula U(S) = e-}'1 S cos A2S, see Eq. (5.6) . By substituting this expression into Eqs. (5.23a) or (5.23b) and assuming that Y > 0, one can find the constants AI and A2. For Eq . (5.23a) they are

A

2

=~J2V-YV+I 2 y '

(5 .24a)

5 Dynamics of Soliton Chains

195

where the wave speed V is negative and restricted from the top, V < - I / 4y (otherwise, the expo nent Al becomes imagi nary). For Eq. (5.23b) these constants are (5 .24b) and - 2yY < V < 2yY. The decay rate of oscillatio ns depends on the para meter V. This is illustrated in Fig . 5. 11. 16

u

u

12 6 4

°-15

4

4

2

2

15°135 t (a)

u

165° 10 t

14 (b)

90 t (c)

Fig. 5.11 Possible shapes of a stationary soliton depending on its velocity for Eq. (5.23b): (a) soliton with aperiodic tails, (b) soliton with oscillatory tails, (c) envelope soliton (Fraunie and Stepanyants, 2002).

Due to the non-monotonous character of the potential functio n U(5), see Eq. (5.6), for the oscillating solitons, interaction between them may be both rep ulsive and attractive depending on the distance between their maxi ma. Respectively, the chain becomes stable or unstable when dista nces between solito ns vary. Boundaries between stable and unstable zones in the soliton chai n are dete rmined by zeros of function U'(5). Considering only two interacting solitons with oscillating tails, one can construct statio nary solutions in the form of stable or unstable bisolitons. Exam ples of such solutions numerically constructed in (Obregon and Stepanyants, 1998) for Eqs . (5.23) with y = I are shown in Fig. 5. 12. Stable biso litons correspond to the cases when the maximum of one soliton is located in one of the local minima C'potential wells") of another soliton (suc h exa mples are shown in Figs. 5. 12b and d), whereas unstable bisolitons correspond to the cases when the maxim um of one soliton is located in one of the local maxima of another soliton (one of such examples is shown in Fig . 5.12c). Similar bisolitons were obtained for Eq . (5.23a) and even observed experimen tally in a specially constructed electromagnetic transmission line (Gors hkov et al., 1979). Analogous solutions in the form of bisolitons were also obtained for 2D models such as Kadomtsev-Petviashvili (KP) equation and its generalisations (Abramyan and Stepanyants, 1985a, b, 1987). In the case of KP equation bisoliton solution and even more complex stationary multisoliton solutio ns were derived analytically (Pelinovs ky and Stepanyants, 1993), but all these constructions were found to be unstable. Biso litons also have osci llatory tails which means that even more co mplex stationary structures - bound states of multisolitons - can be constructed. This state-

K.A. Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

196

4

2

(a)

0 -+--~-~-~---=::::=''--T--+-r+-+---, 0 in both cases . Both these equations are rather universal models applicable for description wave processes in many physical systems . Even in the simplest case of a periodic chain containing only three solitons at the period the chain dynamics may be fairly complex representing an example of Hamiltonian chaos . One may expect even more complex oscillations when the number of solitons is more than three in each period . In conclusion note that all physical processes described and numerically modelled in this section were observed experimentally for electromagnetic waves in nonlinear transmission lines (Gorshkov and Papko, 1977).

5.5 Soliton stochastization and strong wave turbulence in a resonator with external sinusoidal pumping As long as solitons are considered as interacting particles, it is natural to put forth the problem of existence of a "soliton gas," i.e. a stochastic ensemble of solitons. This problem has been discussed since the famous work by Fermi, Pasta and Ulam

5 Dynamics of Soliton Chains

203

(1955), where an expected equipartition of modes failed to be established in an equation of a nonlinear string. After Zabusky and Kruskal (1965) demonstrated the particle-like features of solitons in the KdV equation, this unusual behaviour was related to mode synchronisation with the formation of solitons . The question of possible stochastization of solitons was given a negative answer (Zakharov, 1971): as long as in the KdV equation, solitons are not changed upon collisions, and their energy distribution remains unchanged, which is analogous to the case of a onedimensional ideal gas with pair collisions. This feature was discovered to be generic for all integrable systems. However, even in such simple equation as the KdV, the stochastization is possible if the phase shifts between solitons become significant in the process of their interaction with each other and external fields. An example of such stochastization was considered by (Gorshkov et al., 1977). In that work, a limited-length electromagnetic line consisting of Nnonlinear LC oscillatory circuits was studied. Both ends of the line were completely reflecting (so that the line was an electromagnetic resonator), with a periodic forcing (pump) at one end. Such a system is not completely conservative: a soliton, propagating back and forth along the line, periodically interacts (collides) with the end pump and, depending on the phase of these collisions, increases or decreases its energy. If there are several (or many) solitons in the line, interaction of the trial soliton with all others affects its propagation time and, hence, the phase of interaction with the pump changes at each period which can result in stochastization of the solitons or, in other words, strong wave turbulence. In the same paper, an estimate for the condition of such stochastic "heating" of the "soliton gas" in the resonator was made . The solitons were considered as the analogs of particles moving between two oscillating walls: a model used before in relation to the so called Fermi acceleration [details and further references can be found in (Zaslavsky, 1984, 1985,2005)]. According to these models, the stochastization criterion is K » I, whereK is the "phase stretching coefficient," characterizing change of phase upon to consequent interact ions of a soliton with the end pump. This issue was discussed with George Zaslavsky whose remarks and advices were very useful in the preparation of the paper (Gorshkov et a\., 1977). The experiment with the aforementioned electromagnetic line of N nonlinear cells showed that only regular soliton dynamics occurs when N < 80 [such a dynamic regime was observed earlier (Gorshkov et a\., 1973) and can be called the parametric pulse generation] .

s

(a)

(b)

w

Fig.5.19 A fragment of random soliton sequence (a) in the resonator and its Fourier spectrum (b) when K :::Y I.

204

K.A. Gorshkov, L.A. Ostrovsky and Y.A. Stepanyants

But for N > 100, a random soliton sequence, a "soliton gas," was observed in accordance with the theoretical prediction based on the Chirikov-Zaslavsky criterion (Gorshkov et al., 1977). An example of such soliton gas and corresponding Fourier spectrum are shown in Fig. 5.19 .

5.6 Chains of two-dimensional solitons in positive-dispersion media Description of wave processes in terms of ensembles of interacting solitons appears to be very useful not only in one-dimensional but in multidimensional cases as well. In this section we describe the peculiarities of the dynamics of two-dimensional solitons using the known Kadomtsev-Petviashvili equation as an example: (5.34) Here the equation is presented in the dimensionless form with only one parameter (J' = ± I which controls the dispersion. Namely, (J' = I corresponds to the case of positive dispersion which is considered here . In such a case Eq. (5.34) is dubbed the KP I-equation, whereas in the case of negative-dispersion media «(J' = -I), the corresponding equation is dubbed the KP2-equation. Figure 5.20 illustrates the qualitative difference in the character of dispersion.

1.0

aJ(k)

0.8

0.6

0.4

0.2 0.0 / / / '/

0.0

0.2

0.4

0.6

0.8

1.0

k

Fig. 5.20 Qualitative sketch of the dispersion relation for plane sinusoidal waves of infinitesimal amplitude described by linearized KdV equation (5.1).

5 Dynamics of Soliton Chains

205

It is evident that in the one-dimensional case when propagation of a plane wave is considered, one can let a = 0 and reduce the KP equation (5.34) to the KdV equation (5.1) (note that the character of the dispersion in this case in insignificant - the KdV equation can be presented in the dimensionless form regardless of the dispersion character in the real system). Hence, all known solutions to the KdV equation, including the soliton (5.2), are also particular solutions to the KP equation. It is currently well known that in isotropic nonlinear media, the evolution of multidimensional perturbations essentially depends on the character of the dispersion . In particular, Kadomtsev and Petviashvili (1970) discovered that in the case of positive dispersion, plane solitons moving in the x-direction and described by Eq. (5.2) are unstable with respect to the self-focusing, whereas in the negative-dispersion case, plane solitons (5.2) are stable . This conclusion was later confirmed by Zakharov (I975) who derived an exact formula for the instability growth rate. To be more precise, the self-focusing instability of plane solitons is determined by the "decaying" spectrum of small perturbations, i.e., by dispersion relation which allows the three-wave resonances between quasi-monochromatic waves. The spectrum is indeed decaying in such a sense in isotropic homogeneous media with weak positive dispersion. In anisotropic media, however, the spectrum may be decaying even in the case of negative dispersion (Abramyan et al., 1992). The linear and early nonlinear stages of the self-focusing instability within the framework of the KPI-equation have been studied by many authors [see, e.g., (Pelinovsky and Stepanyants, 1993) and references therein] . As was ultimately clarified , the nonlinear development of small perturbations of a soliton front gives rise, at the intermediate stage of instability, to periodic chains of two-dimensional solitons (the so-called KP lumps), which have larger amplitude and smaller velocity than the initial plane soliton (note that in the positive-dispersion media, solitons of greater amplitude move slowly). The initial plane soliton emitting lumps decreases in amplitude and moves faster than the chain of lumps. A lump is described by the formula (see also Fig. 5.21)

Fig.5.21 Qualitative sketch of the KP lump described by Eq. (5.35).

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K.A. Gorshkov, L.A. Ostrov sky and Y.A. Stepany ants

9+ 16V 2y2 -12V~2

u(~ , y) =48V (9+ 16V2 y2+ 12V~2)2 '

(5.35)

where ~ = x + Vr, and V > 0 is the dimensionless velocity of the lump. Note that Eqs . (5.4) in application to KP lumps (5.35) can be transferred to a completely integrable system, the Calogero-Moser system of particles on a complex plane (Gorshkov et a1. , 1993). The corresponding set of equations follows from the Hamiltonian (Calogero, 1976) H

.2 = '"' i..JZk

k

'"' 2 )2' i..J ( l /'2 . (a) t = 0, (b) t = 15. (c) t = 40; right colum n - YI < /'2 . (d) t = 0, (e) t = 10, (f) t = 25. Numbers indicate maxima of the wave field.

Alternatively. if Yl < Y2, a perturbed chain of lumps separa tes first in the course of the primary decay of the initially modulated plane soliton with the formation of residual small amplitude plane soliton . Then , the chain of lumps decays into two separate chains of different perio ds. This process is illustrated in Figs. 5.23d-f. In the part icular case of Yl = Yz, both scenarios of the plane soliton decay take place simultaneously, and no metastable intermediate structure can be clearly distingu ished in the process of decay.

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K.A . Gorshkov , L.A. Ostrovsky and Y.A. Stepanyants

The results presented above can be generalized for an initial perturbation with any number M of discrete sinusoidal modes . In this case, the development of the self-focusing instability gives rise to M parallel chains of lumps propagating one after another with transverse wave numbers and to a single plane soliton with the smallest possible amplitude and highest velocity. Thus, the process of plane soliton decay for an M-mode perturbation is the M-fold decay of one of the two structures (either the perturbed plane soliton or the perturbed chain of lumps) formed at each intermediate stage of plane soliton decay. The specifics of this process depend on the ratio of the growth rates of unstable modes . These results can be generalized to the case when there is a localised perturbation on the front of the initial plane soliton . In this case the analysis shows (Pelinovsky and Stepanyants, 1993) that asymptotically, when t ----+ 0, the corresponding initial perturbation gives rise not only to the structures described above (plane soliton and chains oflumps), but also to separate two-dimensional solitons following the leading residual plane soliton . The number of such separate solitons, their amplitudes and relative position depends on the intensity of the initial perturbation of the plane soliton. An example of such a process was presented in (Infeld et aI., 1995) on the basis of numerical calculations. A self-focusing mechanism in the case of an arbitrary perturbation may, in general, give rise to a disordered ensemble of individual lumps and periodic chains of lumps as well as a residual plane soliton whose amplitude may be infinitesimal. If the decay of a plane soliton occurs in a closed system, the self-focusing instability results in creation of numerous two-dimensional solitons - lumps. Reflecting from the boundaries, the lumps may undergo multiple elastic collisions similar to collisions of KdV solitons (see, e.g., Ablowitz and Segur, 1981). As a result, the stochastic ensemble of lumps - a "gas" of 2D solitons - may be formed performing a specific sort of strong wave turbulence. This hypothesis, however, has not been examined thus far. To underpin and supplement the aforementioned hypothesis, consider also the evolution of a periodic chain of lumps arranged along the axis y and moving in the x-direction. As mentioned, such a chain is unstable against small transversal perturbations, which is similar to the self-focusing instability of a plane soliton (Burtsev, 1985). As a result of instability development, the chain disintegrates into secondary chains of lumps (Pelinovsky and Stepanyants, 1993). The typical process of chain disintegration for the case of simple periodic perturbation is shown in Fig. 5.24 . The analysis of decaying processes in the simplest case of plane solitary wave instability allows us to understand the problem of plane-wave instability in positivedispersion media . As nonlinear quasi-plane structures decay, the energy is not lost in small oscillations of the medium . Instead , it is condensed in two-dimensional and plane solitons (although in non-integrable systems some portion of energy may be scattered in the non-solitonic form of quasi-linear ripples). Since the model considered is conservative, these processes are reversible ; i.e., the soliton merging is also possible. Note that although all the results presented here are based on the integrability of the KP I-equation, it is believed that for other similar models with positive dispersion where the instability of a plane soliton relative to self-focusing

211

5 Dy namics of Soliton Chains x 9

38. 1

(a)

0

- 9 '--

...J.....+.

o

- 37

37 y

x 9 47.1

(b)

0

- 9 '--

---L_+_

o

- 37

37 y

x

9

(c)

54.2

0

-9

-37

ciJ

13.2

0

00

37 y

Fig. 5.24 Decay of a chain of lumps sin usoida lly modulated in y-direction: (a) (c) I = 20.

I =

0, (b)

I =

10,

was discovered, similar plane soliton decay may be observed provided the inelasticity effects of soliton interaction are negligible. However, the question of the origin

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K.A . Gorshkov, L.A . Ostrovsky and Y.A. Stepanyants

of small oscillations of the medium which can accompany soliton decay needs an independent investigation.

5.7 Conclusion As was shown in this review, in many cases solitons can be treated as particles interacting through the potentials which are determined by asymptotics of their own fields. A specific feature of such particles , in contrast to the Newtonian particles, is that their asymptotics can be rather different (exponential, algebraic , oscillating). Note that in 2D and 3D cases the effective masses of solitons can be tensors rather than scalars (see, e.g., Gorshkov et al., 1993), although such cases were not considered here. A periodic chain of solitons may be stable or unstable depending on the basic governing equation, and soliton interactions within the chain may be elastic or inelastic. In the latter case, their interaction may be accompanied by a weak radiation . It is important to emphasise that the perturbation method described in this paper is equally applicable both to integrable and non-integrable governing equations and even to dissipative equations. Moreover, it can be used not only in the one-dimensional case, but in multidimensional cases as well (Gorshkov et al., 1993; Gorshkov, 2007) . On the basis of this method, soliton dynamics in two- or even three-dimensional lattices may be studied . As was shown in this paper, unstable soliton chains may result in a rather complex soliton dynamics which may even become stochastic under certain conditions. Here we presented only relatively simple examples of quasi-stochastic behaviour of solitons occurring as a result of chain instability . The problem of soliton stochastization in closed or periodic systems seems topical and promising from the point of view of application of the concept described in this review.

Few words in memory of George M. Zaslavsky The authors had numerous scientific and personal contacts with George while in Russia and later in the USA. He always impressed us by combining a seriou s and almost meticulous approach to scientific work with friendly and unbiased personal communications. George regularly participated in the Gorky Scientific Schools on Nonlinear Oscillations and Waves. He was one of the most popular lecturers. His lectures at the Schools provoked great interest, and after the lectures George was usually "attacked" in the lobby by numerous questions (one such after-lecture discussion is shown in the photo Fig. 5.25). On several occasions George visited our laboratory at the Radio physical Research Institute and later at the Institute of Applied Physics of the Russian Academy of Sciences in Nizhny Novgorod . Discussions with him were always interesting and fruitful and we acknowledged his useful

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advice [see, e.g., (Gorshkov et al. 1977)]. We were also pleased to meet George at many other conferences and discuss with him not only scientific problems but general issues in art, literature, history, politics, etc. One of these informal meetings is reflected in the photo Fig. 5.26.

Fig.5.25 G.M. Zaslavsky at the Gorky School on Nonlinear Oscillations and Waves. Village Zholnino (near Gorky), March, 1973.

Fig. 5.26 G.M. Zaslavsky and L.A. Ostrovsky at a conference taking place on board of a ship cruising on the Yenisey River (early 1980s).

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Gerdjikov V.S., Kaup OJ., Uzinov I.M. and Evstatiev E.G., 1996, Asymptotic behavior of N -soliton trains of the nonlinear Schrodinger equation, Phys. Rev. Lett., 77, 3943-3946. Gorshkov KA., 2007, Perturbation Theory in the Soliton Dynam ics, The Doctoral Degree Thesis, Institute of Apllied Physics, Russian Academy of Sciences, Nizhny Novgorod, 252 p (in Russian) . Gorshkov K.A., Ostrovskii L.A . and Papko V.v., 1973, Parametric amplification and generation of pulses in nonlinear distributed systems, lzvetiya VUZov, Radiofizika, 16, 1195-1204 (in Russian. Eng\. trans\. : Radiophys. and Quantum Electronics, 1973, 16, 919-926). Gorshkov K.A., Ostrovsky L.A . and Papko V.v., 1976, Interactions and bound states of solitons as classical particles, Zh. Eksp. Teo r.Fiz; 71, 585-593 (in Russian. Eng\. trans\. : Sov. Phys. JETP , 1976,44,306-311). Gorshkov KA ., Ostrovsky L.A . and Papko V.v., 1977 , Soliton turbulence in the system with weak dispersion, DAN SSSR , 235, 70-73 (in Russian). Gorshkov K.A., Ostrovsky L.A ., Papko V.V. and Pikovsky A.S ., 1979, On the existence of stationary multisolitons, Phys. Lett. A , 74, 177-179. Gorshkov KA. and Papko V. V. , 1977, Dynamic and stochastic oscillations of soliton lattices, Zh. Eksp. TeorFiz., 73, 178-187 (in Russian. Eng\. trans\. : Sov. Phys. JETP, 1977,46,92-97). Gorshkov K.A. and Ostrovsky L.A ., 1981, Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica D, 3, 428-438. Gorshkov KA. , Ostrovsky L.A. , Soustova l.A. and Irisov V.G., 2004, Perturbation theory for kinks and its application for multi soliton interactions in hydrodynamics, Phys. Rev. E, 69,016614. Gorshkov K.A., Pelinovsky D.E. and Stepanyants Y.A., 1993, Normal and anomalous scattering, formation and decay of bound-states of two-dimensional solitons described by the Kadomtsev-Petviashvili equation, Zh. Eksp. Teor.Fiz. , 104, 2704-2720 (in Russian. Engl. transl. : Sov. Phys. JETP, 1993,77, 237-245). Gorshkov KA. and Soustova l.A. , 2001 , Interaction of solitons as compound structures in the Gardner mode\. lzvetiya VUZov, Radiofizika , 44, 502-514 (in Russian. Eng\. transl. : Radiophys. and Quantum Electronics, 2001 , 44, 465-476). Grimshaw R., 1979, Slowly varying solitary waves. I Korteweg-de Vries equation, Proc. Roy. Soc., 368A, 359-375; Slowly varying solitary waves, II Nonlinear Schrodinger equation, Proc. Roy. Soc ., 368A, 377-388. Grimshaw R., Pelinovsky D., Pelinovsky E. and Slyunyaev A., 2002, Generation of large-amplitude solitons in the extended Korteweg-de Vries equation, Chaos, 12, 1070-1076. Grimshaw R., Pelinovsky D. and Talipova T., 1997, The modified Korteweg-de Vries equation in the theory of large-amplitude internal waves, Nonlin. Proc. in Geophy s, 4,237-250. Hasimoto H., 1970, Kagaku, 40, 401-403 (in Japanese). Infeld E., Senatorski A. and Skorupski A.A., 1995, Numerical simulations of Kadomtsev-Petviashvili soliton interactions, Phys. Rev. E, 51,3183-3191 .

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Kadomtsev B.B., Petviashvili VI., 1970, On the stability of solitary waves in weakly dispersive media. DAN SSSR 192, 753-756 (in Russian. Engl. transl. : Sov. Phys. Doklady, 1970, 15, 539-54\). Kakutani T. and Ono H., 1969, Weak non-linear electromagnetic waves in a cold, collision-free plasma, J. Phys. Soc. Japan , 26, 1305-1318. Karpman VI. and Maslov E.M., 1977, Perturbation theory for solitons, Zh. Eksp. Teor.Fiz. 73,537-559 (in Russian. Engl. transl. : Sov. Phys. JETP, 1977,46, 281295). Kaup OJ . and Newell AC., 1978, Soliton as particles, oscillator and in slowly changing media : a singular perturbation theory, Proc. Roy. Soc. London A ., 301, 413-446. Kawahara T., 1972, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33, 260-264. Kawahara T. and Takaoka M., 1988, Chaotic motions in oscillatory soliton lattice, ~ Phys.Soc.Japan,57,3714-3732. Kawahara T. and Takaoka M., 1989, Chaotic behavior of soliton lattice in an unstable dissipative-dispersive nonlinear system, Phys ica D, 39, 43-58. Kawahara T. and Toh S., 1988, Pulse interactions in an unstable dissipative dispersive nonlinear system, Phy s. Fluids, 31, 2103-2111. Keener J.P. and McLaughlin D.W. I977a, Soliton under perturbation, Phys. Rev. A ., 16,777-790. Keener J.P. and McLaughlin D.W. 1977b, Green function for a linear equation associated with solitons, J. Math. Phys., 18, 2008-2013. Kivshar Yu.S. and Malomed B.A, 1989, Dynamics of solitons in nearly integrable systems, Rev. Modern Phys., 61, 763-915. Kuznetsov E.A. and Mikhailov A.V , 1974, Stability of stationary waves in nonlinear weakly dispersive media, Zh. Eksp. TeorFiz., 67, 1019-1027 (in Russian . Engl. transl. : Sov. Phys. JETP, 1974,40,855-859). Lee Ch.- Y. and Beardsley RC; 1974, The generation of long nonlinear intemal waves in a weakly stratified shear flows, J. Geophys. Res., 79,453-457. Lennard-Jones J.E., 1924, On the determination of molecular fields. II. From the equation of state of a gas, Proc. Roy. Soc. A, 106,463-477 (see also Wikipedia: http ://en.wikipedia.org/wi kilLen nard-Jones-potential). Leonov AI., 1981, The effect of Earth rotation on the propagation of weak nonlinear surface and intemal long oceanic waves, Ann . New YorkAcad. Sci., 373, 150-159. Liu Y. and Varlamov V, 2004, Stability of solitary waves and weak rotation limit for the Ostrovsky equation , J. Diff. Eq. , 203, 159-183. Manakov S.V, 1974, Complete integrability and stochastization of discrete dynamical systems, Zh. Eksp. Teor.Fiz., 67,543-555 (in Russian. Engl. transl. : Sov. Phys. JETP, 1974,40,269-274). Miles J.w. , 1977, Obliquely interacting solitary waves, J. Fluid Mech., 79, 157-169. Resonantly interacting solitary waves. ibid. 171-179. Nagashima H., 1979, Experiment on solitary waves in the non-linear transmissionline described by the equation du /dt+udu /d z-d 5u /dz5 = 0, J. Phys. Soc. Japan, 47, 1387-1388.

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Newell A.C. and Redekopp L., 1977, Breakdown of Zakharov-Shabat theory and soliton creation, Phys. Rev. Lett., 38, 377-380. Obregon M.A., 1993, Chaotic dynamics of soliton chains in rotating media with positive dispersion, Master Degree Thesis, NNGU, Nizhny Novgorod . Obregon M.A and Stepanyants Yu.A, 1998, Oblique magneto-acoustic solitons in a rotating plasma, Phys Lett A 249, 315-323. Ostrovsky L.A., 1978, Nonlinear internal waves in a rotating ocean. Okeanologia, 18, 181-191 (in Russian. Engl. transl.: Oceanology, 18, 119-125). Ostrovsky L.A and Gorshkov K.A, 2000, Perturbation theories for nonlinear waves, In: Christiansen P, Soerensen M (eds.) Nonlinear science at the down at the XXI century, 47-65. Elsevier, Amsterdam . Ostrovskii L.A and Pelinovskii E.N., 1972, Method of averaging and the generalized variational principle for nonsinusoidal waves, Prikladnaya Matematika i Mekhanka (PMM), 36, 71-78 (in Russian. Engl. transl.: Appl. Math. Mech., 1972, 36,63-70). Ostrovsky L.A. and Stepanyants YA ., 1989, Do internal solitons exist in the ocean, Rev. Geophy s., 27,293-310. Ostrovsky L.A and Stepanyants Yu.A, 2005, Internal solitons in laboratory experiments: Comparison with theoretical models, Chaos, 15,037111. Pelinovsky D.E. and Stepanyants Yu.A., 1993, New multisoliton solutions of the Kadomtsev-Petviashvili equation, Pis'ma v ZhETF, 57, 25-29 (in Russian. Engl. trans!': JETP Lett ., 1993,57,24-28). Potapov Al. , Pavlov l.S ., Gorshkov K.A and Maugin G.A, 2001, Nonlinear interactions of solitary waves in 2D lattice, Wave Motion, 34, 83-95. Rybak S.A. and Skrynnikov Yu.I., 1990, A solitary wave in a uniformly curved thin rod, Akust Zhurn, 36, 730-732 (in Russian. Engl. transl: A single wave in a thin rod of constant curvature . Sov. Phys. Acoustics, 1990, 36, 410-411). Scott A.C., Chu EYE and McLaughlin D.W., 1973, The soliton: a new concept in applied science, Proc. IEEE, 61, 1443-1483. Slyunyaev A V, 200 I, Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity, Zh. Eksp. Teor.Fiz. , 119,606-612 (in Russian. Eng\. trans\.: JETP, 2001, 92, 529534). Slyunyaev A.V and Pelinovski E.N., 1999, Dynamics of large-amplitude solitons, Zh. Eksp. Teor.Fiz., 116,318-335 (in Russian. Engl. transl.: JETP, 1999,89, 173181.) Stepanyants YA , 2005, Dispersion of long gravity-capillary surface waves and asymptotic equations for solitons, Proc. Russ. Acad. Eng. Sci. Ser. Appl. Math. And Mech, 14, 33-40 (in Russian). Stepanyants YA ., 2006, On stationary solutions of the reduced Ostrovsky equation: periodic waves, compactons and compound solitons, Chaos, Soliton s and Fractals, 28, 193-204. Toda M., 1989, Theory ofNonlinear Lattices 2nd ed., Springer-Verlag, Berlin. Vakhnenko VA, 1999, High-frequency soliton-like waves in a relaxing medium, J. Math. Phys. , 40, 2011-2020 (Early version: Vakhnenko VA, 1991, Peri-

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odic short-wave perturb ations in a relaxing medium , Prep rint, Inst. of Geophy s., Ukrainian Acad, Sci., Kiev). Weisstein E.W., 2003 , CRC Conc ise Encyclop edia of Mathematics 2nd ed., Chapman & Hall/CRC , Boca Raton et al. Whitham G.B., 1965, Nonlinear dispersive waves, Proc. Roy. Soc. A., 283,238-261 . Whitham G.B., 1974, Linear and Nonlinear Waves, Wiley-Interscience, New York. Zabu sky N.J. and Kruskal M.D., 1965, Interaction of "solitons" in a collis ionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15,240-243. Zaitsev A.A., 1983, Formation of stationary nonline ar waves by superposition of soliton s, DAN SSSR , 272,583-587 (in Russian. Engl . transl .: Sov. Phys. Doklady, 1983, 28, 720-722). Zakharo v, 1971, Kinetic equation for solitons, Zh. Eksp. Teor.Fiz., 60,993-1000 (in Russian. Engl. transl.: Sov. Phys. JETP, 1971,33,538-541). Zakharov, 1975, Instability and nonlinear oscillations of solitons, Pis 'ma v ZhETF, 22, 364-367 (in Russian . Eng!. transl .: JETP Lett., 1975,22, 172-173). Zakharov Y.E., Manako v S.Y. , Noviko v S.P. and Pitaevsky L.P., 1980, Theory of Solitons: The Inverse Scatt ering Method, Nauka, Moscow (in Russian. Engl. trans!.: Zakharov Y.B., Manakov S.Y. , Novikov S.P. and Pitaevsky L.P., 1984, Theory ofSolitons, Consultant Bureau, New York). Zaslavsky G.M., 1972, Scattering and transformat ion of nonlinear period ic waves in an inhomo geneou s medium , Zh. Eksp. Teor Fiz., 62,2129-2140 (in Russian. Engl. trans!.: Sov. Phys. JETP, 1972,34, 622-625). Zaslavsky G.M., 1984, Stochastisity of Dynamical Sys tems, Nauka , Moscow (in Russian). Zaslavsky G.M., 1985, Chaos in Dynamic Sys tems, Harwood Academic Publi shers, NY. Zaslavsky G.M., 2005, Hamiltonian Chao s and Fractional Dynamics, Oxford University Press, New York. Zhdanov S.K., Trubnikov B.A., 1984, Pis 'ma Zh. Eksp. Teor. n«, 39, 110-113 (in Russian. Engl. transl.: JETP Lett., 1984,39, 129-132).

Chapter 6

What is Control of Thrbulence in Crossed Fields? - Don't Even Think of Eliminating All Vortexes! Dimitri Volchenkov

Abstract Convective instabil ity in the cross-field system of thermonuclear reacto rs can be overridden by poloidal drifts. While in crossed fields, a long-tim e, large-scale turbulent regime, in which the eddie s of some particular size are destined to persist longer than usual, would come into being. Perhap s, we may keep such vortexes using them as tools for mainta ining the stability of still an illusory con struct of plasma fusion .

Councilor Hamann : Down here, sometimes I think about all those people still plugged into the Matrix and when I look at these machines I can't help thinking that in a way we are plugged into them . Nco: But we control these machines; they don 't control us. Councilor Hamann: Of course not. How could they? The idea is pure nonsense. But ... it does make one wonder... j ust... what is control? Nco: If we wanted, we could shut these machines down. Councilor Hamann: Of course. That 's it. You hit it. That's control , isn't it? If we wanted we could smash them to bits! ... Although, if we did, we'd have to consider what would happen to our lights, our heat, our air... Nco: So we need machines and they need us. Is that your point, Councilor? Councilor Hamann: No. No point. Old men like me don't bother with making points. There's no point. Nco: Is that why there are no young men on the counc il? Councilor Hamann: ... Good point.

"The Matrix Reloaded" , the second film in The Matrix franchise, Writt en and directed by Andy & Larry Wachowski.

Dimitri Volchenkov I he Center 01 Excellence Cogmtlve Interaction Iechnology (Cl lEe), Umverslty 01 BIelefeld, Postfach 100131, D-33501 , Bielefeld, German y, e-mail: [email protected]

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6.1 Introduction The international project on magnetic confinement fusion is designed to make the transition from today's studies of plasma physics to future electricity-producing fusion power plants . A successful fusion device has to contain the particles in a small enough volume for a long enough time for much of the plasma to fuse. Once fusion has begun, neutrons having a vast kinetic energy radiate from the reactive regions of the plasma, crossing magnetic field lines easily due to charge neutrality, barraging, together with charged particles, the wall blanket of the containment chamber, and degrading its structure. The reliable confinement (or control) of very energetic particles is one of the crucial problems arisen in course of the fusion project. Despite lasting efforts , the strategy of effective plasma flow control of a turbulent boundary layer is still mostly unclear that threatens our hopes for the successful implementation of the project in the near future. Here, we show that control of turbulence being understood in the framework of traditional paradigm as elimination of all long-living turbulent fluctuations in plasma flows is by no means compatible with symmetry of the crossed-field system and inevitably breaks down its stability. While trying to gain control over turbulent patterns in crossed fields, we are perhaps plugged into vortexes keeping some of them as tools for maintaining the stability of still an illusory construct of plasma fusion. In the forthcoming section (Sect. 6.2), we demonstrate that while in crossed fields, an alternative long-time, large-scale sate would exist in which the eddies of some particular size are destined to persist during essentially long time. In Sect. 6.3, we investigate the stochastic problem of the long-range turbulent transport in the Scrape-Off Layer of thermonuclear reactors and calculate (in the one-loop approximation) the magnitude of poloidal drift required to override convective instability in the cross-field system. We conclude in the last section. In our study of the stochastic counterparts of models in nonlinear dynamics, deterministic trajectories are replaced by random trial trajectories of some well defined stochastic processes. The proposed approach is closely related to the Nelson stochastic mechanics, the probabilistic interpretation of dynamical equations, and the critical phenomena theory. We thoroughly use the renormalization group (RG) method - one of the most important non-perturbative techniques developed in the framework of the quantum-field theory. Asymptotic solutions for the models in stochastic dynamics are obtained in the form of a perturbation theory which can be studied by means of Feynman functional integrals . Diagram series of the perturbation theory can sometimes be studied by means of renormalization group techniques . In statistical mechanics, the RG (which is, in fact, a semi-group since the transformations are not invertible) forms an ensemble of transformations that map a Hamiltonian into another Hamiltonian by the elimination of degrees of freedom with respect to which the partition function of the system remains invariant. The RG allows calculating the critical exponents related to phase transitions in renormalizable models .

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6.2 Stochastic theory of turbulence in crossed fields: vortexes of all sizes die out, but one A model of the fully developed turbulence based on the stochastic Navier-Stokes equation with an external random force and a model of magneto hydrodynamic equation s supplemented with stochastic force terms can be formulated as the quan tum field theorie s. We use the powerful method s developed in the Quantum Field Theory to investigate the critical regimes in turbulence and their stability. The existence of dissipation minimum in the sub-leading dissipation regime predicts essentially long lifetime for eddie s of some prefer able size.

6.2.1 The method ofrenormalization group Ultraviolet renormalization has been developed in the framework of quantum field theory in 1953. An article by E.e.G. Stueckelberg and A. Peterm an in 1953 and another one by M. Gell-Mann and EE. Low in 1954 opened the field by a study of the fact of invariance of the renorm alized quantum field action under the variation of bare parameters at the subtraction point. In the framework of quantum field theory, the renorm alization group (RG) was developed to its contemporary form in the wellknown book of Bogoliubov and Shirkov, in 1959. The technique was developed further by R. Feynman, J. Schw inger and S.-I. Tomonaga, who received the Nobel prize for their contributions to quantum electrodynamics. However, these techniques have not been implemented in critical phenomena theory until the works of Leo Kadanoff who had proposed a simple blocking procedure in 1966. In 1974-1975, Kenneth Wilson had used it in order to solve the famou s Kondo problem . In 1982, he was awarded by the Nobel Prize for this work . It is important to mention , in concern with the Kondo effect, the work of P. W. Anderson , D.R. Hamman , and A. Yuval (1970) , in which the techniques similar to that of RG had been used in critical phenom ena theory, independently of Wilson 's approach. The old-style RG in particle phy sics was reformulated in 1970 in more physical terms by e.G. Callan and K. Symanzik. Later (1974) , M. Fowler and A. Zawadowski developed the method of multiplicative renorm alization in the framework of quantum-field theory. It is remark able that the mathematical background beyond the RG is quite simple and has been known long time before Peterman and Kadano ff; it is called the compactification procedure. Logarithmic divergence s arise since the integration domain is not comp act. If we find a way how to project the model onto a compact manifold (in d + I dimensional space, the new dimensional parameter f.1 is called a renorm alization mass), we gain finite amplitudes for integral s (see Fig. 6.1). In general , such a "projection" is irrelevant since it breaks the natural physic al scales, however, it may have a sense if the model possesses a property of scaling invariance.

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Fig. 6.1 The compactification procedure .

Let us suppose that we have found a way how to redefine the model on a compact manifold, and therefore the diagram series converges . Physically relevant results should not depend upon fl. The interaction charges have to be independent of u, and the sum should be invariant with respect to uniform dilatations of all its arguments. The Green function has to be an eigenfunction of the dilatation operator belonging to some eigenvalue y. If now we make a simultaneous rescaling of momentum and mass by 1\., then the Green function G would rescale with a power factor. If G has a constant infrared asymptotic (in turbulence, it is called Kolmogorov constant), we obtain the infrared scaling for the long time large scale asymptotic behavior. Iterating the RG transformations R for the particular values of the initial bare parameters, it may be possible to attain a fixed point such that H*

= R(H*) ,

(6 .1)

where H is a Hamiltonian. In critical phenomena theory, the RG transformations R rely upon the rescaling of the system variables described by the Hamiltonian H at the fixed point H * that has the same appearance whatever the scale at which it is considered. This means that the correlation function of q>(x) (e.g. q>(x) may represent the magnetization density in a magnetic system or spins in the Ising model) must be of the form

(q>(r)q> (O))

rv

r- (d-2+1) ),

(6 .2)

i.e. that the system is at the critical point, in which a correlation length ~ = 00 . If we make a change to parameters of the Hamiltonian H, in the vicinity of the fixed point, H

= H * + LgiOi , i

(6.3)

6 What is Control of Turbulence

223

where O, are called operators and gi are called fields, then we can study how the Hamiltonian evolved under the action of the RG transformations. In order to clarify the idea of the method, let us imagine that transformations we like to study forms a continuous group . Then, the fields gi have to obey the equations of motion, (6.4) If we are interested in the stability analysis of the dynamical system described by (6.4), we linearize the function f3 in its r.h.s. It is clear that linearized equations have the solutions (6 .5)

for some parameter yt > 0, so that the field gi increases due to the renormalization transformation; it is said that gi is an essential field (or a "relevant field"). Otherwise, for Yi < 0, the field gi decreases under the action of renormalization transformations, and called an inessential field (or an "irrelevant field"). Finally, if yt = 0, the field gi does not vary in the linear order and is called a marginal field. In the later case, to investigate the stability of the fixed point we need to go beyond the linear order. In critical phenomena theory, temperature and the magnetic field are those fields pertinent near the critical point. Solving the renormalization group equations, we obtain that in the vicinity of critical point the free energy is of the form F

=

(T - T,:-)(2-a ) f

(

H

(T - Tc")y+f3

)

(6.6)

and therefore satisfies the Widom 's hypothes is of homogeneity. More generally, the RG allows to predict all critical exponents pertinent to the system at the fixed point by studying how its Hamiltonian is transformed by the RG at the fixed point. There are many ways to implement the RG techniques for real-world models . In large scales (small moments), the asymptotic behavior predicted by the RG can be modified . The corrections are calculated by means of the Short Distance Expansion method . They are related to scaling behavior of composite operators, the local averages with respect to a point. Namely these quantities can be measured experimentally . If their scaling dimensions are negative, they can alternate the asymptotic behavior. Scaling dimensions are inherent not to composite operators themselves, but to their certain linear combinations which have a physical meaning. If the coupling constant in quantum field theory is not small, we have deal with the essentially non-perturbative regime, and such a theory is said to be asymptotically free for low energies. The non-perturbative regime is difficult to study, because of in addition to the problem of divergences of Feynman diagrams in perturbative series we have to deal with the essentially non-perturbative contributions coming from the instantons which cannot be neglected . In quantum mechanics and quantum field theory, an instanton is a classic solution of equations of motion, i.e. one of local minimums of the action functional , but not the global one. Mathematical methods developed in quantum fields theory are beyond any doubt applicable also in Eu-

224

Dimitri Volchenkov

clidean space to classical problems involving random fields. Models of stochastic nonlinear dynamics can be reformulated as models of quantum fields theory, and then the powerful techniques developed in that can be used.

6.2.2 Phenomenology offully developed isotropic turbulence Despite more than a century of work and a number of important insights, a complete understanding of turbulence remains elusive, as witnessed by the lack of fully satisfactory theories of such basic aspects as transition and the Kolmogorov "5 /3-spectrum". In phenomenological theory of turbulence formulated by A.N. Kolmogorov in 1941, it was conjectured that the correlation functions of velocity in some intermediate scales (called the inertial interval) depend upon the only dimensional parameter, the power of energy pump W . It was supposed that energy comes from large scale eddies which bifurcate due to nonlinear interactions until the small scale vortexes are dissipated in fluid at the minimal scale. The only physically relevant combination of energy pumping rate Wand momentum k gives the Kolmogorov asymptotic for the fully developed turbulence. It follows then that the velocity of fluid has a formal dimension -1 /3 , and the famous I-dimensional energy spectrum is -5 /3 . This result has been justified in the framework of RG techniques by many authors. The recent theoretical, compu tational and experimental results dealing with homogeneous turbulence dynamics have been summarized in (Sagaut and Cambon, 2009) . In the present section, we follow the seminal work (Adzhemyan et al., 1996). To describe the spectral properties of incompressible fluids in the inertial range of developed turbulence, one considers the stochastic Navier-Stokes equation with an external random force (Monin and Yaglom, 1971, 1975; Wyld, 1961). (6.7) here ~i is the vector velocity field, which is transverse due to the incompressibility condition, p and F; are the scalar pressure field and transverse external random force per unit mass (all these quantities depend on x == (t,x)), Vo is the kinematical coefficient of viscosity, and VI is the Galilean-invariant covariant derivative . Equation (6.7) is studied on the entire t axis and is supplemented by the retarda tion condition and by the condition that ~ vanish asymptotically for t ----+ - 0 0 . We take F to be a Gaussian distribution with zero average and correlator

(t - t') (F;() x Fj (x ' )) = 8(2;rr)d where

J () () .( ') dk Fij k dF k exp lk x - x , k.k,

PiIJ-(k) = 8--_ _kI_J IJ 2

(6.8)

(6.9)

225

6 What is Control of Turbulence

is the matrix of transverse projector in the momentum (Fourier) representation, dF(k) is some function of the momentum k == [k] and the model parameters, and d is the dimension of the physical space. The introduction of a random force phenomenologically models the stochastic drive (which in a real situation must arise spontaneously as a consequence of the instability of laminar flow) and, at the same time, the injection of energy into the system owing to the interaction with large-scale eddies. The average power W of the energy injection is related to the function d F in (6.8) by the equation

W

=

d-I 2(2n)"

j'dkdF(k) .

(6.10)

In the stochastic problem we can also do away with specific initial and boundary conditions and directly study homogeneous, fully developed turbulence (Monin and Yaglom, 1971, 1975 ; Wyld, 1961) . The field

0 for small e > O. When the IR-stable fixed point is present , the leading terms of the IR asymptote of the Green functions W;~ of any single charge model satisfy the RG equation with the replacement g ---+ g*. In particular, we obtain (6.32) Canonical scale invariance is expressed by the equations (6.33) in which F

= {t ,x ,,u,v,g ,m} is the set of all arguments of

W;~,

n = {nqJ ,n~} and

d}W are the canonical dimensions of F in the action functional (6.17): d~ d~1

=d+

I, d~',f1

=

I, dt

=

-2, d;

= 2£

=

(0 in the logarithmic theory) , d~

-I ,

=

I,

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Dimitri Volchenkov

d:, = -I , d,C::,/l = 0, dc:l = I, d: = O. The canonical dimensions of w,~ are the sums of canonical dimensions of their arguments. We are interested in the scaling with dilatations of t , x, and m for fixed J1 , v, and g. Substituting the canonical dimensions into (6.32) and (6.33), after eliminating J1a/l and va v we obta in the equation of critical scaling (6.34) with the coefficients (6.35) which are the corresponding critical dimensions. Substituting the known values 'tv = -nqJ + (d + I )nqJI , and d$n = nqJ - nqJ' (the dimension of the connected function w,~ is equal to the sum of the dimensions of its fields) into (6.35), we obtain the following expressions for the critical dimensions:

2£ /3 , d~, =

,1qJ = I -

2£ /3, ,1 r

=

,1qJl

-,1 w

=d-

,1qJ,

= -2+2£/3 .

,1111

=

I,

(6.36)

They do not have terms of order £2 , £ 3 , and so on, and coincide with the Kolmogorov dimensions (6.13) for the real value e = 2. This main result has been reproduced in (De Dominicis and Martin, 1979) for the first time. Influence of weak uniaxial small-scale anisotropy on the stability of inertial range scaling regimes in a model of a passive transverse vector field advected by an incompressible turbulent flow was investigated in (Jurcisinova et al., 2006) by means of the field theoretic renormalization group . Weak anisotropy means that parameters which describe anisotropy are chosen to be close to zero, therefore in all expressions it is enough to leave only linear terms in anisotropy parameters. Turbulent fluctuations of the velocity field are taken to have the Gaussian statistics with zero mean and defined noise with finite correlations in time. In (Jurcisinova et al., 2006) , it was shown that stability of the inertial-range scaling regimes in the three-dimensional case is not destroyed by anisotropy but the corresponding stability of the two-dimensional system can be destroyed even by the presence of weak anisotropy. Critical behavior of a fluid, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group in (Antonov and Ignatieva, 2006) in a simplified model where relaxation stochastic dynamics of a non-conserved scalar order parameter was coupled to a random velocity field with prescribed statistics . Existence of a new, non-equilibrium and strongly anisotropic, type of critical behavior (universality class) was established, and the corresponding critical dimensions were calculated. The scaling behavior appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different.

6 What is Control of Turbulenc e

233

6.2.6 What do the RG representations sum? In the renormalized theory all quantities are calculated as series in the charge g of the type R(g , ... )

=

['gIlR Il(··· ).

(6.37)

11 =1

For any initial value of charge gO, the renormalized value g E [O,g. rv e], and so it can be assumed that g. c::: e. Owing to the smallness of g rv e and the absence of poles in e in the coefficient s of the series (which are elimin ated by the UV renorm alization) , it may seem that in the e scheme there is absolutely no need for any infinite summation of contributions of the series (6.37). This is true, but not for the critical region : direct calculations show that the coefficients RIl involve factors of the type (s-2 e - I)/ e, which are UV-finite (in the limit e ----+ 0, S = canst ) and of order unity for s = k/)1 c::: I, but become of order 1/ e and higher for [s In s] ?: I. The maximum number of such "large" factors of order 1/ e in the terms of the perturbation series never exceeds the number of "small " factors g rv e. These two numbers have to be approximately equal that means at small s we have the new parameter

~ == ~(s-2e -I) ,

e

(6.38)

all powers of which must be summed for each order in e. This is the statement of the first infrared problem in the language of the renorm alized theory with small e > 0.

6.2.7 Stochastic magnetic hydrodynamics In this section , we consider the field-theoretic analysis of several problems in magneto hydrodynamic statistics . These problems include the inertial-range scaling laws in incompressible fully developed turbulence of conductive fluid corre spond to various regime s of physical scaling behavior in a model of magneto hydrod ynamic (MHD) equations supplemented with stochastic force terms and the scaling exponents of some composite operators . The first attempt to study the MHD model has been performed in (Fournier et al., 1982), but it was incomplete (see the details below) . Then, the correct model had been propo sed in (Adzhemyan et al., 1985), but the renorm alization had been made erroneous. Here, we present the correct version of the renormali zed magnetohydrodynamics. The stochastic MHD equations for two transverse vector fields ( v is the velocity field and is the magnetic field) are written as

e

(6.39) (6.40)

234

Dimitri Volchenkov

r

where A = c2 / 4n O" y is the inverse magnetic Prandtl-type constant, and I" and are the Gaussian random force and curl of the random current with mean zero values and covariance

U? (r,t)f! (r' ,t' )) = o(t -

t')D~{3 (r - r'},

a, f3 = v, e.

(6.41)

where e = B/ .j4np, B is the magnetic induction, 0" is the conductivity, and p is the density of medium, p is the scalar pressure field, c is the speed of light. The transversal condition for the velocity field v follows from the incompressibility constraint:

J ·v=O. The purely longitudinal contribution of the pressure in (6.39) can be eliminated by inserting a transverse projection operator I1s = Ois - kikslk2 onto solenoidal vector fields in front of the longitudinal contributing factors : (vJ)v to V'tV and (eJ)e . The correction to (6.39) is given by the Lorenz force which is proportional to [curlB x B]

= (BJ)B - J(B 2 / 2),

where the second term is included into the pressure p, and by (6.40) which follow s from Ohm 's law for a moving medium in the simplest form, j

= O"(E+ [v x B]/ c),

and from Maxwell's equations without allowance for the displacement current. The Fourier transform of D~{3 or force spectrum,D~{3 (k), is necessarily non-negative. The canonical dimensions of these fields are the same, and the inverse magnetic Prandtl constant is dimensionless. In the massless models, the covariances for the random forces (6.41) are chosen in the power-law forms : 1'1' Dis

= gl Y 3 Pis d d.;

(6.42)

VI' ,

= k 4 - d - 2e ,

dee

= k4 - d -2ae ,

d I,e -- k 3 - d - ( I+a)e .

The amplitude factor gi (i = 1,2,3) in the correlation functions play the role of coupling constants (y 3 is separated from the amplitude factors for the future convenience) . The coupling constant g3, in the mixed correlation function can be defined as g3 = ~ .jglg2 where ~ is not in essence a charge but an arbitrary parameter of the theory. This situation is analogous to the gauge parameter found in quantum electrodynamics. ~ is subject to the inequality I~ I:::; I, which follows from the requirement that the matrix of correlation functions be positive. The value e = 2 corresponds in the momentum representation to o(k), this expressing the idea of "pumping of energy from the large-scale motion". The theory is renormalizable and logarithmic for E = O. The positive constant a in the exponent of the magnetic correlation function is an arbitrary parameter of the theory and models the difference between the spectra of magnetic and hydrodynamic energy pumping. Cism is the completely antisymmetric pseudo-tensor. The index structure of the correlators (6.42) is determined by the

6 What is Control of Turbulence

235

requirements that the fields be transverse and spatial parity be conserved: the field Vi is a vector, while i is a pseudovector, and so the mixed correlator is a pseudo-tensor. It is automatically transverse in the indexes i and s. In calculations performed in the spirit of dimensional regularization, the symbols Oik and Cism can formally be used for arbitrary dimension d , but in the final expres sions the symbol Cism, in contrast to Ois , is mean ingful only for the real dimen sion d = 3. In other dimensions, say, in d = 2, there is no pseudo-tensor which is transverse in both indexes, and the mixed correlator must be taken to be zero. The mixed correlator was not introduced in (Fournier et al., 1982) and had been introduced in (Adzhemyan et al., 1985).

e

6.2.8 Renormalization group in magnetic hydrodynamics Stochastic dynamic s allows for the path-integral repre sentation of the probabilit y generating functionals, using the so-called MSR action (Martin et al., 1973), (de Dominicis, 1976), (de Dominicis and Peliti, 1978). The generating function al of renormalized correlation functions takes the form G(A do not get admixed to any other one and are not renormalized, as soon as all the appropriate I-irreducible functions are in reality equal to zero due to the presence of closed loops of the advanced functions . Formally, these diagrams are logarithmic, but in reality, the structure of interactions of action (6.44) provides the removing of one derivative from the loop to the external 1/>' -line and red uces the diagram index of divergence. Thus, the scaling dimension of these operators are equal to the space dimension, L1 (1/>'1/» = 3. The operators, with given derivative, (dl/>I/»i , ca n be reduced to a differential, d( 1/>1/», due to the fact that the fields I/> == {v, e} are transversal. The scaling dimensions of such operators are L1 (d(I/> I/> ))

= L1 ( I/> I/> ) + I ,

I/>

== {v, e} .

The rest of vector operators, FI = Viv2 and F2 = Vie2, are the true tensors, yet F3 = e ie 2 and F4 = ev 2 are pseudotensors, so that these pairs do not be admixed on renormalization . Due to Galilean invariance property of the action, one ca n prove that FI is finite and it does not be admixed to F2, as well as F3 does not be admixed to F4 , (ZII = I, Z21 = 0, Z34 = 0). At the "k inetic " fixed point these operators have different powers of the coupling constant g2. Yet at the "magnetic" point all the non-diagonal elements of a renormalization matrix are equal to zero as being proportional to 1.* = O. Thus, the scaling exponents are determined simply from the diagonal elements of renormalization matrix just as in case of a simple multiplicative renormalization, and we shall say that these dimensions are "associated" with the corresponding composite operators. A deviation of the scaling dimensions from the canonical dimensions are the following :

YII

= 0, 'Y22 = 2(CI - I )y, Y33 = - 6CIY, Y44 = - 2y,

where Y = gJ/ BA - g2/ BA2, CI = (d + 2)(d - 1)(1. + 1) -1 . The scaling exponents are listed in Table 6.2.

6 What is Control of Turbulence

243

Table 6.2 The critic al dimension s of the comp osite operators (1jJ 1jJ )1jJ; Composite Operator V;V

2

e2 e;e 2 V;

e;v2

Kinetic regime

Magnetic regime

3 - 2e 3 - 2(a + 0.507 )e 3 - 3(a + I)e 3 -(a + 0.6S)e

3 3 + ISae 3 +60ae 3 - 2ae

6.2.13 Instability in magnetic hydrodynamics In accordance with the SDE method, the inertial-range asymptote of a I-partical correlation function can be expressed as follows :

< 1/>I(k,t)/fJ2(-k,t)

>=Ak-d- L1

q> I -

L1 \l:2

(I + ~bi (~tF;),

{1/>I ,/fJ2} == {v,e} ,

(6.66) where A is a Kolmogorov-type constant, and .11jl are the scaling dimen sions of the fields. One can see that RG-predicted spectrum is secure as long as .1F; > O. If .1F; is negative, the appropriate contribution changes the scaling asymptote of correlation function in the inertial range . If we are interested in asymptote of static correlation functions.z.e., they do not depend on time , we do not con sider the contributions of tho se operator s that are not Galile an invariant. It is quite clear that such a contribution would depend on the parameter of Galilean tran sformation b(t), but these operators contribute to asymptote of dynamic correlation funct ions . For the real value of E = 2 some Galilean invariant operators have negative scaling exponents at the "kinetic" and "magnetic" fixed point s for some values of the pump parameter a:

Notice, that the last one doe s not contribute to the asymptotes of I-partical static correlation functions at the "k inetic" point. This operator is a pseudotensor, it cannot contribute to real tensor correlation functions, yet its contribution to the mixed I-partical correlation function , < ve >, is O(g2* ) and can be neglected. For the same reason, the operators e2 and eiek do not contribute to asymptotic of I-partical hydrodynamic correlation function , < vv >. Hence, in the static case, for < vv > and < ve > the RG-predicted scaling asymptotes is secure for the both "kinetic" and "magnetic" points. If a > ~ , the exponent for < ee > should be corrected at the "kinetic" point as follows .1( < ee

» = -d -

2.1e - .1 (e 2 ) .

244

Dimitri Volchenkov

This result has a simple physical meaning . At the "kinetic" point the magnetic field is passively advected to hydrodynamics. When the value of the parameter a is comparable to I, the spectrum of magnetic pump is infrared localized . It means that the inertial-range motion is exposed to ambient magnetic field, which depends on the hydrodynamics. In this case, some instabilities arise in MHD system, (Lifschitz, 1989), which are driven by the magnetic pressure gradient J(B 2 /2) . Yet the velocity field correlation function is virtually unaffected by these instabilities, but they contribute to the magnetic field correlation function. Likewise the usual hydrodynamics, at the "kinetic" critical regime the inertialrange asymptotes of dynamic correlation functions have a lot of essential contributions from Galilean non-invariant operators. In this sense, one can say, (Eyink, 1994), that there are infinitely many fixed points in the fully developed turbulence. However, at the "magnetic" critical regime, the scaling asymptotes for the < vv > - and < ee > -functions still have the same value as predicted by RG. The mixed function exponent has a correction associated with the operator i V2 , which becomes essential, while a > ~ . It is important to notice that in this case the contributions from each of the eivll-type operators are also possible. We do not know their scaling dimensions, so that this scaling exponent would be corrected as follows

e

.1 « ve »

= -d -.1e -

.1" - max [.1 (evil)].

6.2.14 Long life to eddies of a preferable size In the present section, we compute the scaling asymptote of the spectral density tensor of energy dissipated in a unit time per unit mass by the magnetic hydrodynamical system being in the "kinetic" critical regime, (Landau and Lifshitz, 1995; Landau et al., 1995):

e = V'O'V+AV [V' x ef,

(6.67)

averaged with respect to the statistic of Gaussian distributed random force f . Here & is a tension tensor of the incompressible fluid. Doing some basic calculations, we arrive at

where the angular brackets denote the average with respect to configurations f(x ,t). The result of RG-transformation acting on a renormalized composite operator (the local average of fields and their derivatives with respect to one point) is always a linear combination of the renormalized composite operators having the same symmetry, structure, and canonical dimension . This fact is known as a miming of composite operators (Collins , 1992). Denoting the renormalized composite operators of a mixing set as FjR, we write the RG equation for them in the form :

6 What is Control of Turbulence

245

(6 .69) where ,1ij is the matrix of critical exponents. The linear combinations 2' {FjR}, for which ,1ij has the diagonal form, have the definite physical meaning and correspond to the certain physical processes. The unique property of the energy dissipation composite operator e is that two different eigenvectors of the RG-operator have the same eigenvalue corresponding to the zero anomalous dimension y = O. Consequently, the relevant critical dimension matrix ,1F can be transformed merely to the Jordan form

DRC2'{FjR} = ,1L2'{FjR} , { DRC2" {FjR} = ,1L2" {FjR} + 2'{FjR} ,

(6 .70)

in which 2' {FjR} is the eigenvector and 2" {FjR} is the adjacent vector. ,1L is the shared critical exponent of 2' and 2". Such a phenomenon has not been discussed neither in the quantum fields theory literature nor in the statistical physics before. In the asymptotic region k] J1 ----+ 0, one can solve (6.70) to obtain (6.71) where C] and C2 are the normalized amplitude factors . Now we calculate the critical exponents ,1L and the relevant linear combination s 2' of renormalized composite operators explicitly. Note that the energy dissipation operator e is a sum of local composite operators of the canonical dimension dF = 4. The result of the cat ion of the DRc-operator on e can be written in the form (6 .72)

where K, and K2 are some linear combinations of the renormalized composite operators V'/V'kG~ , in which dc = 2, and FjR with dF = 4, and there fore K] and K2 are renormalized separately with no mixing. The set of composite operators Gik with dc = 2 reduces to the tensor operator q,iq,k which is a sum of scalar and zero -trace operators,

q,

= {v, tl} .

We have studied the renormalization of this family in the previous subsection. All of them have their own critical dimensions independently of others and contribute to the following part of the energy dissipation function :

The family of composite operators dF

= 4 consists of 7 items :

Dimitri Volchenkov

246

FI= (V4 ),

F2= (v282),

F5 = (8· .18) ,

r"3 = (v-ziv), F4= (V·(8 ·V8 )), F7 = (8 4 ) .

F6 = (8· (8 · Vv) ),

Linear combinations of their renormalized analogs contribute to another part of the energy dissipation function :

The renormalized operators F{ are related to the not renormalized ones , Fk, by the linear equations Fj =ZikF{, in which Zik are the renormalization constants found from the requirements that all correlation functions with one FjR and any number of fields v, 8, v', and 8' are finite as the UV-cutoff A ----+ 00. The problem of computation of the entries Zik can be substantially simplified by the symmetry arguments. For instance, since the model of magnetic hydrodynamics is invariant under the Galilean transformation of fields, the composite operators which break this symmetry is not renormalized being finite. Moreover, they do not mix to any other Galilean invariant operator. Therefore, ZII

=

I,

Zj2

Zk3

Zil

= 0,

= 0,

"Ii> I ,

"Ij > 2,

(6.73)

= Zk4 = 0, 'ik > 4.

Then, one can use the Schwinger functional equations and the Ward identities expressing the Galilean invariance of the MHD model: (6.74) where cp == {v , 8} and cp' == {v', 8 '} , AIJ! and A IJ!I are the relevant source functions . The r.h.s. of (6.74) is UV-finite and has the definite critical exponents independently. Therefore, the operators in the I.h.s. should also be UV-finite having the definite critical exponents:

(VZ, V.L1 V+ Z3V. (8 . V8 )) { ()., vZ28· L1 8 + Z38· (8 · Vv))

< 00, < 00,

(6.75)

where Z, ,2,3 are the renormalization constants of MHD-action. UV-finiteness means that the divergent parts of the renormalization constants subtract each other in the combinations (6.75) , therefore,

Zi3+ a Z i4=0 ,

U:t5 ,6), Z55=Z2 1 ,

Z66=Z3

1

,

247

6 What is Control of Turbulence

where a = g(d - I )/2d£(1 + A )(4n )d/2r(d /2 ). Other nontrivial entries of Z - ikmatrix require evaluation of the diverging parts of the relevant I-irreducible diagram s and remain unknown . Neverthele ss, we show that (6.73) and (6.75) provide us enough information to define the critical exponents of the operator £, . The matrix Z ik appears to be triangular and its diagonal elements give us the complete set of anomalou s exponents, 11k = -2£ · dJg logZik + 0 (£2 ), (6.76) The zero eigenvalue is twice degenerated, y" = Y22 = 0, the entries "tn , Y22, Y43 and Y33 so not equal to zero and still unknown , however, they do not contribute to £], £,

= EWijFj ,

Wij

= diag [O,O, 1,0,1.,0,0] .

i

Denote the matrix which transform s Wij into the Jordan form as U , then, the linear combin ations of renorm alized compo site operators which corre spond to £, in the process of UV-renormalization are

There are two such combinations:

in which a] ,2 are some analytical coefficients expre ssed via the unknown entrie s (which are obviously finite even as E --+ 0). We have to emphasize that is a vector defined in the two-dimen sional eigensubspace of the RG-operator having the single eigenvalue Y= where

Zik]

Lf

°

L'/ = (v·L1 v + Z3Z1]v , ( e · ve) ),

L'{R

= (Abf e· L1e + z:; ' e · (e · vv) + zl ' ( 4 )

(6 .77)

are the linearly independent vectors spanning this eigensubspace. The critical expon ent relevant to is L1 L 1 = 4 - 4£ /3 = 4/3 (for E = 2). The anomalou s exponent corre spond ing to the second combin ation, L~ , is - Y2 that gives L1L2 = 4 - 2£ = (s = 2). The critical dimensions of the compo site operators G ik are L1c = 2 + L1t/J;I!Jk where L1Mk are the critical dimensions of composite operators with d F = 2 studied in the previous subsection. Certa inly, we have:

Lf

°

(6.78) All value s are computed for

e = 2.

248

Dimitri Volchenkov

We conclude this section collecting the results on the critical exponents of the energy dissipation function f. In Fig. 6.4, we have presented the different asymptotic contributions into the energy dissipation function E == f(k) /£o via the dimensionless scaling parameter s == k] j1 where £0 is a constant energy dissipation rate. The sum of all contributions is drawn with the black bold line. The uniform rate £0 is given by the thin gray horizontal line, two power-law asymptotic contributions are represented by the dotted lines. At a decided disadvantage for the small scale eddied (s » I) (in the far-dissipation range) the total dissipation rate increases considerably, therefore, they dissipate very fast. Fig. 6.4 displays that there are two opportunities for the long-time , large-scale asymptotic behavior of f . In the inertial range, indeed, the constant dissipation rate £0 dominates the dissipation process. Nevertheless, in the MHD model, the alternative asymptotic "steady state" exist, and it would come into play when the regime characterized by the constant dissipation rate looses the stability. One can see on Fig. 6.4 that both dissipating regimes meet precisely at the dissipation wave number k" = j1 ~ A and at the pumping scale kj1-1 ~ O. The alternative dissipating regime has a minimum somewhere in between these two points . The existence of dissipation minimum in the sub-leading dissipation regime predicts essentially long lifetime for the eddies of some preferable size €. The linear combination is responsible for the long -time breaking of the size equivalence of eddies in the inertial range . From (6.77) one can see that it describes a kind of feedback controlling loop, accentuating the eddies of particular size and suppressing the turbulence in other scales by shadowing one of the infinitely many unstable periodic orbits embedded in the chaotic turbulent attractor. When the trajectory converges to the optimal orbit, the feedback term vanishes identically,

Lf

The energy dissipation rate 5

4

o

0.5

1.5

S

2

2.5

3

Fig. 6.4 The critical exponents of composite operators of the canonical dimension 4 in the model of magnetic hydrodynamics in the kinetic critical regime .

6 What is Control of Turbulence

249

The latter equation defines the configurations of fields fJ and v relevant to the alternative dissipation regime .

6.3 In search of lost stability The functional formulation of long-range turbulent transport problem in the ScrapeOff Layer (SOL) of thermonuclear reactors reveals convective instability in the cross-field system which can be override by a finely tuned poloidal drift. We also consider a simple qualitative discrete time model of anomalous transport in the SOL which exhibits a surprising qualitative similarity to the actual flux driven anomalous transport events reported in experiments.

6.3.1 Phenomenology of long-range turbulent transport in the scrape-offlayer (SOL) ofthermonuclear reactors Turbulence stabilization in plasma close to the wall blanket of the ITER divertor is the important technical problem determining the performance of the next step device. Long range transport in the scrape-off layer (SOL) provokes the plasma wall interactions in areas that are not designed for this purpose . Evidence of the strong outward bursts of particle density propagating ballistically with rather high velocities far beyond the e-folding length in the SOL has been observed recently in several experiments (Rudakov et aI., 2002 ; Antar et al., 2001) and in the numerical simulations (Ghendrih et al., 2003) . These events do not appear to fit into the standard view of diffusive transport: the probability distribution function (pdf) of the particle flux departs from the Gaussian distribution forming a long tail which dominates at high positive flux of particles (Ghendrih et al., 2003) . Theoretical investigations of the reported phenomena remain an important task. In the forthcoming sections, we consider a variety of two dimensional fluid models based on the interchange instability in plasma studied in (Nedospasov, 1989; Garbet, 1991) and discussed recently in (Ghendrih et al., 2003) exerted to the Gaussian distributed external random forces to get an insight into the properties of turbulent transport in the cross-field system . The E x B drift motion of charged test particle dynamics in the SOL was analyzed to investigate a transport control strategy based on Hamiltonian dynamics in (Ciraolo et al., 2007) . A method of control which is able to create barriers to magnetic field line diffusion by a small modification of the magnetic perturbation has been proposed in (Chandre et al., 2006). This method of control is based on a localized control of chaos in Hamiltonian systems .

Dimitri Volchenkov

250

Neglecting for the dissipation processes in plasma under the constant temperatures T; » Ti, this problem is reduced to the interactions between the normalized particle density field n(x,y,t) and the normalized vorticity field w(x,y,t) related to the electric potential field ep(x,y,t), (6.79) defined in the 20 plane transversal to ez , the direction of axial magnetic field Bo. In (6.79), x and yare the normalized radial and poloidal coordinates respectively. The Poisson's brackets are defined by

When g = 0, Equations (6.79) describe the 2D-rotations of the density and vorticity gradients around the cross-field drift v

= -elBo V'ep x ez,

in which V' == (~r,()y) . Their laminar solutions (with w = 0) are given by any spatially homogeneous electric potential ep = 1/1, (z ) and any stationary particle density distribution n = 1/12 (x,y). Other configurations satisfying (6.79) at g = 0 are characterized by the radially symmetric stationary vorticity fields

with the electric potentials invariant with respect to the Galilean transformation

ep(x) ---., ep(x) +xq>, (r) + C!'2(t) , where the parameters of transformations q>' ,2 (z ) are the arbitrary integrable functions of time decaying at t ---., - 0 0 . The relevant density configurations

n = q>3 (x, y-

[ 00 vy(x,t/) dt')

have the form of profile-preserving waves convected in the poloidal direction by the poloidal cross field drift vy(x,t) . The poloidal component of cross field drift itself remains invariant with respect to the Galilean transformation v y ---., v y + q>, (z ), while its radial component Vx = O. Configurations that satisfy (6.79) for g > 0 have the Boltzmann density distribution of particles in the poloidal direction. In particular, those solutions compatible with the Galilean symmetry discussed above (with v y "* 0) are the solitons (solitary waves) of density convected by the poloidal electric drift,

nocexp-

/

gT x,y

) Iy-jt Vy(x,y,t')dt'l , -

00

6 What is Control of Turbulence

251

where T (x, y) is an arbitrary function twice integrable over its domain. In addition to them, for g > 0, Equation. (6.79) allows forthe radially homogeneous configurations a.rn = 0, IV = U(y) mod2n with v y = 0 which do not fit into the Galilean symmetry, these are the steady waves,

11

Y

n ex: exp -U(y, )dy, . g 0

The latter solution does not possess a reference angle and can be considered as an infinitely degenerated state of the system since the relevant configurations {n, w} can be made equal at any number of points by the appropriate choice of U : U (Yl) = U(Y2) = ... = U(YIl), and Jg 1 U(y')dy' = Jg2 U(y')dy' = ... = Jg" U(y')dy' . For instance, it can be represented by the periodic lattice potential controlled by the spokes of high particle density radiating from the center. With two concurrent symmetries there can occur either the frustration of one of them or the vanishing of both with the consequent appearance of a complicated dynamic picture that is most likely stochastic . The latter case corresponds to a maximally symmetric motion resulting from the destruction of unperturbed symmetries (Sagdeev and Zaslavsky, 1986). In particular, instability in the system (6.79) occurs with respect to any small perturbation either of density or vorticity. Accounting for the dissipation processes in plasma smears the picture, so that the small scale fluctuations would acquire stability. We demonstrate that the small scales fluctuations can be stable provided there exist the reciprocal correlations between the stochastic sources of density and vorticity in the dynamical equations. The large scale stability of a fluctuation can be characterized by the order parameter

in the momentum space where kx and ky are the radial and poloidal components of momenta respectively. For the uncorrelated random forces (under the white noise assumption), a fluctuation with ~ > 0 is unstable with respect to the large scale asymptote in the stochastic problem. The accounting for the convection of particles by the random vortexes introduces a finite reciprocal correlation time 'rc ( [r - r'[) between the density and vorticity random forces. Then there exists the critical scale ~c , in the stochastic model, such that a fluctuation with ~ < ~c vanishes with time, but its amplitude grows up unboundedly with time, for ~ > ~c. Biasing of wall components can locally modify turbulent transport and is considered to be beneficial if one aims to insulate the Tokamak main chambers from the bursts of density (Ghendrih et aI., 2003). Indeed, the generation of a uniform electric drift in the poloidal direction, v y ---+ v y - V, would frustrate one of the symmetries in (6.79) reestablishing the Galilean invariance in the system . For instance, those configurations characterized by the trivial poloidal component of electric drift v y = 0 would be eradicated. We investigate the problem of turbulence stabilization close to the divertor wall in the first order of perturbation theory and shown that there exists a critical value IVc I < 00 of the poloidal electric drift which would suppress the

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large scale instability in the stochastic system with the correlated statistics of random forces, ~c > O. However, for the uncorrelated random sources in the stochastic problem, ~c = 0 and I Vc I ----+ 00 as k ----+ O. Correlations between the unstable fluctuations of density at different points are described by the advanced Green 's functions which are trivial for t > O. In particular, these functions determine the concentration profile of the unstable fluctuation s of density which increases steeply toward the wall. The size of such fluctuations grows linearly with time. In this case, the statistics of the transport events responsible for the long tail of the flux pdf is featured by the distribution of the characteristic wandering times of growing blobs convected by the highly irregular turbulent flow in the close proximity of the divertor wall. In our model, we have replaced this complicated dynamics with the one dimensional (the radial symmetry is implied in the problem) discrete time random walks. Such a discrete time model would have another interpretation: the advanced Green 's function is a kernel of an integral equation which relates the amplitudes of the growing fluctuations apart from the wall with those on the wall, in the stochastic dynamical problem. Indeed , this equation is rather complicated and hardly allows for a rigorous solution. Therefore, being interested in the qualitative understanding of statistics of the turbulent transport in the SOL , we develop a Monte Carlo discrete time simulation procedure which would help us to evaluate the asymptotic solutions of the given integral equation. General approach to the probability distributions of arrival times in such a discrete time model has been developed recently in (Flori ani et aI., 2003) . In general, its statistics can exhibit the multi-variant asymptotic behavior. Referring the reader to (Floriani et aI., 2003) for the details , we have shown that the statistics of arrival times for the unstable fluctuations is either exponential or bounded by the exponentials (in particular, the latter would be true in the case of the randomly roaming wall) that is in a qualitative agreement with the data of numerical simulations and experiments (Ghendrih et aI., 2003).

6.3.2 Stochastic models ofturbulent transport in cross-field systems The stochastic models of cross field turbulent transport used in the forthcoming sections refers to the effectively two-dimensional fluid model of plasma based on the interchange instability in the SOL (Nedospasov, 1989; Garbet, 199 I) recently discussed in (Ghendrih et aI., 2003) . In this model, one assumes the temperatures of ions and electrons to be constant, 'L « Te . Then the problem is reduced to that of two coupled fields, the fluctuations of normalized particle density n(x,y, t) and that of vorticity field w(x,y, t), governed by the following equations t7

_

vIW -

A

UOV Ll l- W -

( _ I )k8k ::l k '-k~ 1 - k - uyn

,,

+ I,IV ,

(6.80)

6 What is Control of Turbulence

253

written in the polar frame of reference with the normalized radial x = (r - a)/ p, and y = Ps poloidal coordinates. Time and space are normalized respectively to Qi- 1, the inverse ion cyclotron frequency, and to Ps, the hybrid Larmor radius. The covariant derivative is

a8/

in which V' == ({Jr , Oy), and il l- is the Laplace operator defined on the plane transversal to the axial magnetic field. The effective drive 0 0 modeling the injection of particles from the divertor core along with the perturbations risen in the system due to the Langmuir probes (Gunn, 200 I; Labombard, 2002). For a simplicity, we assume that the processes of gain and loss of particles are balanced in average therefore (f,,) = O. The stochastic source of particles is used instead of the continuously acting radial Gaussian shaped source (localized at x = 0) studied in the numerical simulations (Ghendrih et al., 2003). Similarly, we impose the random helicity source fw exerting onto the vorticity dynamics in (6.80). Furthermore, the random sources f" and fw account for the internal noise risen due to the microscopic degrees of freedom eliminated from the phenomenological equations (6.80). From the technical point of view, the random forces help to construct a forthright statistical approach to the turbulent transport in the SOL. In particular, it allows for the quantum field theory formulation of the stochastic dynamical problem (6.80) (based on the Martin-Siggia-Rose (MSR) formalism (Martin et al., 1973» that gives a key for the use of advanced analytical methods of modern critical phenomena theory (Ma, 1976). The Gaussian statistics of random forces in (6.80) is determined by their covariances,

DIlIl(r- r', t - t ') == (f,,(r, t )f" (r',t') ) ,

Dww(r - r', t - t')

== \fw(r, t )fw(r', t ') ) ,

Dimitri Volchcnkov

254

where r == (x,y). describing the detailed microscopic properties of the stochastic dynamical system . We discuss the large scale asymptotic behavior of the response functions ( 8n(r ,t) / 81,,(0 ,0) ) and ( 8n(r ,t) / 8fw(0,0) ) quantifying the reaction of system onto the external perturbation and corresponding to the r-distributions of particle density fluctuations expected at time t > 0 in a response to the external disturbances of density and vorticity occurring at the origin at time t = O. The high order response functions are related to the analogous multipoint distribution F.Il ( rl ,tl, "' , rll,tll,. r ' I ,t(, " "' , r Il,tll ) as . f unctions

8" [n(rl,tl )'" n(rll,tll)] ) ( 81" (r'j , t] ) ... 1" (r'll ' til ) -

"( " ') z: F" rl,tl , "' , rll,tll; r '(,tl' "' , r ",t"

permut ation s

with summation over all n! permutations of their arguments rl ,tl , ... , r" , t". We consider a variety of microscopic models for the random forces 1" and tin the stochastic problem (6.80) . Under the statistically simplest "white noise" assumption, these random forces are uncorrelated in space and time, DIlIl (r - r', t - t')

= 1" 8(r' - r) 8(t - t'),

Dww(r - r' , t - t') = T;v 8(r' - r) 8(t - t') ,

(6.81)

in which 1" and T;v are the related Onsager coefficients. Recent studies reported on the statistics of transport events in the cross- field systems (Ghendrih et aI., 2003 ; Carreras, 1996) pointed out the virtual importance of correlations existing between density and vorticity fluctuations in the dynamical problem . In particular, this effect is referred to the formation of large density blobs of particles close to the divertor walls by attracting particles via the cross field flow, the latter being the larger for strong blobs with strong potential gradients (Ghen drih et aI., 2003) . Indeed, in the physically realistic models of turbulent transport in the SOL, it seems natural to assume that the random perturbations enter into the system in a correlated way. To be specific, let us suppose that there exists a finite reciprocal correlation time 'rc (I r' - r I) > 0 between the random sources fw(r, 'rc ) and f,,(r',O) in the stochastic problem (6.80) . For a simplicity, we suppose that the relevant relaxation dynamics is given by the Langevin equation, (6.82) in which f3 ~ ( f~) > O. In the momentum representation, the non-local covariance operator 'r(-:-I can be specified by the pseudo-differential operator with the kernel (6.83) which specifies the characteristic viscoelastic interactions between the "fast" modes of density and vorticity fluctuations. The coupling constant A, > 0 naturally establishes the time scale separation between "fast" and "slow" modes. In the case

255

6 What is Control of Turbulence

of 2y « I , the Langevin equation (6.82) with the kernel (6.83) reproduces the asymptotic dispersion relation typical for the Langmuir waves traveling in plasma, W rv k2 - Th as k ----+ 0 with n, c::: 0.0804 (in three dimensional space) (Pelletier, 1980) . Alternatively, for the exponents 2y ----+ I, it corresponds to the ion-acoustic waves traveling in the collisionless plasma with the velocity /\. v rv JTe / M where M is the ion mass . Intermediate values of y correspond to the various types of interactions between these two types of plasma waves described by the Zakharov 's equations (Zakharov, 1972) . The relaxation dynamics (6.82 - 6.83) establishes the relation between the covariances of random sources in (6.80), I

DIlIl(r,t) = 4;rrf3

J ' , t"

drdt Jo dp

Jo(pr')exp(-/\. v p 2- 2 yt,) , , /\' l - 2y Dww(r -r,t -t) Vp

(6 .84) where r == Ir I, and Jo is the Bessel function of the first kind . In the present section, we choose the covariance of random vorticity source,

\!w(r,t)!w(r',t') ) =

J~:

J(::)2

Dww(w,k) exp [- iW(t - t' )+ik (r - r' )],

k == Ik I, in the form of white noise (6.81), in which the relevant On sager coefficient l,v is found from the following physical reasons. Namely, the instantaneous spectral balance of particle flux, I W(k) = 2

JdW 2;rr ( f,,(k, W)f,,(- k ,W)) ,

(6.85)

derived from (6.84) should be independent from the reciprocal correlation time t"c( k) at any k that is true provided Dww(w , k) ex: /\. k- 2y . Furthermore, the On sager coefficient r w has to fit into the appropriate physical dimension which is assembled from the relevant dimensional parameters, Uo v and k. Collecting these factors, one obtains the Ansatz D ww ex: /\. u6v3 k6- d - 2y , in which d = 2 is the dimension of space . The power law model for the covariance of random helicity force ex: k6 - d - 2y does not meet the white noise assumption since 8(r - r' ) rv kO and therefore calls for another control parameter 2£ > O. Eventually, we use the model (6 .86) with the actual value of regularization parameter 2£ = 4 , for d = 2. Let us note that the Ansatz (6.86) is enough flexible to include the various particular models of particle pump into the SOL. For instance, the alternative to the white noise assumption spatially uniform particle pump for which the covariance D ww c:::

. u5:() k = lim

~ ---;o

J (xPs)-~ dx

-

e ikx

r(d /2) . (J: ) = k- d ~ /2 lim ':> k ps ,

2;rr'

~ ---; o

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Dimitri Volchenkov

in the large scales, can be represented by the Ansatz (6.86) with the actual value 2£ = 3. In the rapid-change limit of the stochastic model, A ----+ 00 (i.e., 't"c ----+ 0), the covariance (6.84) turns into I f, (k

\

Il\

V , m)f,I lI\ - k ,m) ) - Af3 r--;»

k2 -d-2e+2y

'

(6.87)

and recovers the white noise statistics (6.81) along the line e = r, in d = 2. Alternatively, in the case of A ----+ 0 (that corresponds to 't"c ----+ 00), the time integration is effectively withdrawn from (6.84), so that the resulting configuration relevant to (6.86) appears to be static oc k 4- d - 2e and uncorrelated in space (at d = 2) for 2£ = 2. The power-law models for the covariances of random forces has been used in the statistical theory of turbulence (Adzhemyan et al., 1998, see also the references therein). The models of random walks in random environment with long-range correlations based on the Langevin equation (6.82) have been discussed in concern with the problem of anomalous scaling of a passive scalar advected by the synthetic compressible turbulent flow (Antonov, 1999), then in (Volchenkov et al., 2002) , for the purpose of establishing the time scale separation, in the models of self organized criticality (Bak et al., 1987; Bak, 1996). Recently, the renormalization-group methodology have been applied in order to prove the breakdown of magnetic flux conservation for ideal plasmas, by nonlinear effects (Eyink and Hussein, 2006) . The analysis of (Eyink and Hussein, 2006) is based upon an effective equation for magneto-hydrodynamic (MHD) modes ; it is proven that flux-conservation can be violated for an arbitrarily small length-scale that is similar to the decay of magnetic flux through a narrow superconductive ring, by phase-slip of quantized flux lines . Being analogous to Onsagers result on energy dissipation anomaly in hydrodynamic turbulence, this result gives analytical support to and rigorous constraints on theories of fast turbulent reconnection .

6.3.3 Iterative solutions in crossed fields The linearized homogeneous problem, for the fluctuations of density n and vorticity w vanishing at t ----+ 00,

[at - V.1-1-]'X 1l = 8(r)8(t), [at -

Uo

v .1-1- ]'Xw (r , r) + g Iay 'XIl (r , t) = 8( r ) 8 (t ),

is satisfied by the retarded Green 's functions, +-

.1 1l (r,t )

(r4vt 2

e(t) = - exp - - ) , 4nvt

(6.88)

257

6 What is Control of Turbulence k 2 and arisen at the point r E .Q inside the divertor at time t with the particle density 8n( r' , t ' ) of those achieved the divertor wall at some subsequent moment of time t' > t at the point r ' E d.Q:

/ 8n(r ,t' )lc)Q

=1

1 I, but for the compact .Q as IA I I < I. To be specific , let us consider the circle CR of radius R as the relevant domain boundary and suppose for a simplicity that the density of particles incorporated into the growing fluctuations inside the domain is independent of time and maintained at the stationary rate 8no( r) . Then the r-integral in the r.h.s of (6.106) can be calculated at least numerically and gives the growth rate B(R) for those density fluctuations,

8n(R, r )

= -r ·B(R) ,

(6.108)

where -r is the traveling time of the density blob to achieve the divertor wall that can be effectively considered as a random quantity. It is the distribution of such wandering times that determines the anomalous transport statistics described by the flux pdf in our simplified model. The discrete time model we discuss below is similar to the toy model of systems close to a threshold of instability studied in (Flori ani et aI., 2003) recently. Despite its obvious simplicity (the convection of a high density blob of particles by the turbulent flow of the cross field system is substituted by the discrete time I-dimensional (in the radial direction) random walks characterized with some given distribution function) , its exhibits a surprising qualitative similarity to the actual flux driven anomalous transport events reported in (Ghendrih et aI., 2003). We specify the random radial coordinate of a growing fluctuation by the real number x E [0 , I]. Another real number R E [0, I] is for the coordinate of wall. The fluctuation is supposed to be convected by the turbulent flow and grown as long as x < R and is destroyed otherwise (x 2 R). We consider x as a random variable distributed with respect to some given probability distribution function lP' {x < u} = F (u). It is natural to consider the coordinate of wall R as a fixed number, nevertheless, we discuss here a more general case when R is also considered as a random variable distributed over the unit interval with respect to another probability distribution function (pdf) lP' {R < u} = Q(u). In general, F and Q are two arbitrary left-continuous increasing functions satisfying the normalization conditions

F(O)

= Q(O) = O,F(CXl) = Q(CXl) =

1.

Given a fixed real number 11 E [0 , I], we define a discrete time random process in the following way. At time t = 0, the variable x is chosen with respect to pdf F, and R is chosen with respect to pdf Q. If x < R, the process continues and goes to time t = I. Otherwise, provided x 2 R, the process is eliminated. At time t 2 I , the following events happen: i) with probability 11, the random variable x is chosen with pdf F, but the threshold R keeps the value it had at time t - I. Otherwise, ii) with probability I - 11 , the random variable x is chosen with pdf F , and R is chosen with pdf Q.

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6 What is Control of Turbulence

If x :2: R , the process ends; if x

< R, the process continues and goes to time t + I.

Eventually, at some time step r , when the coordinate of the blob, x, drops "beyond" R, the process stops, and the integer value -r resulted from such a random process limits the duration of convectional phase. The new blob then arises within the domain, and the simulation process starts again. While studying the above model, we are interested in the distribution of durations of convection phases PI) (r: F, Q) (denoted as P( r) in the what following) provided the probability distributions F and Q are known, and the control parameter 1] is fixed. The motionless wall corresponds to 1] = O. Alternatively, the position of wall is randomly changed at each time step as 1] = I . The proposed model resembles to the coherent-noise models (Newman and Sneppen, 1996; Sneppen and Newman, 1997) discussed in connection with a standard sandpile model (Bak et al., 1987) in self-organized criticality, where the statistics of avalanche sizes and durations take power law forms . We introduce the generating function of P( r ) such that

P(s) =

L s' P( r ), 00

P(-r)

, =0

= ~ d' P(s) r!

ds'

I

s=o'

(6.109)

and define the following auxiliary functions

K(n) =

1 00

F(u)"dQ(u) , OK(n) = K(n) - K(n+ I),

p(l) = 1]' K(l + I) , q(l) = (1-1])' K(l)'-I , r(l) = 1]' [1] OK(l+ I) + (1-1] )K(l+ I) OK(O)] , P = 1] oK(I) + (I - 1]) K(I )OK(0) .

for I :2: I , for! :2: I , for I :2: I ,

p(O) = 0 , q(O) = 0 , r(O) = 0 , (6 .110)

Then we find ,

P(s)=OK(O)+ps+

s

'( r() [r(s)+pp(s)q(s) +P K(l)q(s)+K(l) q(s)r(s)] , I-psqs

(6.111) where p(s),q(s),r(s) are the generating functions corresponding to p(l) ,q(l) ,r(l) , respectively. In the marginal cases 1] = 0 and 1] = I, the probability P(r) can be readily calculated, (6.112) The above equation shows that in the case of 1] = 0 , for any choice of the pdf F and Q, the probability P(r) decays exponentially. In the opposite case 1] = I, many different types of behavior are possible, depending upon the particular choice of F and Q. To estimate the upper and lower bounds for P(r) for any 1], one can use the fact that

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Dimitri Volchenkov

K(IY' < K(n) < K(I)

and 0

< 8K(n) < K(I),

n E N.

Then the upper bound for K(n) is trivial, since 0 ~ F(u) ~ I for any u E [0 , I]. The upper bound for K (n) exists if the interval of the random variable u is bounded and therefore can be mapped onto [0 , I] (as a consequence of Jensen 's inequality, and of the fact that the function u :----+ u" is convex on the interval ]0, I [ for any integer n). The calculation given in (Floriani et al., 2003) allows for the following estimation for the upper bound,

P1)(r) ~ T/T 8K(r) + (I - T/)K(I) 8K(0) [T/ + (1 - T/) K(l)]T-I +T/K(l) { [T/+(l-T/)K(l)]T-l_T/T-l} ,

(6 .113)

and, for the lower bound,

P1)(r) ~ T/T 8K(r) + (1- T/) K(1)T 8K(0)

= T/TP1) =I(r)+(l-T/)P1)=o(r).

(6.114)

We thus see that, for any 0 ~ T/ < I, the decay of distribution P( r) is bounded by exponentials. Furthermore, the bounds (6.113) and (6.114) turns into exact equalities, in the marginal cases T/ = 0 and T/ = 1. The simpler and explicit expressions can be given for P(r) provided the densities are uniform dF(u) = dQ(u) = du for all u E [0 , I]. Then Eqs . (6.112) give, I

P1) =1 (r ) = (r + 1)(r + 2)

(6.115)

For the intermediate values of T/, the upper and lower bounds are

T/ T

(r+ 1)(r+2)

I (I__T/ + )T

+ (1- T/)2- (HI ) < P( r ) < _ -

- 2

2

(6.116)

The above results are displayed in Fig. 6.10. The accounting for the dissipation processes introduces the order parameter ~ = Ikyl / (k; + k;) and its critical value ~c such that the particle density fluctuation 8n(~) grows unboundedly with time as ~ > ~c and damps out otherwise. We compute the value of ~c, in the first order of perturbation theory developed with respect to the small parameter Ps/ Rii where Ps is the Larmor radius, R is the major radius of torus, and ii is the mean normalized density of particles. Our results demonstrate convincingly that the possible correlations between density and vorticity fluctuations would drastically change the value ~c modifying the stability of model. Characterizing the possible reciprocal correlations between the density and vorticity fluctuations by the specific correlation time rc , we demon strate that any fluctuation of particle density grows up with time in the large scale limit (k ----+ 0) as rc ----+ 00 (the density and vorticity fluctuations are uncorrelated) and therefore ~c = O. Alternatively, ~c > 0 provided rc < 00.

6 What is Control of Turbulence

271

-2 1

P I (1) = (t+I) (/+2)

-4

q=!

-6

q=O. 7

-8 -1

Po

q=O .5

(1) = T(t+!)

-12 -14 q =O

-16 -18

5

10

15

20

25

Fig. 6.10 The distributions of wandering times near the wall in the discrete time model, in the case of the uniform densities dF (u) = dG(u ) = du for all u E [0,00) at different values of control parameter 1].

The reciprocal correlations between the fluctuations in the divertor is of vital importance for a possibility to stabilize the turbulent cross field system, in the large scales, by biasing the limiter surface discussed in the literature before (Ghendrih et al., 2003) . Namely, if ~c > 0, there would be a number of intervals [Vk-l ,Vk] for the uniform electric poloidal drifts V such that all fluctuations arisen in the system are damped out fast. In particular, in the first order of perturbation theory, there exists one threshold value Vc such that the instability in the system is bent down as V > Vc . However, Vc --+ 00 as ~c --+ O. To get an insight into the statistics of growing fluctuations of particle density that appear as high-density blobs of particles close to the reactor wall, we note that their growth rates are determined by the advanced Green's functions analytical in the lower half plain of the frequency space. We replace the rather complicated dynamical process of creation and convection of growing density fluctuations by the turbulent flow with the problem of discrete time random walks concluding at a boundary. Such a substitution can be naturally interpreted as a Monte Carlo simulation procedure for the particle flux. Herewith, the wandering time spectra which determine the pdf of the particle flux in such a toy model are either exponential or bounded by the exponential from above . This observation is in a qualitative agreement with the numerical data reported in (Ghendrih et al., 2003) .

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Dimitri Volchenkov

6.4 Conclusion Applications of methods developed in quantum field theory to the problems of statistical physics and critical phenomena have a long history . These powerful methods became an important tool in studies of nonlinear dynamical systems . In this report, we have developed a strategy of use the RG method in purpose of study the longtime large-scale asymptotic behaviors in stochastic magneto-hydrodynamics. The main conclusion of the study in magneto-hydrodynamics is that the RG transformations are characterized by two different fixed points stable with respect to long-time large-scale asymptotic behavior that can be naturally interpreted as "kinetic" and "magnetic" critical regimes , in which fields and parameters of the MHD theory acquires different critical dimensions. We have investigated long-time large-scale asymptotic behavior of correlation functions and composite operators (the local averages of fields and their derivatives , which can be observed in real experiments) in both critical regimes . The immediate observation of our study is that the MHD system is thoroughly unstable. Perhaps, the most fascinating result of our approach to MHD is the prediction of "optimal size" eddies that could survive in cross-fields much longer than others . In fact, we claim that if the cross-field system losses stability, it becomes transparent for certain plasma vortexes .We have cons idered two -dimensional models of the cross-field turbulent transport close to the "scrape-off layer" (SOL) in thermonuclear reactors. Stochastic perturbations of electron density and vorticity are responsible for the aggregation of electrons into bulbs which then propagate ballistically towards the wall blanket. The operation stability of the "next step" device crucially depends upon correlations between the fluctuations of electron density and vorticity . We have studied possible mechanisms which break the operation stability and proposed a simple discrete-time "toy model" resembling a Markov chain that correctly reproduces the statistics of the burst-like events observed experimentally in the ITER facility, in Cadarache (France).

References Adzhemyan L.Ts ., Vasil'ev A.N . and Pis'mak Yu.M., 1983, Renormalization-group approach in the theory of turbulence: The dimensions of composite operators, Theor. Math. Phys. , 57,1131. Adzhemyan L.Ts ., Vasiliev A.N . and Gnatich M., 1985, Quantum-field renormalization group in the theory of turbulence: Magnetohydrodynamics, Theor. Math. Phys., 64, 777. Adzhemyan L.Ts ., Antonov N.V. and Vasiliev A.N., 1989, Infrared divergences and the renormalization group in the theory of fully developed turbulence, JETP , 68, 1272. Adzhemyan L.Ts ., Volchenkov D. and Nalimov M.Yu., 1996, The renormalization group investigation of correlation functions and composite operators of the model

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ofstohastic magnetic hydrodynamics, Teoret. Mat. Fi z., 107,142. Adzhemyan L.Ts ., Antonov N.V. and Vasiliev A.N ., 1996, Quantum Field Renormalization Group in the Theory of Fully Developed Turbulence, Physics Uspehi, 39 , 1193 (in Russian). Adzhemyan L.Ts ., Antonov N.V. and Vasiliev A.N ., 1998, Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach , London . Adzhemyan L.Ts ., Antonov N.V., Gol'din P.B., Kim T.L. and Komp aniets M.V., 2008 , Renormaliz ation group in the infinite-dimensional turbulence: third-order results, 1. Physics A , 41 , 495002. Antar G.Y., Devynck P., Garbet X. and Luckhardt S.c., 2001 , Intermittency and burst properties in tokamak scrape-off layer, Phys. Plasmas, 8, 1612. Antonov N. v., 1999, Anomalous scaling regimes of a pass ive scalar adveeted by the synth etic velocity field, Phys. Rev. E, 60, 6691 . Antonov N.V. and Ignatieva A.A ., 2006, Critical behav ior of a fluid in a random shea r flow: Renormalization group analysis of a simplified model, J.Phys.A, 39, 13593. Bak P., Tang C. and Wiesenfeld K., 1987, Self-organized criticality : An expl anation of the I/f noise, Phys. Rev. Lett., 59, 381. Bak P., 1996, How nature works, Springer, Berlin. Bau sch R., Jan ssen H.K . and Wagner H., 1976, Renormalized Field-Theory of Critical Dynamics, Z. Phys. B 24, 113. Bogolubov N.N. and Shirkov D. v., 1980 , Introdu ction to the Theory of Quantum Fields, 3rd ed ., Wiley, New York. Camargo S.J., Tasso H., 1992, Renormaliz ation Group in Magneto-hydrodynamic Turbulence, Phys. Fluids B, 4, 1199. Carreras B.A . et al., 1996, Fluctuation-induced flux at the pla sma edge in toroidal devi ces, Phys. Plasma s, 3, 2664. Chandre c., Vittot M., Ciraolo G., Ghendrih Ph. and Lima R., 2006, Control of stochasticity in magnetic field line s, Nucl. Fusion J. of Plasma Phys.: Thermonucl. Fusion, 46, 33 . Ciraolo G., Ghendrih Ph ., Sarazin Y. , Chandre c., Lima R., Vittot M . and Pettini M ., 2007 , Control of test particle transport in a turbulent electrostatic model of the Scrape Off Layer, J. Nucl. Materials, 363-365,550-554. Coll ins J., 1992, Renormalization: An Introduction to Renoramlization, the Renormalization Group, and the Operator -Product Exapansion, Cambridge University Press, Cambridge. Dominicis C.de., 1976 , Techniques de renormalisation de la theorie de s champ s et dynam ique de s phenomene cr itique s, J. Phys., (Par is), 37 , Suppl. CI , 247 . Dom inici s C.de . and Peliti L., 1978, Field-theory renormalization and critical dynamic s above Te·: Helium, antiferromagnets, and liquid-gas systems , Phys. Rev. B., 18, 353 . Dominicis C.De. and Martin P.c., 1979, Energy spectra of certain randomly-stirred fluids, Phys. Rev. A, 19,419. Eyink G.L., 1994 , Renormalization group calculations using statistical hydrodynamics, Phys. Fluids, 6, 3063 .

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Eyink G.L. and Hussein A , 2006, The breakdown of Alfveu's theorem in ideal plasma flows: Necessary conditions and physical conjectures, Physica D, 223, 82 . Floriani E., Volchenkov D. and Lima R., 2003, A System close to a threshold of instability, J. Phys. A, 36 , 4771 . Fournier J.-D., Sulem P.L. and Pouquet A , 1982, Infrared properties of forced magneto-hydrodynamic turbulence, 1. Phys. A , 15, 1393. Garbet X. et al., 1991, A model for the turbulence in the scrape-of-Iayer of tokamaks, Nuc!. Fusion, 31, 967. Ghendrih Ph., Sarazin Y., Attuel G., Benkadda S., Beyer P., Falchetto G., Figarella C; Garbetl X., Grandgirard V. and M . Ottaviani, 2003, Theoretical analysis of the influence of external biasing on long range turbulent transport in the scrapeoff layer, Nuc!. Fusion, 43, 1013-1022. Gunn J., 2001 , Magnetized plasma flow through a small orifice , Phys. Plasmas, 8, 1040. Janssen H.K., 1976, Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties, Z. Phys. B: Condo Mat , 23, 377. Jurcisinova E., Jurcisin M., Remecky R. and Scholtz M., 2006, Influence of weak anisotropy on scaling regimes in a model of advected vector field, The Seventh Small Triangle Meeting, Herlany, September 17-20. Jurcisin M. and Stehlik M., 2006, D-dimensional developed MHO turbulence: Double expansion model, J. Phys. A: Math. Gen., 39,8035 . Kolmogorov A.N ., 1941, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokladi Akademii Nauk USSR, 30, 299 (in Russian, English Version : Kolmogorov AN., 1991, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 434 (1890), pages 9-13. Kraichnan R.H., 1959, The structure of turbulence at very high Reynolds number, J. Fluid Mech., 5, 497. Kraichnan R.H ., 1965, Kolmogorov's Hypotheses and Eulerian Turbulence Theory, Phys. Fluids , 7, 1723. Kraichnan R.H., 1966, Lagrangian history closure approximation for turbulence, Phys. Fluids 8, 575 . Kraichnan R.H., 1966, Isotropic Turbulence and Inertial-Range Structure, Phys. Fluids, 9, 1728. Labombard B. et al., 2000, Cross-field plasma transport and main-chamber recy cling in diverted plasmas on Alcator C-Mod, Nucl. Fusion, 40, 2041 . Labombard B., 2002, An interpretation of fluctuation induced transport derived from electrostatic probe measurements, Phys. Plasmas, 9, 1300. Landau L.D. and Lifsh itz E.M. 1995, Hydrodynamics, in Ser.: Theoretical Physics 5, Butterworth-Heinemenn, Oxford. Landau L.D ., Lifshitz E.M. and Pitaevskii L.P., 1985, Electrodynamics of Continuous Media, Butterworth-Heinemenn, Oxford.

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Lifschitz A.E., 1989, Magnetohydrodynamics and Spectral Theory, Kluwer Academic Publishers, Dordrecht. Ma S.K., 1976, Modern Theory of Critical Phenomena , Benjam in, Reading. Martin P.c., Siggia E.D. and H. A. Rose, 1973, Statistical Dynam ics of Classical Systems, Phys. Rev. A, 8, 423. Monin A.S. and Yaglom A.M., 1971, Statistical Fluid Mechani cs, 1, MIT Press, Cambridge, Mass. Monin A.S. and Yaglom A.M., 1975, Statistical Fluid Mechanics, 2, MIT Press, Cambridge, Mass. Nedospasov A.V. et aI., 1989, Turbulence near wall in tokamaks, Sov. J. Plasma Phys., 15, 659. Newman M.EJ . and Sneppen K., 1996, Avalanches, scaling, and coherent noise, Phys. Rev. E, 54, 6226. Obukhov A.M., 1941, On the distribution of energy in the spectrum of turbulent flow, Dokladi Akademii Nauk USSR, 32, I, 22 (in Russian). Pelletier G., 1980, Langmuir turbulence as a critical phenomenon. II Application of the dynam ical renormalization group method, 1. Plasma Phys., 24 , 421. Phythian R., 1977, The functional formalism of classical statistical dynamics , J. Phys. A , 10, 777. Rudakov D.L., l .A . Boedo, R.A. Moyer et aI., 2002, Fluctuation-driven transport in the DIII-D boundar y, Plasma Phys. Control Fusion, 44, 717. Sagaut P. and Cambon c., 2009, Homo geneous Turbulence Dynam ics, Cambridge University Press, Cambrige . Sagdeev R.Z. and Zaslavsky G.M., 1986, in Nonlin ear Phenomena in Plasma Physics and hydrodynami cs, Ed. R.Z. Sagdeev, Mir Publishers, Moscow. Sneppen K. and Newman M.EJ., 1997, Coherent noise, scale invariance and intermittency in large systems, Physica D, 110,209. Stangeby P.c. and McCracken G.M., 1990, Plasma boundary phenomen a in tokamaks, Nucl. Fusion, 30, 1225. Vasil'ev A.N., 1998, Functional Methods in Quantum Field theory and Statistics, Gordom and Breach, New York. Volchenkov D., 1997, Composite operators of the canonical dimension d=3 in magneto-h ydrodynamic turbulence, Acts ofSt.-Peterburg University: Physics and Chemistry, 2, 9-16 (in Russian). Volchenkov D., 2000, Field-theoretic approach to a stochastic magnetoh ydrodynam ics: the dimensions of composite operators, Phys. Lett. A , 265, 122-117. Volchenkov D., 200 I, The Bending Instability in the Vorticity Transport Through a Turbulent Flow, Intern. Jour. ofMod. Phys. B, 15, 1147-1164. Volchenkov D., Cessac B. and Blanchard Ph., 2002, Quantum field theory renormalization group approach to self-organized criticality : the case of random boundaries, Int. 1. Mod. Phys. B, 16, 1171. Volchenkov D., 2005, Stochastic models of edge turbulent transport in the thermonuclear reactors, 1. Phys.: Con! Ser., 7, 214-226. Wyld H.W., 1961, Formulation of the theory of turbulence in an incompressible fluid, Ann. Phys., 14, 143.

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Zakharov Y.E., 1972, Collapse of Langmuir waves, Sov. Phys. JETP, 35, 908. Zhang Y-c., 1989, Scaling theory of self-organized criticality, Phys. Rev. Lett., 63, 470. Zinn-Justin J., 1990, Quantum Field Theory and Critical Phenomena, Clarendon, Oxford .

Chapter 7

Entropy and Transport in Billiards M. Courbage and S.M. Saberi Fathi

Abstract Recent progress of the theory of dynamical systems and billiards sheds new light on the nonequilibrium statistical mechanics. Mixing, weak mixing and continuous spectrum are associated to relaxation to equilibrium via entropy increase. The properties of the relaxation time are reflected in the transport properties, which could be anomalous both in Sinai billiard with infinite horizon and in the barrier billiard. Numerical simulations are presented to corroborate these properties.

7.1 Introduction Recent progress in the ergodic theory of dynamical systems allowed to reconsider some old long debated problems as entropy increase and transport in conservative motions . The motion in billiards is, in this respect, the most studied model. Divergence of trajectories, mixing and weak mixing are among the main properties that are leading to an increase of the Gibbs coarse-graining entropy . These properties studied in the billiard by Krylov and Sinai were the main motivations for a series of simulations of the time evolution of the Gibbs entropy . We shall present some of them in the first part of this chapter. On the other hand, anomalous transport was discovered as one of the most important properties of chaotic and pseudo-chaotic motion in billiards, a field to which M. Courbage Laboratoire Matiere et Systemcs Complexes (MSC), UMR 7057 CNRS et Universite Paris 7- Denis Diderot , Case 7056, Batirnent Condorcct, 10, rue Alice Domon et Lonie Duquet , 75205 Paris Cedex 13, France, e-mail : [email protected] .fr S. M. Saberi Fathi Department of Physics , University of Wisconsin-Milwaukee, 1900 E. Kenwood Blvd ., Milwaukee, WI 53211 , USA, e-mail: [email protected]

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the prominent physicist George Zaslavsky greatly contributed. In fact, two types of irregular motions can be found in billiards, the first corresponding to exponentially diverging trajectories, and the second corresponding to linearly diverging trajectories. We shall present a series of simulations of the transport in the second part of the chapter.

7.2 Entropy 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 Shanonn 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 distribu tion!r(x) as

S(!r(x)) = -

J

!r(x)log(!r(x)dx.

(7.1)

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 Htheorem . 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 at 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 only allowed transitions are from X = m to X = m - I, with probability m /2N, or from X = m to X = m + I with the complementary probability. Mark Kac gave an exhau stive solution of this model in his book (Kac, 1959). Briefly speaking, it is possible to find a unique stationary probability distribution {)1i}, i = 0, I, ... , n = 2N; such that any initial distribution {Vi(t)} converges to {)1i}' The non-equilibrium entropy of the distribution {Vi(t)} with density !r = is given by the Boltzmann like formula :

vj;l

279

7 Entropy and Transport in Billiards

J

- it log itdu = -

L Vi(t) l oVi(t) g- . i

J.!i

(7 .2)

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 to distinct regions, {.9'i} . The set of .9'/ s form a partition of the phase space . However, it is obvious that strictly speaking the process X (z ) 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 OS where to each trajectory is attached two manifolds expanding and dilating in transversal directions. So, any domain of the phase space will be squeezed and 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. 7.1). The importance of mixing and exponential instability of trajectories for obtaining H -theorem has been discussed by (Krylov, 1950). 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 J.!, invariant under T, i.e. J.!(T - 1E) = J.!(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 that (7 .3)

Then, the H -theorem means the existence of a negative entropy functional S( VI) which increases monotonically with t to zero, being attained only for v = J.!. The existence of such functional in conservative dynamical systems has been the object of several investigations during last decades see (Courbage, 1983; Courbage and Prigogine, 1983; Garrido et al., 2004 ; Goldstein and Penrose, 1981; Misra et aI., 1979; Courbage and Misra, 1980; Goldstein et aI., 1981; Sinai', 1994). Here we study this problem for the Lorentz gas and hard disks . Starting from the non-equilibrium initial distribution v, and denoting by .9' a partition of the phase space formed by cells (.9'1,.9'2, ..., .9'11 ) and by Vi(t) = V O T - I (.9'i), the probability at time t for the system to be in the cell .9'i and such that V(.9'i) :f. J.!( .9'i) for some i, the approach to equilibrium implies that Vi(t) ---+ J.!i as

280

M. Courbage and S.M. Saberi Fathi

1.0

1.0

0.8

0.8

"

~ 0.6

'"

' ;(

< 0.4

~

0.2

0.2

0.0

0.0 0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

X Axis Title

0.4

0.6

0.8

1.0

0.8

1.0

X Axis Title

1.0

1.0 0.8

0.8

~" 0.6

" ~0.6

'" ' ;(

'" ';(

< 0.4

:

1.4 ~

1.2

~J

.~

10

~

\-....

1.0 t-~-~~-,.--~-~--.-~

o

500 000 X-coordinates

I 000 000

Fig.7.20 Dependence of the central moments on (nlnn) for D

=

I, a = 0.473 after 106 collisions.

7.3.2 Transport in the barrier billiard The goal of this section is to provide renewal results of massive computations and to show that some well defined transport properties of particles do not follow the Gaussian law, exhibit superdiffusion, and, for the time of observation, do not displaya "normal" approach to the limit distribution. We consider two models: SBIH (periodic Lorentz gas) and stadium (Bunimovich) billiard (Bunimovich, 1979). We interpret these observations as a result of particles long "flights" in the corridors and, as a result of the flights, persistent fluctuations (Zaslavsky and Edelman, 2004) that do not have a finite time of relaxation as it exists for the Gaussian fluctuations. It is now established that the exponential instability of trajectories leads to strong ergodic and stochastic properties, diffusion and H-theorem . Recently, numerical computations have confirmed a relation between the rate of entropy increase in the H-theorem and the positive Lyapunov exponents (Courbage and Saberi Fathi, 2008) . Yet the transport properties of these systems are still unclear (see references in (Courbage et al., 2008». The situation is worst as to such properties in dynamical systems which does not exhibit exponential instability . The problem of a kind of mixing and stochastic properties of systems with zero Lyapounov exponents is a challenging problem that has been the object of many publications (see (Gutkin and

302

M. Courbage and S.M. Saberi Fathi

Katok, 1989; Horvat et al., 2009) and references therein) . Here we consider a simple model of a particle colliding elastically with horizontal bars periodically distributed on the plane . This model, some times called a barrier billiard , was studied by several authors. It has been used as a model of a pseudo-integrable system (Richens and Berry, 1981; Hannay and McCraw, 1990; Wiersig, 2000) and a model of free particle dynamics along field surfaces in plasma physics (Zaslavsky and Edelman, 2004). It is also a model of a gas of non-interacting particles moving with constant velocity on the plane and undergoing collisions only with those scatterers. The nonequilibrium statistical description is thus reduced to the dynamics of the one particle probability distribution. The motion of the particle along x-coordinate is simply a uniform translation, while the motion along y-coordinate is apparently a random walk as seen in the Fig. 7.11. The transport properties have been studied (Zaslavsky and Edelman, 2004) in considering the motion of bunch of trajectories with closed initial conditions. It was shown that the transport is anomalous and seems of the following form : (7 .37)

with ,11(1) > 1. The superdiffusion comes , as in the case the Sinai Billiard with infinite horizon , from the existence of corridors between the arrays of scatterers allowing arbitrarily long free trajectories (Courbage et al., 2008) . In this section we provide new computations in order to test the convergence of the displacement along y-coordinate to a gaussian distribution. We will study the reduced central moments of order I, 2, 3 and 4. A possible divergence with respect to the gaussian moment is then clearly displayed. The displacement along y-direction is given by n

y(n,x,c) = LCi(X,C).

(7 .38)

i= l

The mean value of y(n) with respect to the invariant measure is zero and the second moment for a rational a has been studied in (Courbage, 2005) where it is shown that

f (~~(x,e))

2 dp

(x,e) '" en'.

(7 .39)

So the motion is ballistic . In the next section we will study the case of irrational Note that, on account of the symmetry, all odd moments are zeros.

a.

Here n is the time scale We first simulate the convergence of the moments of the y-displacement. We shall take the first moment equal to O. The m th raw moments for an arbitrary variable y at the time n is defined by: (7.40)

7 Entropy and Transport in Billiards

303

a=:;r

3.0

2.5

2.0

1.5

1.0

o

500000

1000000 Time

1500000

2000000

Fig.7.21 The ratio of second and fourth moment s, M,~(II) , versus time for a = M2 (II)

7L

where Yk(n ) is the kth-experiment at time n, N is the total number of exper iments (trajectories). Let us now test the convergence of the norma lized moments to Gaussian moments . This can be done throug h the fourt h moment of the normalized displacement, MY}'(' l ) which should converge to 3. This should be the asymptotic of 2

Il

Fig. 7.21 shows

M,~

M2

M~ .

M,-

versus time . Table 7.3. shows that the limit is not Gaussian . -

Looking now for the second moment, we will compute o and B such that (7.41)

By drawing Log-Log diagram of M~ (n), we obtain B by linear fitting of the following relation (7.42) logM~(n) = Blogn +A, (A = log o ").

Table 7.3 The values of B, A , second and fourth moments and their ratio for different values of at n max = 106. The value of (j2 is equal to I(YI

a

B

A

M2(n max)

M4(n max)

V2

1.51 1.69 1.13 2.00

- 3.3 1 - 1.49

1.07 1.00 0.97 1.00

1.68 1.90 2.28 1.18

e 7r

7/ 3

1.69 - 1.03

M~ (nmax )

M;2(n max) 1.45 1.89 2.43 1.18

a

304

M. Courbage and S.M. Saberi Fathi

LogM;(n) IE11

-:

9E10 8E10 7E10

a

=

7/3

6E10 5EIO 4E10

M; =A +B Log n B =2.00 A = -1.026

3E10

2E10

500000

600000

700000

800000 900000 1000000

Logn LogM;(n)

600000 500000

400000

M;=A +B Log n B = 1.51 A = -3 .31

300000

200000 +--------.-------r---....----....-------r700000 800000 900000 1000000 500000 600000 Logn Fig. 7.22 Log-Log diagra m of MH n ) versus time, by linear fitting for y-variable, a = 7/3 and a = V2 and number of trajectorie s is N = 1000. The fit was done for time between 5.3 x 105 and 106 . The results for other values of a are given in Table 7.1.

7 Entro py and Transport in Billiards

305

M2 (n) 3.0 a =e

2.5 &

2.0

\

•

1.5

1.0 0

200000

400000

600000

800000 1000000

Time M4(n) 12 a =e

10 8 6 4 2 0

0

200000

400000

600000

Time Fig. 7.23 The moment M2(n ) and M4(n ) versus time for a

=

e.

800000 1000000

306

M. Courbage and S.M. Saberi Fathi

Fig. 7.22 shows a fitting for the case ex = v2. In the Table 7.3. we see some values of A and B. For ex = 7/3 we check that our result for B = 2.00 is adequate with the theoretical theorem stating that limll-.oo B = 2 (Courbage, 2005). Now, the m th reduced central moments is defined by (7.43) where N is the number of trajectory. For second central moments we have obviously limll-. ooM2(n) = I. We test again if the transport is" Gaussian" , then M4 should be equal to "3" . Table 7.3 shows the values of central moments for M2(n max ) and M4(n max ) . These values show that the transport is not Gaussian . This table shows that the B-value is I < B :::; 2. So we have a super diffusion . Fig. 7.23 represents the second and fourth normalized moments for ex = e and their zooms which show that they do not converge.

7.4 Concluding remarks The computations of the evolution of the entropy amount of some given nonequilibrium initial distributions in the Lorentz gas show an exponential type increase during initial stage after which the entropy increases slowly and fluctuates near its maximal value. These computations confirm the existence of a relaxation time generally assumed in the derivation of kinetic equations (Balescu, 1975) which is at the origin of the rapid increase of the entropy as due to the number of collisions. The dispersive nature of the obstacles is responsible of the exponential type increase. This exponential type increase has been demonstrated for the Sinai" entropy functional (Sinai", 1994) in hyperbolic automorphisms of the torus. On the other hand, the relation of the entropy increase to Lyapounov exponents can be understood through Pesin relation and Ruelle inequality. In fact, the rate of entropy increase should be bounded by the Kolmogorov-Sinai entropy and such bound have been found by Goldstein and Penrose for measure-theoretical dynamical systems under some assumptions (Goldstein and Penrose, 1981). An open question is to characterize the initial invariant probability measures reaching the upper bound. New simulations of transport of the barrier billiard confirm previous Zaslavsky and Edelman simulations showing a superdiffusion (Zaslavsky and Edelman, 2001). Our results show moreover the non gaussian nature of the transport. The strong dependence on the irrational velocity angle ex show however that the transport in this billiard is still unclear. The entropy results for irrational ex show mixing for the relaxation of the velocity direction. But, the coarse- graining entropy has relaxation only in time average reflecting some weak mixing for some distinguished observabies . The Gibbs coarse-graining entropy is not a completely monotonic function of time . A completely monotonic entropy functional has been obtained when the map

7 Entropy and Transport in Billiards

307

T on the space X is a Bernoulli system or, slightly more generally, a K-system (Courbage, 1983; Misra et al., 1979; Courbage and Misra, 1980; Goldstein et aI., 1981). That is to say : there is an invariant measure /l and some partition ~o of X such that T ~o becomes finer than ~o ( we denote it: T ~o ?: ~o). Using the notation : T"~o = ~Il' we obtain a family of increasingly refined partitions, in the sense of the above order of the partitions. Moreover, ~11 tends, as n --+ 00, to the finest partition of X into points, and ~Il tends , as n --+ - 0 0 , to the most coarse partition, into one set of measure I and another set of measure zero . A physical prototype of a Bernoulli and a K-system is the above Sinai' billiard (Sinai', 1970; Gallavotti et aI., 1974) . A geometric prototype of a Bernoulli and a K-system is uniformly hyperbolic system with Sinai' invariant measure (Sinai', 1972). Heuristically, the monotonic entropy increase corresponds to the process of dilation of expanding fibers.

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Index

Barrier biliards Billi ards Brain dynamic s Brownian motion Chaotic motion s Coarse-graining Diffu sion Dilut ed networks Discrete map with memory Emotion -cogni tion interaction Entropy Ergodicit y Finite- size effects Fractional derivatives Fractional equations Fractional proces ses Fractionally stable distributions Henon map HMFmodel Inequivalence of ensembles Inhib ition Vlasov equation Kink K-S entropy Levy motion Long living eddies in crossed-field systems Long-range interactions Lorentz gas

Lump Markov processes Ment al disorder Mixing Non-equil ibrium entropy Non-equilibrium ensembles Normal hyperbo licit y Phase transitions Quasi-stationary states Random graphs Relaxation Repli ca method Sinai Billiard s Soliton Soliton chains Soliton interaction Stable heteroclinic channel Stochastic counterparts of nonlinear dynamics Stochasticity Structural stability Symmetry of the crossed-field system Transient behavior Transport Wave turbulen ce Weak mixing Winne rless competition Zasl avsky map

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