Complex Systems: Theory and Applications 9781536108606, 9781536108712

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MATHEMATICS RESEARCH DEVELOPMENTS

COMPLEX SYSTEMS THEORY AND APPLICATIONS

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MATHEMATICS RESEARCH DEVELOPMENTS

COMPLEX SYSTEMS THEORY AND APPLICATIONS

REBECCA MARTINEZ EDITOR

New York

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Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface Chapter 1

vii A Process Algebra Approach to Quantum Electrodynamics: Physics from the Top Up William Sulis

Chapter 2

Realizing Success for Complex Converging Systems Geerten Van de Kaa

Chapter 3

Development of the Generalized Nonlinear Schrödinger Equation of Rotating Cosmogonical Body Formation Alexander M. Krot

Chapter 4

Chapter 5

Index

The Application of Neural Network Modeling in Organizing a Hierarchical Teaching System Based on Mentorship A. Dashkina and D. Tarkhov Modelling Organisation Networks Collaborating on Health and Environment within ASEAN P. Mazzega and C. Lajaunie

1 43

49

95

117 149

PREFACE This books provides new research on the theories and applications of complex systems. Chapter One reviews the process algebra approach to quantum electrodynamics. Chapter Two describes a specific aspect of complex systems and the fact that they may consist of established subsystems or components that originate from converging industries. Chapter Three examines the development of the generalized nonlinear Schrödinger equation of rotating cosmogonical body formation. Chapter Four analyzes the application of neural network modeling in organizing a hierarchical teaching system based on mentorship. The final chapter presents two methods to evaluate the collaborative potential of a network of 16 organizations and identifies measures to promote their coordination. Chapter 1 – The process algebra was developed to study information flow and emergence in complex systems. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM), providing a realist model of quantum mechanics which appears to be free of the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this chapter, the process algebra model is reviewed, extended to the relativistic setting and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The

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process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two. Chapter 2 – In this chapter a specific aspect of complex systems, the fact that they may consist of established subsystems or components that originate from converging industries, will be described. To realize such complex systems, common standards with which the components of such systems can be interconnected are essential. A specific type of standard will be described which can be used to realize such complex systems; subsystem standards. It will be determined which factors affect the success of subsystem standards by studying a specific example of a subsystem standard; USB. This specific case illustrates the importance of flexibility; standards should guarantee a certain amount of flexibility so that it is possible to adapt them to changing requirements that inevitably emerge when components that originate from multiple converging industries are connected. Second, the case illustrates the importance of network diversity in that subsystem standards should be supported by a diverse network in terms of stakeholder representation. The paper concludes with recommendations for future research directions. Chapter 3 – This chapter considers the statistical theory of gravitating spheroidal bodies to derive and develop a new generalized nonlinear Schrödinger equation of a gravitating cosmogonical body formation. Previously, the statistical theory for a cosmogonical body forming (so-called spheroidal body with fuzzy boundaries) has been proposed. As shown, interactions of oscillating particles inside a spheroidal body lead to a gravitational condensation increasing with the time. In this connection, the notions of an antidiffusion mass flow density as well as an antidiffusion particle velocity in a rotating spheroidal body have been introduced. The generalized nonlinear time-dependent Schrödinger equation describing a gravitational formation of a rotating cosmogonical body is derived. This paper considers different dynamical states of a gravitating spheroidal body and respective forms of the generalized nonlinear time-dependent Schrödinger equation including the virial mechanical equilibrium, the quasi-equilibrium and the gravitational instability cases. Besides, the last case involves the avalanche gravitational compression increasing (when the parameter of gravitational condensation grows exponentially with the time) among them the case of unlimited gravitational compression leading to a collapse of a spheroidal body. Within framework of oscillating interactions of particles, a

Preface

ix

frequency interpretation of the gravitational potential and the gravitational strength of a forming spheroidal body is considered in detail. In particular, the authors explain how Alfvén’s oscillating force modifies the forms of planetary orbits within the framework of the statistical theory of gravitating spheroidal bodies. They find that temporal deviation of the gravitational compression function of a rotating cosmogonical body induces the Alfvén additional periodic force. An oscillating behavior of the derivative of the gravitational compression function implies the special case when the additional periodic force becomes counterbalance to the gravitational force thus realizing the principle of an anchoring mechanism in exoplanetary systems. Chapter 4 – The authors consider the application of the hierarchical teaching systems and mentorship. We illustrate that they have many advantages over the conventional system of organizing the learning process within a group of learners with different levels of knowledge. The process of interaction between a teacher and students is complex and non-linear. If the knowledge is transferred not only from a teacher to learners, but also from one student to another, the system can be referred to as a hierarchical one. Neural networks prove to be an appropriate tool for creating models of such systems, so we apply them here. They have conducted a number of practical experiments which involved application of mentorship and the hierarchical teaching systems within some groups of learners. The results of the experiments were processed by neural networks. It allowed us to create a sociodynamic model of the learning process and to forecast and maximize its further results. At the end of this chapter, the authors formulated the conclusion and gave practical recommendations. Chapter 5 – The emergence of infectious diseases is related to environmental factors such as biodiversity loss, land use and land cover changes, and regional impacts of climate change. The threat of worldwide pandemics led to the development of prevention and mitigation strategies implemented via public policies and national/international legal instruments. These measures involve networks of organisations engaged in a wide range of quite disparate activities. The authors present two methods to evaluate the collaborative potential of a network of 16 organisations and identify measures to promote their coordination: 1) the first method uses Galois lattices to identify groups of organisations and issues forming a nexus able to tackle with the environment and health challenges. It also allows pointing out the divide between some sub-issues that should be considered together in order to develop an integrated approach of these problems as recommended for example by the One Health initiative; 2) the second method, inspired by the

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functioning of networks of cerebral cortex areas for the realisation of highlevel cognitive functions, analyses the graphs induced by mutual information functions between organisations. Here they evaluate these functions based on the composition of the governing boards (altogether involving 91 organisations) and partnerships (altogether involving 263 organisations) of these organisations. The approach gives the opportunity to assess a priori the effects induced by a change in the profile of the collaborating organisations. The contributions of these two methods are then discussed and compared to other approaches developed for the analysis of social intelligence or sociocognitive artificial systems. Both methods are illustrated by the analysis of a network of several organisations involved in the management of health and environmental issues in Southeast Asia (hot spot for emerging infectious diseases and for biodiversity). Taken as a whole, our findings show that regional governance in the health-environment sector is polycentric and entangled, and provide guidance for improving governance on the basis of the competences and collaborations of participating organisations.

In: Complex Systems Editor: Rebecca Martinez

ISBN: 978-1-53610-860-6 c 2017 Nova Science Publishers, Inc.

Chapter 1

A P ROCESS A LGEBRA A PPROACH TO Q UANTUM E LECTRODYNAMICS : P HYSICS FROM THE T OP U P William Sulis ∗ McMaster University, Hamilton, Canada

Abstract The process algebra was developed to study information flow and emergence in complex systems. From this perspective, fundamental phenomena are viewed as emerging from primitive informational elements generated by processes. The process algebra has been shown to successfully reproduce scalar non-relativistic quantum mechanics (NRQM), providing a realist model of quantum mechanics which appears to be free of the usual paradoxes and dualities. NRQM appears as an effective theory which emerges under specific asymptotic limits. Space-time, scalar particle wave functions and the Born rule are all emergent in this framework. In this paper, the process algebra model is reviewed, extended to the relativistic setting and then applied to the problem of electrodynamics. A semiclassical version is presented in which a Minkowski-like space-time emerges as well as a vector potential that is discrete and photon-like at small scales and near-continuous and wave-like at large scales. QED is ∗

Email: [email protected]

2

William Sulis viewed as an effective theory at small scales while Maxwell theory becomes an effective theory at large scales. The process algebra version of quantum electrodynamics is intuitive and realist, free from divergences and eliminates the distinction between particle, field and wave. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two.

1.

Introduction

Throughout much of its history, physics has been guided by the idea of reductionism, the belief that we can explain all physical phenomena by searching the smallest scales of space and time to find the fundamental elements of reality, discovering and explicating the fundamental principles governing their individual behavior and interactions, and then working our way upwards using these elements likes block of LEGO (to borrow an idea of Trofimova [1]) to build ever more complicated entities. In the 1980’s a set of alternative ideas was put forward, first, based upon ideas derived from the study of nonlinear dynamical systems [2] and later upon ideas of complexity [3]. The rise of complexity science marked a sea change in our thinking about reality, although in physics these ideas have been embraced mostly by condensed matter theorists [4]. An impassioned statement against the reductionist viewpoint was given by the biophysicist Robert Rosen [5]: “. . . the basis on which theoretical physics has developed for the past three centuries is, in several crucial respects, too narrow and that, far from being universal, the conceptual foundation of what we presently call theoretical physics is still very special; indeed, far too much to accommodate organic phenomena (and much else besides). That is, I will argue that it is physics, and not biology, which is special; that, far from contemporary physics swallowing biology as the reductionists believe, biology forces physics to transform itself, perhaps ultimately out of all present recognition.” (pg. 315) The key insight behind Rosen’s statement is the realization that at ever larger spatial and temporal scales, entities do not merely become more complicated,

A Process Algebra Approach to Quantum Electrodynamics

3

they become complex. One of the key properties of complex systems is that they exhibit emergence. Many different notions of emergence have been developed over the years but here the focus is on emergence as understood within the framework of archetypal dynamics [6]. There an emergent situation is described as one in which the system in question admits description in terms of multiple mutually irreducible semantic frames. A semantic frame generalizes the concept of a reference frame in physics, including within it definitions of entities, their actions and interactions, as well as specifying the manner in which an observer must interact with such entities so as to preserve their integrity. No single semantic frame frame is capable of expressing everything that is essential for the system. A complex system is thus contextual in a very deep and fundamental way. In physics it is assumed that physical entities are described by one semantic frame which exists in different versions termed frames of reference. These frames of reference are all reducible to one another through symmetry transformations. Every observer uses the same semantic frame albeit it may be expressed as possibly different frames of reference. The archetypal dynamics approach denies this assumption, and instead asserts that virtually all entities are emergent. From an archetypal dynamics perspective, the inability to reconcile quantum mechanics and general relativity suggests that the universe is, in fact, an emergent entity admitting mutually irreducible descriptions. From the viewpoint of archetypal dynamics, the quest to unify quantum mechanics and general relativity is misguided. A more useful approach would be to explicitly express the semantic frames underlying these theories, to determine their respective entities and their effective ranges, and to search for phenomena that might lie within regions of overlap between the two. Within archetypal dynamics, there is no universal theory of everything, only theories of some things, that together weave a patchwork quilt or tapestry of understanding. The Process Algebra approach to be described here turns the usual reductionist paradigm on its head. Following suggestions of Rosen [5], Trofimova [7] and others, it seeks to apply ideas of complexity and emergence at the lowest spatial and temporal scales and to explore the possibilities that follow. In particular, this initial application of the Process Algebra program seeks to develop a realist theory of fundamental phenomena which is a true completion of quantum

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mechanics, not merely a reformulation [8, 9, 10, 11]. It proposes a discrete, finite and generated space-time in which fundamental entities such as particles and fields appear as emergent phenomena. These fundamental entities are generated by processes utilizing only causally local information. For that reason such theories are termed quasi-non-local. The theory is compatible with special relativity. Measurement is viewed as an interaction between a process and a specialized measurement process without requiring a separate theory. There is no measurement problem in the process algebra approach. The primitive elements of the theory possess definite properties, although the set of properties is incomplete, as they are only those imparted to them by their generating process and, for reasons described below, no process can assign definite values to all possible attributes. Furthermore, the process of measurement involves an interaction between processes, whose outcome is dependent upon the nature of that interaction. For these reasons process algebra theories are termed quasi-noncontextual. Issues related to non-locality and contextuality are beyond the scope of this paper but have been addressed in detail elsewhere [8, 9] The key insight behind the process algebra approach is the following. The standard Hilbert space formulation of quantum mechanics suggests decomposing a wave function Ψ(z) into a sum of eigenfunctions Ψn (z) of some measurement operator, usually the energy H. That is, Ψ(z) =

X

an Ψn (z)

n

where the coefficients an give rise to the probability that the system will be in eigenstate n when a measurement is performed. There is, however, an alternative way of decomposing a function using a method known in von Neumann’s time as a Cardinal function expansion. The idea is to expand each basis function as Ψn (z) =

X

Ψn (mnk )Tmnk g(z)

k

{mnk }

where each set Jn = consists of a sampling of elements of some spacetime manifold M, g(z) is a universal interpolation function (in simple settings

A Process Algebra Approach to Quantum Electrodynamics

5

a sinc function), T is a translation operator (Tm g(z) = g(z − m)). Then ψ(z) =

XX n

an Ψn (mnk)Tmnk g(z)

k

The next step is to create each set Jn in several generations (roughly corresponding to distinct times), so Jn = ∪i Jni = ∪i {(i) mnk } and to propagate information forward from one generation, Jni , to the next, Jni+1 . The local wave function value for some k ∈ Jni+1 is generated from the local values at j ∈ Jni by Ψn ((i+1) mnk ) =

X

K(k, j)Ψn((i)mnk )∆

j∈Jni

where K is a propagator defined on the set of samplings, ∆ is measure of local volume, usually taken to be lP3 . This is a discrete version of the usual propagator equation Z 0 0 Ψ(x , t ) = K(x0 , t0 ; x, t)Ψ(x, t)dx M−

Note that the amount of information required to define the wave function has been reduced to the set of local wave function values defined on the discrete set of points I = ∪n Jn , which effectively quantizes space-time and also the wave function. Moreover, interpolation theory (in particular sinc interpolation theory) shows that a suitably large finite subset of I is sufficient to generate the wave function to a high degree of accuracy [9]. The idea of generation is key here. If the set of space-time elements, I was generated, then some measurement apparatus could interact with such an element while it was in the process of being generated, which would result in the appearance of a finite and localized event, while the wave function, in its emergent totality, would have the appearance of a spatially extended entity. This would resolve both the measurement problem and the issue of wave-particle duality. Whether the system is viewed as a particle or as a wave is no longer an ontological question but is seen to be a consequence of the level of observation - particle at small scales and wave at large scales. Moreover, whether a measurement process interacts with a system process would depend upon the

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local wave function value associated with the element being generated. This contributes to determining the compatibility [7] between the measurement and system processes, which in turn determines whether or not they couple and interact. This gives rise to an emergent probability structure (compatible with the Born rule) which possesses non-Kolmogorov [12] features of the type seen in quantum mechanics. Note that it is possible for the sets Jn of points to be mutually disjoint, so that in the expansion of the wave function over the basis states, ψ(z) =

XX ( Ψn (mnk )Tmnk g(z)) n

k

the space-time elements corresponding to each basis function are entirely distinct from one another. No space-time element ever possesses attributes from two distinct basis states. Although the basis states superpose, this occurs only at the emergent level. At the fundamental ontological level there are no superpositions. Thus there are no Schr¨odinger cats in this model. The quantum realm thus becomes an emergent situation admitting two distinct semantic frames, one corresponding to the description based on the sampling set, and one based on the description corresponding to the wave functions. Note that these descriptions are approximately equivalent: it is possible to map from the wave function to the sampling space, but mapping in the reverse direction depends upon the choice of interpolation function. Not all interpolation functions will yield equivalent wave function. It is important to note that in the non-relativistic setting, the information that enters into the generation of space-time elements and their local process strengths is propagated in a causally local manner. The space-time elements are generated independently of one another and in a non-contiguous manner. This is termed quasi-non-locality since it occurs only at the level of the processes, which, being generators of space-time, exist independent of space-time. That this does not conflict with relativity is discussed below. It is also important to note that each space-time element mnk is assigned a definite wave function value Ψn (mnk ), implying that the element of reality associated to mnk possesses a definite physical attribute. The local wave function value is generated by a process, and as will be shown below, a process will not generate all possible properties, merely a subset. Such a situation is termed

A Process Algebra Approach to Quantum Electrodynamics

7

quasi-non-contextual, meaning that physical entities possess a set of properties having definite values, but this set is never complete. There is an ontological wave function, but it is not the wave function of non-relativistic quantum mechanics except in special cases. The standard quantum mechanical wave function is a computational tool derived from these ontological wave functions via a process covering map (discussed below). It is for this reason that the process algebra model is proposed as a true completion of quantum mechanics [8]. Previous work focused on scalar non-relativistic quantum mechanics, which the process model approximates to a high degree of accuracy (10−33 or better) [9, 10, 11]. The process algebra approach appears to avoid many of the usual paradoxes and being discrete and finite is intrinsically divergence free with a natural ultraviolet cutoff [9]. Unlike orthodox thinking, in the process algebra model quantum mechanics arises in the limit in which Planck’s constant (~), Planck time (tP ) and Planck length (lP ) all trend to zero. Classicality is determined, not by scale, but rather by the complexity of the interactions among the generating processes.

2.

Object Versus Process

A fundamental difference between the complexity sciences and the traditional sciences, beyond that between linear reductionism and nonlinear complexity, is between the concepts of object and of process. For more than two millennia, science has followed a program of objectification of reality. An object is a mental abstraction. Some elements of reality behave like objects but many more do not. Adopting an attitude of objectification influences all subsequent relationships to the entities of reality regardless of whether or not they actually behave like objects. An object is considered to have an existence and properties that are independent of the environment within which it is embedded, and in particular, independent of any specific observer. An object endures, that is, it persists across time, and often is treated as if it were eternal. An object may consist of component objects but if so then this set of components is a fixed property of the object. This continuity of structure over time gives an object a property of individuality. This is not to be confused with identity, since it is quite possible for two individual objects to be indistinguishable by any conceivable means.

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An object may undergo change, but only in the spatial configuration of its constituents or in certain attributes or states, such as those that describe position and orientation. Note that properties are considered intrinsic and independent whereas attributes are changeable and extrinsic, and thus may be environmentally dependent. Nevertheless, attributes are thought of as being possessed by the object even if not wholly specified by the object. For example the capacity to hold a location in space or a duration in time is possessed by the object even if the precise specification of such may depend upon details of the embedding environment. An object does not act, it reacts. An object cannot set goals or pursue goals. An object can transmit information but neither creates nor interprets it. An object cannot intend, apprehend, prehend, or comprehend. The concept of an object explicitly and specifically precludes any aspects that could be considered subjective. Objects are localized, and their interactions too are local; being independent of both environment and observers, objects are non-contextual. An archetypal expression of the concept of object is a rock. It has a fixed form and constitution. It has properties, like hardness, color and constitution, that are intrinsic and fixed, and attributes, like temperature, orientation, position, momentum and angular momentum, that are variable and relative. It reacts to its environment but does not act. It has no intentions, no goals. The concept of object lies at the heart of virtually all mathematical and physical constructs. Indeed, mathematical entities are the only true, ideal objects because they can be considered to exist outside of reality within some universe of Platonic ideals. All actual entities that exist within reality are at best approximations to the concept of object. Even that archetypal object, the rock, appear less objective when examined near the Planck scale. Physics may be considered to be the study of those elements of reality that are objective or can be approximated as objects. For physics, the objectification of reality has proven to be highly successful. Success, however, can lead one to misattribute a physical reality to those ideal mathematical objects. This conceptual error can then become a hidden impediment to further study as explored by Mermin [13] in an article on the dangers inherent in the reification of mathematical structures. Objectification really only works when applied to a relatively limited range of natural phenomena, primarily inanimate matter observed at macroscopic scales

A Process Algebra Approach to Quantum Electrodynamics

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and isolated from its natural environment. When applied indiscriminately it leads to conceptual confusion, explanatory restrictions and paradox. This has been abundantly clear in quantum mechanics, resulting in the measurement problem, wave function collapse, wave-particle duality, Schr¨odinger’s cats, multiverses and the like. The appearance of these conceptual paradoxes should have led scientists to question the universal applicability of the concept of object, but instead they chose to question the existence of reality itself. Life provides a plethora of counterexamples to the concept of object. Organisms, societies, languages, cultures, minds, organizations all fail to possess one or more fundamental characteristics of an object. In truth even the most fundamental entities in nature, the fundamental particles, fail to fulfill all of the conditions of an object. All naturally occurring complex systems [14] are fundamentally transient in nature: they arise, they develop, and they fade away. They interact with their environment openly, exchanging matter and energy. They have neither fixed, stable components, nor a fixed form. Properties as well as attributes may depend upon context. In spite of this they are still capable of individuality and spatiotemporal coherence. They are generators, carriers and transformers of information and meaning. They have agency, acting upon their environment and not merely reacting to it. They intend, prehend, apprehend, and comprehend. They create, seek and follow goals. An alternative to the concept of object had been proposed more than 2000 years earlier by the Greek philosopher Heraclitus and the Indian psychologist/philosopher Siddhartha Gautama. Both proposed a metaphysics in which reality consisted of an ever changing flux of phenomena organized into coherence by some form of underlying subjectivity. These ideas flourished in the East but languished in Western thinking. The first Western philosopher to seriously revisit this metaphysics was Alfred North Whitehead, particularly in his difficult book Process and Reality [15]. Whitehead’s philosophy has been described as a philosophy of organism and re-introduced a concept of subjectivity into metaphysics in the notion of prehension. Whitehead conceived of a process as a sequence of events having a coherent temporal structure in which relations between the events are considered more fundamental than the events themselves. Whitehead viewed process as being ontologically prior to substance and becoming to be a fundamental aspect of being. This stands in contradistinction to both modern physics and mathematics, which have no notion of process and no

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notion of becoming. Whitehead’s process theory considers all physical phenomena as emergent from an ultimate reality of information laden entities called actual occasions. Expressed in modern physical language, the basic postulates are that: 1. everything in reality is generated by process; 2. everything that we observe is emergent from an ultimate lowest level that itself is inherently unobservable; 3. individual events of reality come into existence in a discrete but nonlocalized form, and what we observe is a diffuse avatar, which extends over space and time and constitutes a wave function; 4. this wave function is an emergent effect but it is through such wave functions that observable physical phenomena arise. Reality possesses two aspects - actual occasions, which are the primitive experiential elements, and processes, which generate the actual occasions. Manifest actual occasions in turn influence which processes are active and interacting. All physical entities are emergent from these actual occasions. This implies that individual actual occasions and processes can be inferred but not directly observed. Processes are distinguished by informational parameters which impute properties to the actual occasions that they generate, and which determine the types of interactions permissible between processes. Processes generate actual occasions that manifest space and time, but they themselves exist outside of space and time. The quantum nature of reality is a consequence of the discreteness manifesting at the fundamental level while the wave nature of reality is a consequence of the inability to resolve these fundamental events in space and time, thus physical reality acquires emergent wave like aspects. Process theory places emphasis upon three important characteristics: 1. the unfolding of process in the manifestation of actual occasions is strongly determined by the context generated by all participating processes and by the dynamics of their interactions; 2. observable aspects of physical reality - continuity, space-time, physical entities, symmetries - are emergent;

A Process Algebra Approach to Quantum Electrodynamics

11

3. interactions between processes are triggered by actual occasions and follow patterns expressed algebraically in the process algebra. In [9] it was shown that the use of a process algebra approach led to a realist version of non-relativistic quantum mechanics which nevertheless still exhibited the three characteristics typical of quantum mechanical situations: 1. the quantization of exchange in interactions; 2. the existence of Non-Kolmogorovian probability; 3. the existence of non-local influences.

3.

The Formal Process Algebra Framework

This section provides a concise mathematical formulation of the process algebra approach. The reader interested in more detail, especially in relation to the issues of non-locality, contextuality, non-Kolmogorov probability and hidden variable theorems, is invited to examine previous publications [8, 9, 10].

3.1.

Informons and Causal Tapestry

The actual occasions of Whitehead’s Process Theory are modeled as informons, which are generated by processes. A single complete generation of informons resulting from the action of a process is termed a causal tapestry [16]. Successive processes Pi result in a coherent ordered set of causal tapestries Ii , forming a causal sequence P

P

P

I0 →1 I1 →2 I2 →3 · · · . An external observer interprets the process situation via a semantic frame consisting of a causal manifold [17] M (causal distance d) and a Hilbert space H(M) over M. An informon n is described as a symbol string [n] < mn , φn (z); Γn, pn > {Gn }. The term [n] is merely a label. An informon possesses two extrinsic components, mn ∈ M (interpreted as a space-time point or causal interpretation), φn (z) ∈ H(M) (interpreted as a local interpolation

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contribution to a wave function or local Hilbert space interpretation), and three intrinsic components, Γn (local strength of the generating process), pn (set of local properties inherited from the generating process), and Gn (causally ordered set of informons, causally prior to n, causal distance ρ) called the content or information of n). Given a causal tapestry I, the set GI = ∪n∈I Gn forms a coherent causally ordered set with a causal distance function ρ which is space-like on I and timelike or null on the edges of Gn . The causal interpretation forms a causal embedding of GI into M and ρ(n, m) = d(mn, mm). Information used for the construction of an informon n must reside within Gn . It is viewed as propagating causally along the edges in Gn , thus never in violation of special relativity. Each local Hilbert space contribution from an informon n has the form φn = Γn Tmn g(z) where g(z) is an interpolation kernel on M and Tmn is a translation operatorP (eg. in one dimension Tx g(z) = g(z − x)). The interpolation sum Φ(z) = n Φn (z) over the causal tapestry I is an emergent wave on M. The local properties are usually (relatively) conserved quantities such as energy, momentum, angular momentum, mass, charge.

3.2.

Process Algebra

The process algebra describes how processes interact. A process exists in one of two states, active or inactive. An active process creates a single generation of informons in a series of N rounds, each consisting of r short rounds. Each generation forms a causal tapestry. Interactions between processes, as well as changes of state, are triggered by the creation of individual informons. For mathematical convenience, such transitions will be attributed to the completion of a generation by simply rendering the original process inactive and letting the partial causal tapestry created stand for a complete generation. Informons may be generated nonlocally although only locally causal information is used for their construction (termed quasi-non-locality in [9]). A primitive process non-deterministically [18] generates only one informon per round (R = 1). Complex processes (R > 1) are formed from algebraic combinations of primitive processes. Processes may act concurrently (denoted as a product) or sequentially (denoted as a sum). Processes may contribute to the generation of a single informon (free) or only to distinct informons (exclusive). Processes may act independently or interact. These three conditions result

A Process Algebra Approach to Quantum Electrodynamics

13

in eight distinct different sums and products of processes: ⊕ (sequential, excluˆ (sequential, free, independent), ⊗ (concurrent, exclusive, sive, independent), ⊕ ˆ independent), ⊗ (concurrent, free, independent), and corresponding interactive operations. The zero process, O, is the process that does nothing. Processes may be concatenated, which describes the succession of processes following each complete generation. Note that sums and products are naturally Abelian, while concatenation is naturally non-Abelian. As a general rule, processes corresponding to different states of a single entity combine through the exclusive sum, while processes forming a single state of a single entity combine using the free sum. The holistic nature of a single state is expressed in the superposition of information permitted with the free sum. This is not permitted in the exclusive sum which serves to separate distinct states at an information level. Process algebras possess a convenient heuristic representation as an algebra of combinatorial games, termed “reality games” [9, 18, 19]. An example of a realty game is given below. Standard quantum mechanics offers a dual theory in the form of the momentum representation. There is a dual theory in the process framework but it is much less obvious, being based on the order theoretic dual to the causal tapestry and developing from there a generalized momentum space. The astute reader might also have noticed that processes may be assigned definite values of quantities such as energy, momentum and angular momentum. This does not violate the Heisenberg uncertainty relations although the argument is subtle. Knowledge of momentum or position requires some measurement, which in the process algebra model depends upon the generation of and is triggered by informons. The non-continuous generation of informons gives rise to an apparent distribution of momentum values even though the generating process may have a specific momentum value. These issues, as well as the process approach to the paradoxes, are quite technical and beyond the scope of this paper.

3.3.

The Process Covering Map

A process P applied to an initial causal tapestry I0 will generate a succeeding causal tapestry I1 . If P is applied again to I0 it may yield a different tapestry I10 . Each such causal tapestry represents an alternate possible reality. In order to carry out computations, it is necessary to take this non-determinism into account

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William Sulis

in the form of a set valued map [20] called the process covering map (PCM). Consider a primitive process P. Given some prior causal tapestry I0 , P generates a causal tapestry I1 in a series of rounds, corresponding to a sequence of partial causal tapestries ∅, I10 , I20 , I30 . . ., each formed from the previous tapestry by the 0 inclusion of an informon, Ii0 = Ii−1 ∪ {ni }. The sequence of partial tapestries forms an ordered set (termed a generation chain) with edges labelled by inforn 0 mons n1 , n2 , . . ., nk , . . .(Ii−1 →i Ii0 ) and having a maximal element, the final causal tapestry I1 . Applying P to I again may generate a different ordered set of tapestries with edges n01 , n02 , . . . , n0k , . . .. Two distinct global Hilbert space interpretations will be generated X X Φ1 (z) = φni (z) and Φ2 (z) = φn0i (z) ni

n0i

The union of all possible generation chains forms the process sequence tree of P with initial causal tapestry I0 , denoted Σ(P, I). Associate to Σ(P, I) a set HP of elements of H(M) consisting of all global H(M)-interpretations constructed from every maximal causal tapestry in the sequence tree. Define P(P, I) = HP . For fixed I, and some primitive process P, define the process covering map PI : Π → P(H(M)) by PI (P) = HP , where P(H(M)) is the power set on H(M) and Π is the space of primitive processes. In the limit N, r → ∞ (that is, an infinite number of informons and complete information transfer) the theory of sinc interpolation [21, 22] (and Feichtinger-Gr¨ochenig theory more generally [23]) shows that HP → {ΦP (z)}, a singleton set. This forms the link to standard non-relativistic quantum mechanics [9] and shows why NRQM is considered to be an effective theory from the process algebra point of view. Fixing the process P and varying over the space C of causal tapestries yields a tapestry covering map PP : C → P(H(M)) by PP (I) = P(P, I) = HP . Associating each causal tapestry I with its global Hilbert space interpretation ΦI , we may define a generalized operator OP setting OP (ΦI ) = HP . In the continuum limit, HP becomes a singleton and we obtain an operator representation of the primitive process. General processes are built using the operations of the process algebra. First, define the formal product wP to mean that the local process strengths generated by the actions of P have their value multiplied by w. Then a simple argument shows that

A Process Algebra Approach to Quantum Electrodynamics PI (⊕i wi Pi) =

X i

15

ˆ i w i Pi ) wi PI (Pi ) = PI (⊕

where for two sets of functions A, B the sum A + B = {f + g|f ∈ A, g ∈ B} which extends the map to ΣΠ , the sum algebra over Π. Recall that a co-product of two functions f, g is simply the formal (not pointwise) sum of the two functions f ⊕ g. The co-product of two sets A, B is denoted A t B. Consider a generic product P = Πi Pi of primitive processes Pi . Construct the process sequence tree Σ(I, P), noting that in this case at each round i the product process will generate a correlated set of informons Ai = {n1i , . . ., nni } (nki generated by Pk ). Partition the new causal tapestry I 0 into subsets of informons generated by the individual subprocesses, i.e. I 0 = ∪i Ii0 (Ii0 = {ni1 , ni2 , . . .}). The global Hilbert space interpretation ΦI 0 (z) is decomposed onto each of these component sets yielding Φ(z)0i on Ii0 (an element of PI (Pi )), and thus the co-product sum ΦI 0 (z) = Φ01 (z)⊕Φ02 (z)⊕· · ·⊕Φ0j (z). Define ˆ i Pi ) = PI (P1 ) t PI (P2 ) t · · · t PI (Pn ) PI (⊗ and PI (⊗i Pi ) = PI (P1 ) ⊕ PI (P2 ) ⊕ · · · ⊕ PI (Pn ) . where the set sum ⊕ permits functions to sum. Interactive sums and products must be dealt with case by case.

3.4.

Configuration Space Process Covering Map (PCMC )

The process covering maps yield information about distinct possible realities. For computational purposes a different map is needed, the configuration space PCM (PCM)C . There are two reasons why this is necessary. Suppose that we have a generic product P = Πi Pi of primitive processes Pi . At each round the product process will generate a correlated set of informons Ai = {n1i , . . . , nni } (nki generated by Pk ). Partition the new causal tapestry I 0 into subsets of

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William Sulis

informons generated by the individual subprocesses, i.e. I 0 = ∪i Ii0 (Ii0 = {ni1 , ni2 , . . .}). Note first, if an informon nkj is generated in round j by Pk , then it Q will never be generated in any other round. This implies that I 0 = ∪i Ii0 6= i Ii0 so that a single process action cannot generate all possible correlated sets of informons. Second, although the informons within each set Ak are statistically correlated by virtue of the global generating process P, each Ak and the informons within it are generated independently of one another by the individual subprocesses Pi . The specific actions of each Pi are determined solely by Pi itself. Thus we may artificially extend the sequence process tree. Take a maximal causal set 1 I 0 and extend its generating sequence by an edge obtained from a different path, ensuring that if we wish to add n2 to component 1 Ii0 of 1 I 0 and there exists in component 1 Ii0 of I1 an n1 such that pn1 = pn2 and mn1 = mn2 then Γn1 = Γn2 . Such an informon is said to be admissible in 1 I 0 . Repeatedly extend the sequence tree by adding admissible elements to maximal tapestries until no further extensions are possible. The resulting tree is called the process 2 I 0 , . . .} configuration sequence tree ΣC (I, P). The maximal tapestries {1M I 0 , M of ΣC (I, P) have sufficient numbers of informon combinations to calculate correlations and so it makes sense to define the global configurational H(M) interpretation on one of these maximal tapestries as ΦC i I 0 (z) = M

X

Γn1 · · · Γnn Tmn1 g(z1 ) · · · Tmnn g(zn )

{n1 ,...,nn }⊂iM I 0

where the sum is over the edge sets along the path forming iM I 0 . Given ΣC (I, P), let IM denote the set of all of its maximal causal ΣC (I,P) tapestries. We define the configuration process covering map, PCMC , denoted PC I (P) = to be {

X

Γn1 · · · Γnn Tmn1 g(z1 ) · · · Tmnn g(zn)}

(n1 ,...,nn )∈ iM I 0

taken over all iM I 0 ∈ IM . If no tuples (n1 , . . . , nn ) are excluded this may ΣC (I,P) be written as

A Process Algebra Approach to Quantum Electrodynamics {

X

Γn1 Tmn1 g(z1 ) · · ·

j n1 ∈M I10

X

17

Γnn Tmnn g(zn)}

j 0 nn ∈M In

resembling the more familiar configuration space construction. The linkage between the process algebra model and standard quantum mechanics is through the process and configuration space process covering maps. A single play of a reality game generates a single generation of informons corresponding to a physical system, and thus an ontological depiction of an actualized reality. Quantum mechanics does not provide this, instead providing a probabilistic description of all possible actualized realities. The covering maps show how one moves from the ontological depiction of the causal tapestry to the computational description of quantum mechanics. In [8] it was argued that quantum mechanics arises as an asymptotic limit of a (set valued) quotient map, so that the process algebra model can be understood as a true completion of quantum mechanics, at least in the case of non-relativistic quantum mechanics.

3.5.

The Reality Game

Process algebras possess a convenient representation as an algebra of combinatorial games. This is a powerful heuristic tool in which one complete play of such a game results in the generation of a causal tapestry. These games are termed “reality games” and have been described in detail elsewhere [9], where it was shown how standard non-relativistic quantum mechanics arises. The basic idea is to have two players act cooperatively in order to construct the informons that form a causal tapestry. Each informon is constructed in a series of short rounds, during each of which, information from a single prior informon is incorporated into the nascent informon, preserving all causal relationships. Its information, in the form of its local strength, is incorporated into the strength of the nascent informon as follows. To each generating process P, there corresponds a propagator K P which technically is defined on the edges of the content set of the informons of the causal tapestry. More precisely, this propagator is a function of the causal distance between informons (possibly together with a factor which determines whether or not propagation may actually take place). There P defined on the causal manifold M such will be a corresponding propagator KM P 0 P that K (n, n ) = KM (mn, mn0 ). Then during a short round the strength of the

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William Sulis

nascent informon n0 will be modified according to Γn0 → Γn0 + lP3 K P (n0 , n)Γn . which states that the current value of the strength of the new informon n0 is replaced by that value plus a contribution propagated from the prior informon n. The informon n is then incorporated into the content set of n0 with appropriate order relations and causal distances. In the simplest case in which the causal manifold interpretation maps each informon to an element on an lP4 lattice, the Hilbert space representation is obtained via sinc interpolation. The sinc function is defined as S(x) =

sin x x

. In two dimensions this takes the form Φn0 (z) = Γn0 Tmn0 StP ,lP (z) where mn0 = (mlP , ntP ), K(λ, x) = πx/λ, and Tmn0 StP ,lP (z) = TmlP S(K(lP , x))TntP S(K(tP , t)) and analogously for higher dimensions. The global Hilbert space interpolation Φ(z) on a causal tapestry I is obtained by summing the contributions from each of the informons of the causal tapestry and can be shown, under suitable limits, to converge to the usual quantum mechanical wave function. Using sinc interpolation theory [21, 22] it can be shown [9, 10] that the global Hilbert space interpretation Φ(z) =

X n∈I

X X

Φn (z) =

X

Γn Tmn sinctP lP (z) =

n∈I

lP3 K P (m, n)ΓmTmn sinctP lP (z) =

n∈I m∈Gn

X X

n∈I m∈Gn

P lP3 KM (mm, mn )Γm Tmn sinctP lP (z)

tP ,lP →∞

−→

A Process Algebra Approach to Quantum Electrodynamics XZ P KM (mm, mn)Γm dmmTmn sinctP lP (z) = Ψ(z)

19

n∈I

the usual quantum mechanical wave function.

3.6.

Process Approach to Quantum Field Theory

Classically, particles and fields are considered to be distinct classes of physical entities both phenomenologically and mathematically. This distinction persists in quantum mechanics where particles are described by Hilbert spaces and quantization of Hamiltonians, while fields are described by Fock spaces and second quantization of Lagrangians. In the process algebra approach there is no fundamental difference between particles and fields. These are simply descriptions for two limiting cases. The term particle is used for the limit of small quanta numbers while the term field is used to describe the infinite limit. In the latter case, simultaneous measurements at many spatially separated sites will yield results with high probability, thus satisfying Feynman’s notion of a field as “... a set of numbers we specify in such a way that what happens at a point depends only on the numbers at that point” (sic) [24]. In the process algebra model there is no wave-particle duality. Every physical entity has discrete (interpreted via M) and continuous (interpreted via H(M)) aspects. Let Π denote a collection of primitive processes representing the distinct pure states of a single entity. Let P denote the process algebra generated by Π ˆ combinations and let ΣP denote the subalgebra of P formed by taking all ⊕, ⊕ of elements of P. The space of bosonic-like fields is defined as the subalgebra of P of the form ˆ P) ⊕ (ΣP⊗Σ ˆ P⊗Σ ˆ P) ⊕ · · · ΣP ⊕ (ΣP⊗Σ while the space of fermionic-like fields is defined as the subalgebra of P of the form ΣP ⊕ (ΣP ⊗ ΣP) ⊕ (ΣP ⊗ ΣP ⊗ ΣP) ⊕ · · · . These are the process algebra analogues of the Fock space. Unlike the Fock space, an element of either of these subalgebras is just a complicated process, not a wholly new mathematical form.

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The process algebra representation of a field is phenomenological in character. A process P in state k generating n informons per round (n particle state) can be written as Pnk = Pk ⊗ · · · ⊗ Pk . A general field can then be written in the form n

n1 j ⊕∞ n=1 {wk1···kj Pk1 × · · · ⊗ Pkj |Σini = n}

where the ki run over different states. Each term corresponds to a field having exactly n quanta distributed across different states. The sum describes the case in which the field is in a superposition of different definite quanta states. Note that in the process algebra model, every informon, meaning every space-time point, is associated with a local contribution to the wave function of a process. In the field context, this means that every space-time point is associated with a local contribution to some field. In the process algebra model there is always a nonzero field element at a space-time point because every space-time point is identified with a field (or particle) contribution. In fact there is no space-time without something to mark it. Vacuum fluctuations are not mysterious here. Since the number of informons is finite (though large), field calculations still yield definite values. Divergences appear only in the idealized continuum approximation, not in the causal tapestry.

4.

Process Electrodynamics

The process algebra model is quite general. Its applications depend upon specifying the subalgebra of interest and its representation as a combinatorial game. That in turns requires specifying the strategies to be used by the players of the game [9]. The causal tapestry structure is also very general. The only definite requirement is that the content set of each informon must be a causal set and that these causal sets must be coherent across the causal tapestry. Previous work on the process algebra model focused on the theory of non-relativistic scalar particles, which arguably forms the simplest case. In order to advance the theory further it is necessary to tackle two issues - compatibility with at least the special theory of relativity, and the ability to describe the creation and annihilation of particles and fields. The latter capability is already a feature of the process

A Process Algebra Approach to Quantum Electrodynamics

21

algebra, expressed in the idea of active and inactive states which correspond to creation and annihilation. The main issue is thus compatibility with relativity.

4.1.

Relativistic Considerations

The first problem is to define what is meant by a causal tapestry and its process dynamic being compatible with relativity. It is obvious that, being a discrete and finite structure, it is not directly compatible with relativity, whose symmetries are continuous. However, to an observer, every causal tapestry possesses interpretations to a causal manifold and to a Hilbert space defined on that manifold. That is, there exists a causal embedding m:I →M which preserves both causal order and distance and a local Hilbert space interpretation function φ : I → H(M) mapping each informon to an interpolation contribution. Suppose that a second observer appears, whose mapping of M differs from that of the first observer by a Lorentz rotation or boost L. This second observer will then have interpretations of the form Lm and ψ = Lφ. Since m is a causal embedding, so is Lm, since L by definition preserves causal order and distance. Consider Lφn (z) = L(Γn Tmn g(z)) for some informon n and interpolation function g. L acts on the translation operator LTmn → TLmn . It acts on g as Lg(z) = g(L−1 z). Suppose that φn = Γn Tmn g(z) for some informon n. Then Lφn = L(Γn Tmn g(z)) = LΓn L(Tmn )(Lg(z)) = LΓn TLmn g(L−1 z). In the P generation of Γn one sums terms of the form lP3 KM (mn0 , mn)Γn0 If we apply 3 P P L to this we obtain lP KM (Lmn0 , Lmn )Γn0 . But KM depends only on the P P (Lm 0 , causal distance between informons, so therefore KM(mn0 , mn ) = KM n Lmn ). Thus the process strength is invariant under L. Thus Lφn = L(Γn Tmn g(z)) = Γn TLmn g(L−1 z)

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Thus we see that under a Lorentz rotation or boost L, an informon of the form [n] < mn , φn (z); Γn , pn > {Gn } should map to an informon of the form [n] < Lmn , L(ΓnTmn g(z)); LΓn, pn > {LGn } = [n] < Lmn , Γn TLmn g(L−1 z); Γn , pn > {LGn } so that Γn TLmn g(L−1 (Lmn )) = Γn g(L−1 (Lmn − Lmn )) = Γn . Thus ψ(Lmn) = φ(mn ). Thus the intrinsic features of the informons tapestry are preserved and only the extrinsic features of informons are affected by the Lorentz transformation. The causal tapestry and its process dynamics remain invariant under Lorentz tranformations. It is true that using sinc interpolation in the product form one will not obtain a Hilbert space interpretation that is relativistically invariant. This is not a problem because the Hilbert space interpretation is just that, an interpretation created by an external observer for their own heuristic purposes. They are free to use any interpolation or interpretation method that suits their purposes. The process dynamics is determined by and represented in the intrinsic features of the informons and these remain invariant under Lorentz rotations and boosts. Two observers will see the same dynamics play out even if they label the various components of the dynamics differently. One can always choose interpolation functions which are themselves Lorentz invariant but as they play no essential role in the dynamics or in any calculations made on the causal tapestry this is really unnecessary. They serve merely as a convenient heuristic tool to link the causal tapestry to more standard forms of representing reality. Reality in the process approach is therefore what philosophers term a compound present formed from prior informons that have not yet transferred all of their information, together with nascent informons currently under construction. When the process has completed its construction activity, the nascent informons become prior and construction begins anew with either the same or a new process. The complete action of a process constitutes a generation. At first glance this would appear to contradict the usual understanding of the relativistic prohibition on any notion of simultaneity. Detailed arguments have

A Process Algebra Approach to Quantum Electrodynamics

23

been presented elsewhere [9] but there are two essential points. First of all, processes generate informons according to a causal structure which determines the causal distance between points and their causal ordering, not their space-time coordinates. There is no presumption that the simultaneity of generation of informons leads to their being assigned the same value for their time co-ordinate. The informons that comprise a single generation are all space-like separated from one another. No information ever passes between the informons within a causal tapestry so it is irrelevant what space-time co-ordinates are assigned to them. The only important information is contained in the causal structure which we already know to be invariant. The special theory of relativity asserts that it is impossible for two observers in relative motion to one another to create individual frames of reference that can agree in their assignments of time and space. So what? That information is unimportant anyway. The fact that we, as observers, are unable to agree as to the existence of a universal proper time, is also irrelevant to the question of its existence. This is an example of the principle of misplaced omniscience [9], the naive belief that we should be capable of knowing everything about reality, and that if we can’t know it then it must not exist. An observer is, by definition, an emergent entity which persists across many different causal tapestries and so must exist at an ontological level beyond that of the individual informons upon which it supervenes. It is simply not possible for an observer to bear witness to its own emergence, and hence to one of its underlying causal tapestries. An observer exists in the flow of information across tapestries which ensures its coherence as an entity. An observer can therefore only observe events at the emergent level, that is, at the level of causal manifold and Hilbert space interpretation. Thus the observer observes only the emergent 4 dimensional space-time and its events, and not the causal distance between causal tapestries that give rise to this emergent space-time. Another point which cannot be over-emphasized is the fact that the informons of a causal tapestry do not move. Informons come into existence and fade out of existence. Their presence briefly marks the existence of a (rather fuzzy) space-time element. They create an emergent space-time and so cannot move in space-time. Only the information of which informons are comprised actually “moves”, although strictly speaking this information does not actually propagate in space-time unless it happens to pass along a causal chain of informons.

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In the process algebra model, information may perfectly well “jaunt” from one informon to another without passing through any intervening space-time since these informons are space-time. Lorentz symmetries apply at the emergent level to observers, who are formed from complex interactions among simpler processes. Thus these symmetries more accurately apply to processes and their interactions rather than to the informons that the processes generate. It is for this reason that while these transformation may affect our perception of reality they do not actually affect reality itself. Thus the process algebra model offers a promise to rid physics of many of the paradoxes seemingly created by special relativity, just as it appears to rid quantum mechanics of many of its paradoxes. The existence of a transient now can be inferred so long as we know the spatial distance between space-time points. All we need do is wait for signals to arrive from ever distant points and we can then reconstruct how that transient now was constituted. We simply can never observe it while it occurs. There are many things in life whose existence must be inferred. That does not deter us. The clinging to the idea of a block universe results in more conceptual problems then it solves. The process model of reality appears to yield more or less the same results while painting a much simpler picture of reality.

5.

The Quantum Conundrums

Since its inception, quantum mechanics has been plagued with philosophical problems, paradoxes, and computational issues such as divergences. These matters have been discussed at some length in [9, 8] but a few words are in order here. The process algebra model appears to offer a way out of most, if not all, of these problems. First of all, the model is finite and discrete, which immediately eliminates the divergences and provides the model with a natural ultraviolet cutoff. This also places an upper limit on the possible masses of fundamental particles. The matter of wave-particle duality is resolved. In the process algebra model this becomes a matter of perspective. Fundamental particles are emergent entities which appear particle-like at small scales and wave-like at large scales. It is not a case of either-or. The emergent reality of these fundamental particles is determined by their generating processes, independent of any observer. Different

A Process Algebra Approach to Quantum Electrodynamics

25

observers may have different perceptions of these entities but their observation does not alter, in any way, the essential nature of these particles unless there is a significant interaction with the particles. The observer in the process algebra model does not determine or create reality. Likewise, there is no measurement problem. Unlike in standard quantum mechanics, there is no separate theory to describe the process of measurement. A measurement is simply an interaction between the observed system and a specialized process called a measurement apparatus process. This interaction is described in the process algebra model just like any other interaction. A measurement takes place when an informon of the system process and an informon of the measurement apparatus process are generated, a mutual compatibility is found [7], and the processes then enter into an interaction. The outcome of the interaction may or may not include an altered generating process for the system but it certainly involves a series of transitions involving the measurement apparatus process that eventually results in the creation of a constellation of informons that can be interpreted as a sign, and ultimately as indicating a measurement value. There is no wave function collapse. Recall that in the process algebra model, different states of a system possess disjoint sets of informons serving as interpolation points. Only one of these will trigger off an interaction with the measurement apparatus process and so only one measurement value will be obtained. The precise value will depend upon the compatibility and this in turn will depend upon the local process strengths of the system process and measurement apparatus process informons. From these local process strengths it is possible to derive a probability structure, which is seen to be emergent, non-Kolomogorov, and to follow the Born interpretation. Information flow within a causal tapestry is causal, and therefore compatible with special relativity. It may be non-local but it never involves information flow between space-like separated informons. In particular, information never flows between the informons of a given causal tapestry, only from prior causal tapestries to subsequent causal tapestries. The only serious non-locality appears in the manner in which informons are distributed relative to one another. There is no requirement that informons be generated in such a manner that their causal manifold interpretations should be contiguous to one another. In fact they may created at widely separated locations. This is possible because processes are generators of space-time. They themselves are thought of as existing outside

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William Sulis

of space-time. Prohibitions on non-locality are relevant only to events and information that occurs within space-time, that is, between the entities that are emergent and manifest within space-time. This is not the case for the generating processes. It is for this reason that the process algebra model is said to be quasi-non-local. The process algebra model is also said to be quasi-non-contextual. By this is meant the fact that generating processes impart specific properties to the informons that they generate. No process, however, possesses definite values for all possible properties that may be measured. This can be seen in the fact that the concatenation of processes is a non-commutative operation. Each process corresponds to a generalized operator on the Hilbert space and the concatenation of processes corresponds to the multiplication of these operators. Since multiplication of operators on a Hilbert space is known to be a (generally) noncommutative operation, it follows that process concatenation must also be noncommutative. Note, in contrast, that process sums and products are commutative. This non-commutativity implies that a process will generate a set of informons possessing definite properties, but any interaction that changes the generating process, such as occurs in an interaction with a measurement apparatus process, will result in a new process generating informons with possibly quite different properties or values. Thus the set of properties and their values depends upon the conditions that exist at the time of generation of the informons that carry their properties, and so these properties are contextual in character. Nevertheless note that it is not the measurement act which imparts these values to the informons of a system. It is the generating process of the system that does this. Informons carry a definite, albeit incomplete set of properties which exist independent from any possible measurement act. There is no wave function collapse, no observer dependence for reality. In [9] it is shown how the non-Kolmogorov nature of the emergent probability structure of the process algebra model is compatible with all of the hidden variable theorems. It is shown that it is indeed possible for reality at the fundamental level to be local (at least quasi-non-local at worst) so that the local hidden variable theories are shown to be false. The problem with those theories is that they all make the assumption that local reality or reality at the classical level must be Kolmogorov in structure, and this assumption is simpy false [12].

A Process Algebra Approach to Quantum Electrodynamics

27

Once that fact is appreciated, reality become ”real” again.

6.

Process Approach to Potentials

The potential plays a central role in classical physics whether it is incorporated into the Hamiltonian or the Lagrangian. The opposite has been true in electrodynamics, where force has played a central role the in the Maxwell equations. The discovery of gauge symmetry suggested that the potential of electromagnetism was merely a heuristic device rather than an ontological entity since an infinite number of these potentials could give rise to the same electromagnetic field and there appeared to be no way to distinguish among them. Quantum mechanics reversed these roles again. The Schr¨odinger equation for the interaction of a particle with an electromagnetic field treats the vector potential, like the scalar potential, as a multiplicative operator, but entwined with the derivative operator to yield   1 ∂Ψ(x, t) (−i∇ − qA)2 + qφ Ψ(x, t) = 2m ∂t ∇ + iqA is called the covariant derivative, which appears whenever the electromagnetic field is involved. The discovery of the Aharonov-Bohm effect showed that the vector potential has an independent reality after all, and thus in quantum mechanics the vector potential becomes the more fundamental object. In quantum field theory a process of second quantization is undertaken, in which the field variables become formal operators and not merely functions. For example the formula for an electromagnetic field might take the form A(x, t) =

X

(2ωΩ)−1/2(ak (t)eik·x + a†k (t)e−ik·x )

where ak (t) and a†k (t) are the usual annihilation and creation operators. While classical and standard quantum mechanics utilize a vector potential that connects fairly readily to an observable reality, the same cannot be said for the operator formalism of quantum field theory. Fields do not take on definite values at a point but rather become endowed with a wave function at each point. From the process algebra point of view, this arises because quantum mechanics confuses the ontological wave, given as the Hilbert space interpretation

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William Sulis

of a single causal tapestry, with the heuristic wave function(s) arising from the process covering map and especially the configuration space covering map. In the process algebra one begins by defining a particular instance of reality and then uses the process sequence tree to generate more complex heuristic representations which can be used to carry out correlational analyses. The operator interpretation of process for single particles arising from the PCM suggests that there will be a corresponding operator interpretation in the case of fields, which are the infinite limit of the multi-particle case. The technical challenges involved in demonstrating this are great so for the moment this remains a conjecture. The process algebra model does suggest that the vector potential, being a function of space-time, should be an intrinsic feature of informons. The only possibility is for it to be part of the process strength. The simplest model to begin with is a semi-classical model, in which we take the process strength to be the classical vector potential. Quantization appears as a result of the discrete and finite nature of the causal tapestries. Stochasticity arises from the non-determinism of the process action in generating informons. An individual informon will then act as an instantiation of a photon, which will trigger off an interaction of the field process with any other process to which it is algebraically coupled. According to the measurement theory, if the field corresponds to a single photon field, only a single informon will ever be generated in a single round so any interaction will appear to be the result of a single photon. This interaction could appear anywhere in which the field in present, and the non-deterministic and non-local generation of these informons guarantees that it will indeed behave like a single photon field. Let us consider two models to illustrate this approach.

6.1.

Process Approach to Electrodynamics

The approach to quantum electrodynamics taken here is semi-classical. This is arguably the simplest approach which illustrates the methodology and also provides a baseline from which to develop more accurate and sophisticated models. By semi-classical I mean that we shall use the classical kernel for the wave equation for the four potential based upon Maxwell’s equations and the asymptotic case will be the classical function for the four potential. Quantization becomes a natural consequence of the finite and discrete character of the underlying causal tapestry.

A Process Algebra Approach to Quantum Electrodynamics

29

Photons are interesting because they have an number of interesting properties. They are massless, which implies that they propagate at the maximal possible speed, c. This in turn implies that the causal distance between successive instances of a photon is zero. From the process algebra perspective, this implies that the causal distance between successive causal tapestries representing photon generation must be zero. This in turn implies that the content set of any photon informon must consist solely of informons from the immediate prior causal tapestry and no others. In such a case the order theoretic structure of the content set is that of a causal anti-chain. A causal tapestry in which all of the content sets consist of an anti-chain is called a thin causal tapestry. Although not relevant here, if a causal tapestry has informons whose content sets have no trivial order relations (hence are not anti-chains), then it is called a thick causal tapestry. It is conjectured that causal tapestries corresponding to the evolution of massive entities must be thick casual tapestries. The fact that the causal distance between a prior and nascent photon informons must be zero further implies that the embedding hypersurface of the causal tapestry in the causal manifold must be three dimensional, usually taking the form of a 3-sphere. It is conjectured that a thick causal tapestry will embed into a 4-dimensional hypersurface of the causal manifold, hence the name thick. There is no requirement that the content sets of a thick causal tapestry must have the same order structure, so that the 4-dimensional contributions arising from each prior informon need not bear any relationship to one another. In such a case the local 4-dimensional regions generated from each prior informon need not be isomorphic, giving rise to a tattered 4-space. Three other features are important. Photons in the same state are indistinguishable from one another. Photons may superimpose upon one another. Photons do not interact with one another. Consistent with the hope for a quasi-non-local model, it is assumed that information travels no faster than the entity to whose generation it contributes. This does not preclude the possibility of informons being generated at nonspatially contiguous sites as in the case of non-relativistic quantum mechanics. It does, however, preclude information passing from prior to nascent informons if the distance exceeds the maximum allowable for the entity. If prior informons

30

William Sulis

are localized in space, a nascent informon may still be generated in some noncontiguous spatial region, but in that case it cannot receive information from those prior informons. This understanding is not necessary for what follows but nevertheless worth noting for future reference.

6.2.

The General Reality Game Model

The simplest model which illustrates the basic ideas of the process algebra approach is a regular lattice model in which we can use the theory of sinc interpolation to analyze the global Hilbert space interpretation. It is not particularly realistic although it does work in situations in which a plane wave approximation can be employed. Let us first examine the case of a process generating a single photon. Consider a reality game model for some primitive process Pn (R=1). In the setting of a semi-classical electrodynamics take as process strength a 4vector Γn = (A1 , A2 , A3 , φ), corresponding to the scalar and vector potentials, and as properties pn = (k1 , k2, k3 , h) where the momentum is (k1 , k2, k3 ), and helicity h. For simplicity we shall ignore helicity. The causal manifold M is taken to be a flat 4-dimensional Minkowski space-time. The simplest reality game to consider is based on a non-deterministic radiative kernel strategy [9] in which the causal tapestries embed into a discrete sublattice of the causal manifold M. Assume that this sublattice is a regular lattice in which the spatial spacing is given by lP (Planck length) and the temporal spacing by tP (Planck time). This choice imposes a natural ultraviolet cutoff. The use of a discrete lattice is not unreasonable in the case of photons since they propagate at a fixed speed c. We imagine that a process takes time tP to generate an informon. This example is merely illustrative of the approach. There are many possible strategies and further research is required to determine which strategy is most accurate and definitive. As shown in [9], many strategies may give rise to the same global Hilbert space interpretation, even with different interpolation points and local process strengths. These are termed epistemologically equivalent strategies. Such strategies produce different pictures of reality at the lowest levels but cannot be distinguished at the phenomenological level of observers. This is reminiscent of the phenomenon of gauge invariance. Until it can be shown that these strategies are distinguishable, they can be utilized for mathe-

A Process Algebra Approach to Quantum Electrodynamics

31

matical and computational convenience. Let each informon be labelled by an integer 4-vector, so that n = (x, y, z, t) where x, y, z, t ∈ Z and let it embed to the point (xlP , ylP , zlP , ttP ) in M. Let the causal distance on the causal tapestry be given as ρ(n, n0 ) = c2 (t − t0 )2 − (x − x0 )2 − (y − y 0 )2 − (z − z 0 )2 = d(mn, mn0 ). Let I0 denote the current causal tapestry. Assume that the informons of I0 embed into the sublattice having fixed temporal index t0 . Suppose that an informon n ∈ I0 embeds at the site (x, y, z, t0). Define Fn , the forward front from n as {(x + ilP , y + jlP , z + klP , t0 tP + tP )|i2 + j 2 + k2 = lP2 } These are the lattices points that can be reached at speed c in time step tP . We shall embed the nascent causal tapestry I1 into the sublattice with fixed temporal index t0 + tP . To each lattice element n = (x, y, z, t0 + tP ) we may assign a backward front Bn : {(x + ilP , y + jlP , z + klP , t0 tP )|i2 + j 2 + k2 = lP2 } which consists of all prior lattice sites which could pass information forward to site n at speed c in time step tP . The process strength is the classical four vector A = (A1 , A2 , A3 , φ). Each term in this four vector satisfies the wave equation 1 ∂2 ) )Ai = 0 c2 ∂t2 subject to whatever boundary conditions may be present. In the present example we shall consider only the case in free space in the absence of charges, so that A4 = φ = 0. The Green’s function for this equation has the form (∇2 − (

K(z0 , z) =

1 δ(|z0 − z| − c(t0 − t)) 4π|z0 − z|

The Green’s function only takes into account information arising from the backward cone of each nascent information and each prior informon can only propagate information to the elements of its forward cone, so that there is no point in having r > 6 since no further information exchange can take place. Each round creates an informon to be included in the nascent causal tapestry and

32

William Sulis

possibly deletes an informon from the current tapestry. Each short round propagates information from a current informon in the backward front of the nascent informon currently being generated. A short round of the game is played as follows: 1. The game is initialized with current tapestry I0 and nascent tapestry I1 = ∅. 2. If this is the first short round of a round, Player I selects label n not previously used (which may be taken to be the string for an unoccupied lattice site (x, y, z, t0 + 1)) and initiates a new informon [n] < (, , λ) > {} together with a token counter Tn = ∅. Otherwise Player I selects the nascent informon n currently being generated. 3. Player II selects a current informon m which embeds into the backward front of n. That site is then deleted from the backward front and n is deleted from the forward front of m. 4. Player I assigns the nascent informon n a location mn in the causal manifold M, which for simplicity here we take as having the same lattice values. 5. Player II then provides or updates the content set Gn , replacing Gn by Gn ∪ {m} ∪ Gm 6. Player I then updates the local process strength Γn of n by adding a contribution from n. 7. The short round ends and a new short round begins. If the backward front is empty, then the round stops and the nascent informon n is placed in the nascent causal tapestry, so that I1 is replaced by I1 ∪ {n}. If the forward front of m is empty, then m is deleted from the current causal tapestry, so that I0 is replaced by I0 \ {m}. 8. If a round ends then the entire process repeats with Player I picking a new nascent informon and the rounds continue until either no current informons remain or the limit N in the number of rounds has been reached.

A Process Algebra Approach to Quantum Electrodynamics

33

At the end of the round it should be clear that the content set Gn = Bn , the backward front of n. For n ∈ Gn0 , the propagated information has the l2P form 4π Γn . The game ensures that each informon only contributes once to the nascent informon. The tokens will therefore consist of the set {

lP2 Γn |n ∈ Gn0 = Bn0 } 4π

b denote a global Hilbert space interpretation and A the classical funcLet A tion for the same initial conditions. Then the global Hilbert space interpretation will take the form b 2 (z) = A X

{

X

{

X l2 P Γn }Tmn0 StP ,lP (z) = 4π

n0 ∈I1 n∈Bn0

X l2 P b A1 (mn )}Tmn0 StP ,lP (z) 4π

n0 ∈I1 n∈Bn0

The temporal difference between a current and a nascent informon is tP . Since the causal distance between these informons must be 0, their spatial separation must be lP . If we set the temporal interval between the current and nascent causal tapestries to be tP , then the propagator takes the form 1 δ(|z0 − z| − c(t0 − t)) = 4π|z0 − z| 1 1 δ(|z0 − z| − ctP ) = δ(|z0 − z| − lP ) 0 4π|z − z| 4π|z0 − z| It is clear, therefore, that the only informons that contribute to the local process strength, and therefore the four potential, of an informon n0 are those informons that lie within its backward cone Bn0 . Hence K(z0 , t0 ; z, t) =

X X

n0 ∈I1 n∈I0

X X

n0 ∈I1 n∈I 0

lP3

P b 1 (mn)Tm 0 St ,l (z) = lP3 KM (mn0 , mn )A P P n

1 b 1(mn )Tm 0 St ,l (z) = δ(|mn0 − mn | − lP )A P P n 4πlP

34

William Sulis X X l2 P b A1 (mn)Tmn0 StP ,lP (z) 4π 0

n ∈I1 n∈Bn0

which is identical to the global Hilbert space interpretation determined by the reality game. This in turn implies that

X X

n0 ∈I1 n∈I0

X Z

n0 ∈I1

Mt

b 2 (z) = A

P b 1 (mn)Tm 0 St ,l (z) ≈ lP3 KM (mn0 , mn )A P P n P b 1 (mn)dmn Tm 0 St ,l (z) KM (mn0 , mn )A P P n

becomes exact as tP , lP → 0, hence X Z P = KM (mn0 , mn )A1 (mn)dmn Tmn0 StP ,lP (z) = n0 ∈I1

Mt

X

A2 (mn )Tmn0 StP ,lP (z)

n0 ∈I1

The theory of sinc interpolation [23] shows that when n → ∞ = A2 (z) b 2 (z) = A2 (z). so that A Thus we obtain the classical four potential as an asymptotic limit as N, lP , tP → ∞. However, reality is held to correspond to the case in which tP and lP take their true values, so that the classical case, as the case of non-relativistic quantum mechanics, may be viewed as providing merely an effective theory. The process algebra model describes a finite and discrete reality, in which the four potential is quantized by virtue of this discreteness. Any interaction between this field and another process will only be triggered by informons as they are generated, in the present example only by single informons. Thus one obtains a field represented by the global Hilbert space interpretation corresponding to the classical four potential, but this is quantized by virtue of being emergent from a

A Process Algebra Approach to Quantum Electrodynamics

35

collection of discrete informons, and so corresponds to a single state of a single photon. Notice that the propagating four potential is spatially subdivided but the individual informons remain indivisible just as the phenomenology of photons requires. A single photon may exist in a superposition of single photon states. In such a case the superposition will be represented as an exclusive sum P1 ⊕ P2 ⊕ · · · ⊕ Pn which ensures that informons respect ontological distinctions. In some cases it may be possible to describe a spatial extension of a single photon as a free sum of identical copies of the same photon state, hence ˆ ⊕P ˆ ⊕ ˆ · · · ⊕P ˆ P⊕P . The free sum is used because these individual subprocesses all contribute to the same photon state but repeating them forces spatial separation among their respective informons. This situations bears some resemblance to ideas of collective electrodynamics [24]

7.

The Problem of Initial Conditions

The process algebra model appears capable of generating a classical four potential as an emergent interpolation from a quantized fundamental level of reality. The preceding section provides a demonstration in principle of the process algebra approach but it is still far from being the final answer. One problem is that the model relies on the use of the classical propagator for the wave equation. Missing from the above model is a specification of the initial conditions which are required to give an exact solution to the wave equation. These are incorporated into the initial local process strength values but how are these to be specified? Since the process algebra is a local generative model, it is not permitted to incorporate into the initial conditions effects from boundary conditions that the generating process has not yet interacted with. To do otherwise would assume the possibility of non-local connections which the process algebra model is meant to avoid.

36

William Sulis The solution to the wave equation

1 ∂2 ) )Ai = 0 c2 ∂t2 can be obtained in several ways. The simplest is to note that (∇2 − (

f1 (k · x + cωt) + f2 (k · x − cωt) is a solution for any twice differentiable functions f1 , f2 where kx2 + ky2 + kz2 − ω 2 = 0. These describe spatial waves which propagate in three space in the k direction. The problem though is from where did these spatial forms originate? They are defined on the real line which seems to imply an infinite amount of information is required for their specification. If they are band limited then they could be expressed as a Shannon-Paley-Weiner series of the form fi (x) =

∞ X

fi (n)TnS(ωx)

n=−∞

for some suitable frequency ω. If we accept a modest truncation error then we may truncate the series at some value L and obtain fi (x) =

L X

fi (n)Tn S(ωx)

n=−L

This has the form of a global Hilbert space interpretation of some process PI . Unlike the classical situation in which a photon simply exists, in the process algebra model a photon is generated. The nature and conditions related to this generation will need to be specified. One could imagine two processes acting in succession - an initialization process PI followed by a propagation process PP . The propagation process simply transfers the local process strength from an informon embedding to (z, t) to the nascent informon embedding to (z + b t + tP ) where k b is a unit vector in the k direction for function 1 and to a lP k, b t + tP ) for function 2. nascent informon embedding to (z − lP k, One possible method for implementing the initialization process might be to use the Green’s function for the inhomogeneous wave function representing the presence of a current of charges. The four potential is generated by the presence of charges so it is a reasonable place to look for a model of photon generation.

A Process Algebra Approach to Quantum Electrodynamics

37

If we assume that there is no four potential (hence no photon) initially, then the four potential takes the form 1 A(z , t ) = 4π 0

0

Z

3

d z

Z

∞

dt

−∞

1 δ(|z 0 − z| − (t0 − t))J(z, t) |z 0 − z|

One then implements this as a process. This can be done using a semiclassical approximation for the four current J but a more accurate version awaits the extension of the process algebra model to the fermion sector. Unlike the boson sector for which the theory of sinc interpolation over vectors is fairly well developed, the theory of sinc interpolation over spinors is vastly underdeveloped. Note that the four current J(z, t) will be generated by some process PJ . Thus there will be an initialization process PI for the four potential together with the process for the four current, PJ . Strictly speaking, the formula above represents an interaction between the processes PI and PJ and may be used to determine the correct local process strength to associate with an informon of the four potential causal tapestry given a triggering by an informon of the four current tapestry. Thus the formula permits one to determine informons for the interactive product PI  PJ . The initialization and subsequent propagation can therefore be described as PI  PJ → PP → · · · PP Left is specifying the conditions under which the initialization process becomes inactive and the propagation process active. That must be left for future research. A more standard approach to finding general solutions to the wave equation is to first find a set {φn (z)} of eigenfunctions (eigenvalue λn to the spatial part of the equation subject to the boundary conditions) and then write the solution as a sum [25]

38

William Sulis X

√ λn ct

an ei

√

φn (z) + bn e−i

λn ct

φn (z)

n

.

The constants an , bn can be calculated from certain integrals over an initial wave function and its time derivative. Note though that the coefficients may also be viewed as coefficients of a collection of processes. That is, one associates √ ± ±i λct a process Pλn with each function e φn (z) and then the solution takes the form X

− ˆ an P+ λn ⊕bn Pλn

n

The devil, of course, remains in the details. If we take this approach then the processes take a rather simple form, in which each current informon n embedded at (z, t) simply propagates its information to a nascent informon n0 em√ bedded at (z, t + tP ). The local process strength Γn0 = e±i λn ctP Γn . In such a process system, information does not propagate through space, only through time. This arises because this particular expansion depends upon a separation of space and time variables, which means that one has sought out stationary solutions. This kind of model works well in a mathematical universe in which everything simply is, but it does not work in a process based universe in which everything is in the process of becoming (or fading away). In such a model one is still left with the problem of how to generate the original spatial configuration upon which the subsequent propagation depends. So again we end up having to consider a two stage model in which there must be an initial process of generation of the spatial configuration and its subsequent propagation. Another possible approach might be to consider an iterative procedure (analogous to the Born approximation) and generate the wave from the repeated application of a suitable radiating generating process starting from some localized initial seed. If the time scale for initialization is much shorter than that for propagation then this might achieve a similar result as the two stage model. As a final thought, recall that a Green’s function may often be written as a sum X φ∗ (z0 )φn (z) n

n

λn

A Process Algebra Approach to Quantum Electrodynamics

39

so that any solution can be written as Z X φ∗ (z0 )φn (z) 0 0 n Ψ(z, t) Ψ(z , t ) = d3 z λn n Each term in this series can be interpreted as a process Pn , so if there is an initial process which generates Ψ(z, t), then these processes may act subsequently to generate the future evolution. That is, if there exists a process PI which generates a causal tapestry I whose global Hilbert space interpretation P corresponds to Ψ(z, t), then one may allow n Pn to act successively starting from I so as to generate the wave function at future times. Thus the future evolution takes the form P

I0 →I I

P

Pn

P

Pn

n n → I1 → I2 · · ·

The individual subprocesses could be thought of as representing “virtual particles”. Boundary conditions create similar problems. They may be stated in advance, but how does an initially localized process become aware of these boundaries until it has propagated to and interacted with them? The wave equation essentially describes the situation after it has occurred, when all of the various conditions that extended over space and time can be taken into consideration. This runs counter to the spirit of the process algebra approach in which everything is generated, including space and time. One possible approach to the generation of a spatially extended entity is to consider a free sum (meaning all part of a single state) of identical copies of a single process, that is ˆ ⊕P ˆ ⊕P ˆ ⊕P ˆ ··· P⊕P The successive processes force the global process to extend further into the spatial dimension. The use of quaternion modifiers might permit the individual subprocesses to be identified indirectly with particular spatial locations even though they themselves do not exist and cannot be associated with specific spatio-temporal locations. Nevertheless the use of a quaternion multiplier could impart a relative spatio-temporal displacement between individual subprocesses

40

William Sulis

and their informons, particularly in the case of subprocesses that propagate information only forward in time without any propagation in space. These ideas are merely conjectures at present but appear of sufficient promise to warrant future study.

Conclusion The process algebra paradigm posits the existence of a finite, discrete and transient reality generated by processes. Each generation is described formally as a causal tapestry, whose informons can be interpreted as elements in an emergent space-time and as contributing to an emergent wave function. This approach has shown promise to provide a non-relativistic quantum mechanics of scalar particles which appears to be devoid of paradox and divergences. It provides a realist interpretation of the wave function, eliminates wave function collapse, simplifies measurement theory and reproduces NRQM to a high degree of accuracy. In this paper the theory has been extended to provide a semiclassical model of quantum electrodynamics. Just as NRQM appears as an effective theory in a continuum limit, this semi-classical quantum electrodynamics also appears as an effective theory. It is consistent with the notion of a collective electrodynamics. The present model provides a phenomenological representation but more work needs to be done to develop an effective computational theory. In particular problems remain involving the idea of photon generation, the generation of the four potential in the presence of charges, both moving and stationary, the problem of how to incorporate spatial anisotropy, the form of the process covering map and configuration space covering map and their connection to more standard quantum field theoretic formulations of quantum electrodynamics. Work also needs to be carried out on an extension of the process algebra model to massive bosons and ultimately to fermions. Calculations of specific situations using the process algebra model need to be carried out to determine which specific game strategies produce results consistent with experiment and to provide situations in which predictions could be tested against those of standard theory. Although much work needs to be done, the fact that the process algebra model appears to place quantum mechanics back on a foundation of a paradox free, mostly local, partly contextual, observer free reality is a significant advance over standard quantum mechanics. It demonstrates the usefulness

A Process Algebra Approach to Quantum Electrodynamics

41

of taking an emergent, complex systems approach to the study of the fundamental levels of reality, and how physics can benefit by being carried out “from the top up”.

References [1] I. Trofimova, Phenomena of Functional Differentiation (FD) and Fractal Functionality (FF). Int. J. Design & Nature and Ecodynamics 11(4) 508 (2016). [2] R. Devaney, An Introduction to Chaotic Dynamical Systems. AddisonWesley (1989). [3] J. Cohen and I. Stewart, The Collapse of Chaos. Viking press (1994). [4] R. Laughlin, A Different Universe: Reinventing Physics from the Bottom Down. Basic Books (2005). [5] R. Rosen, Quantum Implications: Essays in Honor of David Bohm. B.J. Hiley & F.D. Peat (eds) Routledge, London (1987). [6] W. Sulis, Archetypal Dynamics, Emergent Situations, and the Reality Game. Nonlinear Dynamics, Psychology and Life Sciences. 14(3), 209 (2010) [7] I. Trofimova, Sociability, Diversity and Compatibility in Developing Systems: EVS Approach. In J. Nation, I. Trofimova, J. Rand and W. Sulis, Formal Descriptions of Developing Systems. Kluwer (2001). [8] W. Sulis, Completing Quantum Mechanics. In K. Sienicki, Quantum Mechanics Interpretations, Open Academic Press, 350 (2016). [9] W. Sulis, Ph.D. Thesis. A Process Model of Non-relativistic Quantum Mechanics, University of Waterloo (2014) [10] W. Sulis, A Process Algebra Model of Quantum Mechanics Journal of Modern Physics. 5(6) 1789 (2014). [11] W. Sulis, Quantum Mechanics Without Observers. arXiv: 1302-4156.

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[12] A. Khrennikov, Ubiquitous Quantum Structure. Springer-Verlag, Berlin (2010). [13] D. Mermin, What’s Wrong With This Habit? Physics Today, Reference Frame, May (2009). [14] W. Sulis, Naturally Occurring Computational Systems. World Futures 39(4) 225 (1993). [15] A.N. Whitehead, Process and Reality. The Free Press (1978). [16] W. Sulis, Causal Tapestries for Psychology and Physics. Nonlinear Dynamics, Psychology and Life Sciences, 16(2), 113 (2012). [17] H.J. Borchers and R.N. Sen, Mathematical Implications of Einstein-Weyl Causality. Springer-Verlag, New York (2006). [18] J.H. Conway, On Numbers and Games. A.K. Peters, Natick (2001). [19] R. Hirsch and I. Hodkinson, Relation Algebras by Games. North-Holland, Amsterdam, (2002). [20] J.P. Aubin and H. Frankowska, Set Valued Analysis. Birkhauser, Boston (2009). [21] F. Stenger. Handbook of Sinc Numerical Methods. CRC Press (2011) [22] R. Marks. Introduction to Shannon Sampling and Interpolation Theory. Springer-Verlag (1991) [23] A. Zayed. Advances in Shannon’s Sampling Theory. CRC Press (1993) [24] C. Mead, Collective Electrodynamics: Quantum Foundations of Electromagnetism. MIT Press, Boston (2002). [25] F. Byron and R. Fuller, Mathematics of Classical and Quantum Physics, Vol. II. Addison-Wesley (1970)

In: Complex Systems Editor: Rebecca Martinez

ISBN: 978-1-53610-860-6 © 2017 Nova Science Publishers, Inc.

Chapter 2

REALIZING SUCCESS FOR COMPLEX CONVERGING SYSTEMS Geerten Van de Kaa Faculty of Technology, Policy, and Management, Delft University of Technology, Delft, the Netherlands

ABSTRACT In this chapter a specific aspect of complex systems, the fact that they may consist of established subsystems or components that originate from converging industries, will be described. To realize such complex systems, common standards with which the components of such systems can be interconnected are essential. A specific type of standard will be described which can be used to realize such complex systems; subsystem standards. It will be determined which factors affect the success of subsystem standards by studying a specific example of a subsystem standard; USB. This specific case illustrates the importance of flexibility; standards should guarantee a certain amount of flexibility so that it is possible to adapt them to changing requirements that inevitably emerge when components that originate from multiple converging industries are connected. Second, the case illustrates the importance of network diversity in that subsystem standards should be supported by a diverse



Corresponding Author Address: Faculty of Technology, Policy, and Management, Delft University of Technology, Jaffalaan, 5, 2628BX, Delft, the Netherlands. Tel: 31.15. 2786789, Tel: 31.15.2786789, Email: [email protected].

44

Geerten Van de Kaa network in terms of stakeholder representation. The paper concludes with recommendations for future research directions.

INTRODUCTION Systems such as home networks or the smart energy grid are complex in that they consist of components that originate from multiple industries that are often converging with each other (G. Van de Kaa, 2009; G. Van de Kaa, Den Hartog, & De Vries, 2009). For example, the smart energy grid is the result of the convergence between the energy industry and the information technology industry. Such systems are currently emerging but to realize them standards are needed that guarantee a sufficient amount of compatibility so that the components can actually communicate with each other. In this specific situation, two types of standards exist; (1) standards that are developed within a component and that define communication within that component, (2) standards that are newly developed to enable communication between components that make up the complex system (G. Van de Kaa et al., 2009). In this short communication the focus lies on the first type of standard; the ‘subsystem standard’ (G. Van de Kaa et al., 2009). Often, various options of subsystem standards are available in the market and these type of standards may also be used to connect industry specific components to components from other industries (effectively realizing the complex system). We will focus on the factors that affect the chances that subsystem standards achieve success. Throughout this manuscript we will use the example of USB. USB is a typical example of a subsystem standard that originally was used within one industry (information technology) to enable interconnection between the personal computer and peripheral equipment. Eventually it became a successful standard to realize communication between an information technology industry specific component (the personal computer) and components from other industries including telecommunications (smart phones) and consumer electronics (television). First, the literature that has focused on factors for standard success in general will be reviewed, and, subsequently, it will be argued which factors are especially important for subsystem standards applying empirical material that is available in the literature.

Realizing Success for Complex Converging Systems

45

THEORETICAL BACKGROUND Before attempting to give an answer to the question which factors affect the success of subsystem standards it is first important to realize that various scholars from a diverse range of backgrounds (including e.g., evolutionary economics, industrial economics, and technology management) have attempted to explain the outcome of standards battles (Suarez, 2004). These scholars stress the importance of accumulating a sufficient installed base of users in order to achieve success (Shapiro & Varian, 1999). Indeed, installed base may very well be one of the most important factors affecting the success of compatibility standards (Shapiro & Varian, 1999). The underlying rationale is that markets in which standards battles are fought are characterized by increasing returns and due to the existence of network effects in such markets the products in which the standards are implemented increase in value the more they are used (Farrell & Saloner, 1985; Katz & Shapiro, 1985). Scholars in the area of technology management and standardization have come up with various factors that affect installed base such as strategic manoeuvring, availability of complementary goods, complementary assets, and firm resources (Gallagher, 2012; Gallagher & Park, 2002). These factors are brought together in theoretical frameworks which can be used to explain the outcome of standards battles (Schilling, 1998; Shapiro & Varian, 1999; Suarez, 2004; G. Van de Kaa, De Vries, Van Heck, & Van den Ende, 2007; G. Van de Kaa, Van den Ende, De Vries, & Van Heck, 2011). The factors are, in part, inductively arrived at through case studies of standards battles such as Blu-ray vs HD DVD, WiFi vs HomeRF, USB vs Firewire, Chipper vs Chipknip (Den Hartigh, Ortt, Van de Kaa, & Stolwijk, 2016; Gallagher, 2012; Gallagher & Park, 2002; G. Van de Kaa & De Bruijn, 2015; G. Van de Kaa & Greeven, in press; G. Van de Kaa, Greeven, & van Puijenbroek, 2013; G. Van de Kaa, Van den Ende, & De Vries, 2015). Additionally, frameworks of factors for standard dominance have been applied to various cases to assess whether the factors included in the frameworks are relevant and whether the frameworks are complete (G Van de Kaa & De Vries, 2015). Finally, weights for the factors have been assessed by applying the factors to various cases of standards battles (G. Van de Kaa, De Vries, & Rezaei, 2014; G. Van de Kaa, Rezaei, Kamp, & De Winter, 2014; G. Van de Kaa, Van Heck, De Vries, Van den Ende, & Rezaei, 2014).

46

Geerten Van de Kaa

FACTORS FOR STANDARD SUCCESS FOR SUBSYSTEM STANDARDS First, subsystem standards should enable a certain degree of flexibility in that they can be changed to changing compatibility requirements that inevitable emerge when components that originate from various converging industries get interconnected. In that respect it seems logical to conclude that flexibility in itself in an important factor that affects the success of subsystem standards. Indeed, USB was quite flexible and could easily be changed to changing requirements which in part affected its success over Firewire which was less flexible (G Van de Kaa & De Vries, 2015; Van den Ende, Van de Kaa, Den Uyl, & De Vries, 2012). Second, subsystem standards should be supported by a diverse network of actors that originate from the various industries that are converging. This will increase the installed base since the standard can make use of the potential installed base of the industries involved. Indeed, many studies that have focused on USB point to the importance of network diversity (G Van de Kaa & De Vries, 2015; Van den Ende et al., 2012).

CONCLUSION AND DISCUSSION This manuscript has explored factors for subsystem standard success for complex systems. It has reviewed the extant literature on factors for standard success and by using insights from the case of USB, a typical example of a subsystem standard, it has determined two factors that seem to be especially important. This is a first indication of the relevance of these two factors and it is based on literature that has paid attention to only one case. We recommend future researchers to study more cases of standards battles where subsystem standards are competing with other subsystem standards, or with system standards, and to determine the key factors that affect success of the subsystem standard. Then, it may be assessed whether the factors mentioned in this manuscript are sufficient or necessary conditions for success. This may be a fruitful area for future research.

Realizing Success for Complex Converging Systems

47

REFERENCES Den Hartigh, E., Ortt, J. R., Van de Kaa, G., & Stolwijk, C. C. M. (2016). Platform control during battles for market dominance: The case of Apple versus IBM in the early personal computer industry. Technovation, 48-49, 4-12. Farrell, J., & Saloner, G. (1985). Standardization, compatibility, and innovation. The Rand Journal of Economics, 16(1), 70-83. Gallagher, S. R. (2012). The battle of the blue laser DVDs: The significance of corporate strategy in standards battles. Technovation, 32(2), 90-98. Gallagher, S. R., & Park, S. H. (2002). Innovation and competition in standard-based industries: a historical analysis of the U. S. home video game market. IEEE Transactions on Engineering Management, 49(1), 6782. Katz, M. L., & Shapiro, C. (1985). Network externalities, competition, and compatibility. American Economic Review, 75(3), 424-440. Retrieved from http://links.jstor.org/sici?sici=0002-8282%28198506%2975%3A3% 3C424% 3ANECAC%3E2.0.CO%3B2-M. Schilling, M. A. (1998). Technological lockout: An integrative model of the economic and strategic factors driving technology success and failure. Academy of Management Review, 23(2), 267-284. Retrieved from http:// links.jstor.org/sici?sici=0363-7425%28199804%2923%3A2%3C267%3 ATLAIMO%3E2.0.CO%3B2-F. Shapiro, C., & Varian, H. R. (1999). Information rules, a strategic guide to the network economy. Boston, Massachusetts: Harvard Business School Press. Suarez, F. F. (2004). Battles for technological dominance: An integrative framework. Research Policy, 33(2), 271-286. Retrieved from http://www. sciencedirect.com/science/article/B6V77-49SFH5C-1/2/6ac467f16758fde 3d35b8edf195c 27b. Van de Kaa, G. (2009). Standards Battles for Complex Systems, Empirical Research on the Home Network. Rotterdam: Erasmus Research Institute of Management. Van de Kaa, G., & De Bruijn, J. A. (2015). Platforms and incentives for consensus building on complex ICT systems: the development of WiFi. Telecommunication Policy, 39, 580-589. Van de Kaa, G., & De Vries, H. (2015). Factors for winning format battles: a comparative case study. Technological Forecasting & Social Change, 91(2), 222-235.

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Van de Kaa, G., De Vries, H. J., & Rezaei, J. (2014). Platform Selection for Complex Systems: Building Automation Systems. Journal of Systems Science and Systems Engineering, 23(4), 415-438. Van de Kaa, G., De Vries, H. J., Van Heck, H. W. G. M., & Van den Ende, J. C. M. (2007). The emergence of standards - A meta-analysis. Paper presented at the 40th Hawaii International Conference on System Sciences, Hawaii, US. Van de Kaa, G., Den Hartog, F., & De Vries, H. J. (2009). Mapping standards for home networking. Computer Standards & Interfaces, 31(6), 11751181. Van de Kaa, G., & Greeven, M. (in press). Mobile telecommunication standardization in Japan, China, the Unites States, and Europe: a comparison of regulatory and industrial regimes. Telecommunications Systems. Van de Kaa, G., Greeven, M., & van Puijenbroek, G. (2013). Standards battles in China: opening up the black-box of the Chinese government. Technology Analysis & Strategic Management, 25(5), 567-581. Van de Kaa, G., Rezaei, J., Kamp, L., & De Winter, A. (2014). Photovoltaic Technology Selection: A Fuzzy MCDM Approach. Renewable and Sustainable Energy Reviews, 32, 662-670. Van de Kaa, G., Van den Ende, J., & De Vries, H. J. (2015). Strategies in network industries: the importance of inter-organisational networks, complementary goods, and commitment. Technology Analysis & Strategic Management, 27(1), 73-86. Van de Kaa, G., Van den Ende, J., De Vries, H. J., & Van Heck, E. (2011). Factors for winning interface format battles: A review and synthesis of the literature. Technological Forecasting & Social Change, 78(8), 1397-1411. Van de Kaa, G., Van Heck, H. W. G. M., De Vries, H. J., Van den Ende, J. C. M., & Rezaei, J. (2014). Supporting Decision-Making in Technology Standards Battles Based on a Fuzzy Analytic Hierarchy Process. IEEE Transactions on Engineering Management, 61(2), 336-348. Van den Ende, J., Van de Kaa, G., Den Uyl, S., & De Vries, H. (2012). The paradox of standard flexibility: the effects of co-evolution between standard and interorganizational network. Organization studies, 33(5-6), 705-736.

In: Complex Systems Editor: Rebecca Martinez

ISBN: 978-1-53610-860-6 © 2017 Nova Science Publishers, Inc.

Chapter 3

DEVELOPMENT OF THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION OF ROTATING COSMOGONICAL BODY FORMATION Alexander M. Krot* Laboratory of Self-Organization System Modeling, United Institute of Informatics Problems of National Academy of Sciences of Belarus, Minsk, Belarus

ABSTRACT This chapter considers the statistical theory of gravitating spheroidal bodies to derive and develop a new generalized nonlinear Schrödinger equation of a gravitating cosmogonical body formation. Previously, the statistical theory for a cosmogonical body forming (so-called spheroidal body with fuzzy boundaries) has been proposed. As shown, interactions of oscillating particles inside a spheroidal body lead to a gravitational condensation increasing with the time. In this connection, the notions of an antidiffusion mass flow density as well as an antidiffusion particle velocity in a rotating spheroidal body have been introduced. The generalized nonlinear time-dependent Schrödinger equation describing a gravitational formation of a rotating cosmogonical body is derived. This chapter considers different dynamical states of a gravitating spheroidal *

Corresponding Author address. Email: [email protected].

50

Alexander M. Krot body and respective forms of the generalized nonlinear time-dependent Schrödinger equation including the virial mechanical equilibrium, the quasi-equilibrium and the gravitational instability cases. Besides, the last case involves the avalanche gravitational compression increasing (when the parameter of gravitational condensation grows exponentially with the time) among them the case of unlimited gravitational compression leading to a collapse of a spheroidal body. Within framework of oscillating interactions of particles, a frequency interpretation of the gravitational potential and the gravitational strength of a forming spheroidal body is considered in detail. In particular, we explain how Alfvén’s oscillating force modifies the forms of planetary orbits within the framework of the statistical theory of gravitating spheroidal bodies. We find that temporal deviation of the gravitational compression function of a rotating cosmogonical body induces the Alfvén additional periodic force. An oscillating behavior of the derivative of the gravitational compression function implies the special case when the additional periodic force becomes counterbalance to the gravitational force thus realizing the principle of an anchoring mechanism in exoplanetary systems.

Keywords: molecular clouds, slow-flowing gravitational condensation, antidiffusion velocity, generalized nonlinear Schrödinger equation, orbital oscillations

1. INTRODUCTION A statistical theory of a gravitating cosmogonical body has been proposed in our previous works [1–6]. Within framework of this theory, the forming cosmogonical bodies are shown to have fuzzy contours and are represented by spheroidal forms (unlike ordinary macroscopic bodies having distinct contours). In such spheroidal bodies, under the condition of critical values of mass density (or parameter of gravitational condensation  [1, 2]) the centrally symmetric gravitational field arises. In this connection, the main problem in the statistical model understanding is what mechanism of a particle interaction leads to slow-flowing process of an initial gravitational condensation of a spheroidal body. For the first time the problem of gravitational condensation was investigated by J. Jeans [7, 8]. Let us note that the gravitational condensation problem of an infinitely distributed substance is directly connected with the gravitational instability problem, see for example Ref. [7–11]. Really, the

Development of the Generalized Nonlinear Schrödinger Equation …

51

linearized theory of gravitational instability leads to the well-known Jeans criterion:

ω 2  4 ,

(1)

where  is Newtonian gravitational constant, ω is a circular frequency of oscillating disturbances,  is a mass density of an infinitely distributed substance. Nevertheless, some works [12, 13] point to that an infinite homogeneous non-rotating substance cannot be in an equilibrium state, therefore, small disturbances do not manage to form any dense bunches. However, process of planet forming is very long with the time, in this connection Newtonian consideration of locally equilibrium system becomes preferable, as pointed by Safronov [10]. The main difficulty of Jeans’ theory is connected with a gravitational paradox [10]: for an infinite homogeneous substance there exists no potential of gravitational field

 g  2

 2 g x 2



 g in accord with Poisson equation  2 g y 2



 2 g z 2

 4 .

(2)

If a mass density value   0 then according to Eq. (2) both gravitational potential

 g and, therefore, specific gravitational force f g grow unlimited

with a distance [10]. As A. Einstein also noted in connection with the cosmological constant problem [14, 15], a constant limit existing for

 g in

spatial infinity leads to the representation that a mass density of substance tends to zero in infinity. This difficulty is reduced within framework of Jeans’ theory and its next modifications by means of a supposition that Poisson equation cannot be applied to an infinite substance in whole but to disturbances  from a mean value  only. It is also supposed there is no gravitational force in an infinite homogeneous immovable substance because a gradient of acceleration and pressure is absent. Otherwise, it could not be in rest [10]. It was shown by Jeans [7, 8], an important law of statistical mechanics can be obtained from equation for evolution of a distribution function of a gas-dust substance.

 wdt , u 

52

Alexander M. Krot

Evidently, a cosmogonical system of gas and dust particles (forming a prestellar or gas-dust protoplanetary cloud) has relaxation time longer than a laboratory system of gas molecules. In this connection, if influence of inner forces on this gas-dust cloud can be ignored then this cloud will not change form and sizes with time noticeably. Such state is a mechanical equilibrium state [16]. Fluctuation interactions of subsystems of this system can violate this equilibrium permanently though rarely. In other words, gas-dust cloud can accept a mechanical equilibrium state in a time moment, i.e., evolution of a gas-dust cloud consists in a replacement of equilibrium states [1]. Choosing a frame of reference into a gas-dust cloud let us introduce a distribution function   of coordinates ( x, y, z )  r and velocities (u, , w)  v as follows:

( x, y, z, u, , w, t )dxdydzduddwdt . Now let us consider an ensemble of particles whose number is defined by a distribution function  multiplied by N ( N is a general number of particles into a molecular cloud). These particles have coordinates x, y, z and velocity components u, , w in a time moment t . After a time interval dt these particles have new coordinates x  udt , y  dt , z  wdt and new

u

 g

dt , 

 g

dt , w 

 g

dt

x y z velocity components . Since investigate the same ensemble of particles, the following equality is true: N   ( x  udt , y  dt , z  wdt , u 

 g x

dt ,  

 g y

dt , w 

 g z

 g x

dt ,  

 g y

dt , w 

 g z

we

dt , t  dt )dxdydzduddwdt 

dt , t  dt )dxdydzduddwdt 

 N  ( x, y, z, u, , w, t )dxdydzduddwdt .

(3)

Now if we represent function  by series in Eq. (3) then we obtain the Jeans’ equation [8]:

u

     g   g   g   w        0, x y z u x  y w z t (4)

Development of the Generalized Nonlinear Schrödinger Equation …

53

which can be considered as a continuity equation (if  g  0 ) or a continuity equation with variable mass (if  g  0 ). However, because of the mentioned gravitational paradox the main problem of self-condensation of an infinitely distributed substance was not solved by Jeans’ theory. L. Nottale [17–22] has developed his theory of the scale relativity to describe both deterministic and stochastic behavior of a particle in gravitational field of a cosmogonical body. In Nottale’s model both direct and reverse Wiener stochastic processes are considered in parallel, that leads to the introduction of a twin Wiener (backward and forward) process as a single complex process [17, 21]. For the first time, the backward and the forward derivatives for the Wiener process were introduced within framework of the statistical mechanics of Nelson [23, 24]. Both Nelson’s statistical mechanics and Nottale’s scale relativistic theory investigate families of virtual particle trajectories which being continuous but nondifferentiable. The important point in Nelson’s works [23, 24] is that a diffusion process can be described in terms of a Schrödinger-type equation, with help of the hypothesis that any particle in the empty space, under the influence of any interaction field, is also subject to a universal Brownian motion (i.e., from the mathematical view-point, a Markov–Wiener process) based on the hypothesis on quantum nature of space-time or on quantum fluctuations on cosmic scale [24–28]. However, in spite of important role of Nelson’s and Nottale’s theories the general equation of gravitational condensation has not been obtained. In this work, we derive the generalized nonlinear Schrödinger equation of a rotating cosmogonical body formation within framework of the statistical theory [1–6]. This work also explains an initial slowly evolving process of gravitational condensation of a rotating cosmogonical body from an infinitely distributed substance hence solved the gravitational paradox problem.

2. THE DISTRIBUTION OF MASS DENSITY AND POTENTIAL AS A RESULT OF AN INITIAL GRAVITATIONAL CONDENSATION OF A MOLECULAR CLOUD Problems of gravitational condensation of a molecular cloud within framework of Jeans’ theory are the following [16]:

54

Alexander M. Krot   

a noncontradictory model of gravitational condensation of an infinitely distributed substance has not been developed up to now; a gravitational potential for an infinitely distributed homogeneous immovable substance is not defined analytically; difficulties occur in finding a general (not particular) solution of Jeans’ equation (4) in view of impossibility to determine an analytical expression for gravitational potential of a molecular cloud.

To solve the mentioned problems of gravitational condensation of a molecular cloud we will use the statistical theory [1–6]. We consider the statistical theory beginning from the derivation of a distribution function of particles in a space filled in homogeneous and isotropic gaseous nebula. In other words, the question is about the distribution of particles in a space. The statistical aspect of the problem results from the fact that the considered body is a system containing a large number N of particles interacting among themselves by oscillations. In microphysics, the cosmic vacuum represents a ground energetic state of quantum oscillations, and its experimental manifestation is Casimir effect [29 p. 1154]. The similar oscillations modifying forms of particle trajectories have been considered by Nelson [23, 24] and later on Nottale [17, 21], so that we can say about the initial oscillatory interactions of particles. On the other hand, in macrophysics it is alleged that the cosmological constant describes the cosmic vacuum [15, 29], i.e., its experimental manifestation on cosmic scales are the fluctuations stipulated by Alfvén–Arrhenius oscillations [30, 31]. Really, it is known [30, 31] that due to the radial and the axial oscillations the moving solid bodies in gravitational field of a central body have elliptic and inclined orbits. So, according to our previous work [1–4] we consider the spatial distribution function of particles Φ satisfying the equation in cylindrical coordinates:

dN h, , z N

 Φh,  , z hdhddz ,

(5)

where dN h , , z is a number of particles whose coordinates belong to the elementary intervals [h, h  dh] , [ ,   d ] and [ z, z  dz ] close to h ,



and z respectively at the given moment of time t  t 0 , N is the total number of particles of gaseous body. As shown in [1, 3, 4], the derived volume

Development of the Generalized Nonlinear Schrödinger Equation …

55

probability density function describing a particle distribution into a rotating gaseous body (being in a state of relative mechanical equilibrium) can be expressed in cylindrical coordinates:

(h, z)  ( / 2 )3 / 2 (1   02 )e  ( h

2

(1 02 )  z 2 ) / 2

,

(6a)

as well as in Cartesian coordinates:

( x, y, z)  ( / 2 )3 / 2 (1   02 )e ( x

2

(1 02 )  y 2 (1 02 )  z 2 ) / 2

(6b)

and in spherical coordinates:

(r,  )  ( / 2 )3 / 2 (1   02 )er

2

(1 02 sin2  ) / 2

,

(6c)

 are polar and azimuth angles,  is a parameter of gravitational condensation,  0 is a constant of stabilization of the variable  . where  and

Obviously, when  0  0 then the equation (6c) goes to the volume probability 2

density equilibrium function for the non-rotational case (or slowly rotational one) [1, 4]:

 (r )  ( / 2 ) 3 / 2 e r

2

/2

.

(6d)

Now let us note that the relationship (5) describes the distribution of particles relative to a distance from the geometrical center and to a direction in a space. Taking into account that an elementary volume in cylindrical coordinates is dV  hdhd dz we can transform Eq. (5) into

dN h, , z dV

 Nh,  , z   N ( / 2 ) 3 / 2 (1   02 )e  ( h dN h, , z

2

(1 02 )  z 2 ) / 2

. (7)

 nh, , z is a local concentration [32] of particles near a dV point with coordinates h,  , z  . Considering Eq. (7) we have:

The value

m0 N ( / 2 )3 / 2 (1   02 )e  ( h

56

Alexander M. Krot

nh, z   N ( / 2 )3 / 2 (1   02 )e  ( h

2

(1 02 )  z 2 ) / 2

.

(8)

If all particles are like and have mass m0 , then, by multiplying both sides of relation (8) by m0 , one obtains [1, 3, 4]:

 h, z   m0 N ( / 2 )3 / 2 (1   02 )e  ( h 2

(1 02 )  z 2 ) / 2

 M ( / 2 )3 / 2 (1   02 )e  ( h

2

2

(1 02 )  z 2 ) / 2

(1 02 )  z 2 ) / 2

 M ( / 2 )3 / 2 (1   02 )e  ( h

 M(h, z) ,

2

(9)

where   m0 n is a mass density, M  m0 N is a mass of the gaseous body. By denoting 0  M ( / 2 )

3/ 2

the mass density for a rotating and

gravitating gaseous body can be written in cylindrical, Cartesian, and spherical coordinate system respectively [1, 3, 4, 33]:

 (h, z)  0 (1   02 )e ( h

2

(1 02 )  z 2 ) / 2

 ( x, y, z)  0 (1   02 )e ( x  (r, )  0 (1   02 )er

2

2

,

(10a)

(1 02 )  y 2 (1 02 )  z 2 ) / 2

(1 02 sin 2  ) / 2

.

,

(10b)

(10c)

Obviously, the iso-surfaces (isostera) of mass density (10a-c) are flattened ellipsoidal ones, and

 02 is a parameter of their flatness (  0 is the eccentricity

of ellipse). As a rule  0  1 , so that these mass density iso-surfaces become spheroidal surfaces. Thus, under the influence of the initial oscillations of particles an isolated gaseous cloud can be transformed to the spheroid-like gaseous body or, simply say, spheroidal body [1–6]. Let us consider the important particular case of spheroidal body which is sphere-like gaseous body. Really, we can see that if

 02  0 then the equation

(10c) becomes the mass density function for slowly rotating or immovable spheroidal body [1–6, 33]:

(1 02 )  z 2 ) / 2

 M(h, z)

Development of the Generalized Nonlinear Schrödinger Equation …

 (r )   0 e r

2

/2

where 0  M ( / 2 )

, 3/ 2

57 (11)

, M is a mass of spheroidal body. As appears from

Eq. (11), under the influence of gravitational interactions of particles stipulated by the initial oscillations, there arises a substance mass density inhomogeneous along the radial coordinate r . Because of a mass density value strictly depends on  in Eq. (11) this positive parameter defines a measure of gravitational interactions of particles in a cloud, therefore it is called the parameter of gravitational condensation [1, 4, 33]. In such a way, under the action of becoming gravitational forces, a great number of particles form a sort of spheroidal body whose density is identical in all directions at the same distance from mass center if a rotation is absent. Obviously, the iso-surface of mass density for this case is a sphere. The greatest mass density is concentrated in the interval [0, r* ] , where r*  1 /  is a point of the mass density bending [1, 4], outside of which it decreases quickly. Let us note because of r* ~ R , where R is a mean radius of gaseous cloud, then

 ~ 1/ R 2 is a very small positive parameter of gravitational condensation. A mean radius R of a forming gaseous cloud can also be choosing the most

2 /   2r* close to which there is the greatest

probable distance rpr 

number of particles of spheroidal body. Let us note this expression for a mass density (11) can be obtained (see [1, 4]) within framework of Einstein’s General Relativity if we take into account that d /   dV / V under condition of M  const where dV  e

/2

dV0 is a space volume element in

the central-symmetric metric, dV0  r 2 sin drdd and  is some function of distance r and time t [34]. On the contrary, if the squared eccentricity  0  1 then Eq. (10a) can 2

describe a mass density of a flattened gaseous disk:

 (h, z)  c (h)ez

2

/2

,

(12)

where  c (h)  ( / 2 ) 3 / 2 lim M (1   02 ) is a value of mass density in a 2  0 1

M 

central flat of this gaseous disk, M is a total mass of an initial prestellar

58

Alexander M. Krot

molecular cloud, i.e., mass of star plus mass of gaseous protoplanetary disk. It is interesting to note that this equation (12) coincides completely the known barometric formula (for a flat rotating disk) obtained with the usage of hydrostatic mechanical equilibrium condition [10 p. 36, 35 p. 769] (or the same formula of mass density distribution in the disk “standard” model derived on the basis of the hydrostatic equilibrium condition jointly with the ideal gas state equation [36 p. 19]). Besides, Gurevich and Lebedinsky used the designation

 max (h) instead of c (h) . The function of mass density (10a-

c) characterizes a flatness process: from initial spherical forms (for a nonrotational spheroidal body case) through flattened ellipsoidal forms (for a rotating and gravitating spheroidal body) to fuzzy contour disks when the squared eccentricity  0 varies from 0 till 1. This means that the derived 2

function (10a-c) is appropriate to describe evolution of a protoplanetary gaseous (gas-dust) disk around a star. Thus, initial gravitational interactions of particles stipulated by their quantum oscillations in an isolated gaseous cloud form a spheroidal body and lead to gravitational field becoming. Namely, there is a threshold (critical) value  c that if    c then gravitational field arises in a spheroidal body [1, 4, 33]. In the simplest case (11) we can seek a spherically symmetric solution

 g depending on r only, therefore Poisson equation (2) becomes:

1   g r   2 r 2

Since

  d  2 d g r    r2   4 0 e 2 .   r dr dr   

 g is a function of r alone, one obtains: r

d g r  dr

(13)

 4 0

2 x e



 x2 2

0

dx ,

r2

(14)

whence we can calculate the gravitational potential:

 g r   4 0 

r



 x2 1 2 2 x e dxdr  C , r 2 0

(15)

Development of the Generalized Nonlinear Schrödinger Equation …

59

where C is the integration constant defined from the condition that the potential is equal to zero on the infinity:

 g   0 . With the aim of

simplifying Eq. (15), we can transform the indefinite integral on the basis of the integration formula by parts:      r r r  x2  x2 1 1  1  2 x2 1  2 r2  1  2 r2 1 2 2 x e dx dr   e dx  r e  e   e    r 2 0 0 2 dx r  0    r  . (16)

From the condition

 g   0 and formulas (15), (16) we obtain

[1, 4, 6]:

4 0 2 x 2  g r    e dx . r 0 r

(17a)

x

2 2 Using the error function erf x   e  s ds , we represent Eq. (17a) in   0

the following way:

 g r   

4 0 2 r 



r  /2

e 0

s 2

ds  

M r





erf r  / 2 .

(17b)



Since lim erf r  / 2  1 , then for large r the last expression turns into r 

 g r   

M r

.

(18a)

The relation (18a), as known, describes the gravitational potential of a field produced by one particle (or a spherical body) of mass M .

60

Alexander M. Krot

In the case of small r , the function e



 2

r2

1

 2

r 2 which leads to the

transformation of formula (17a):

 g r   

4 0 r   2  4 0   3  2 0  2 6  1  x dx   r  r   r    r 0  2  r  6  3   . (18b)

In expression (18b), the higher order values of smallness of r were ignored [4, 6]. Thus, expression (18b) describes the gravitational potential in the near zone of the field, while Eq. (18a) describes that in the remote one of immovable gravitating spheroidal body. In the case of a rotating spheroidal body, the axial rotation of the spheroidal body creates a flattening of its core, therefore the gravitational potential in a near zone of uniformly rotating spheroidal body is described by the following expression in cylindrical coordinates h,  , z  :

 g h, z   2 0



1   02

2 1   02



2

 4(1   02 )  2  2 2 2 2  1 0 h  z   . (19a)  1  





The potential of gravitational field in a remote zone of a rotating spheroidal body deviates from the 1 / r –gravitational field potential (18a), so that for large r it estimated by the relation in spherical coordinates r , ,   [3, 4]:

2 M  g r,  |r r*      r 1   0 2 sin 2  2 M    r 1   0 2 sin 2 

r 1 0 2 sin 2  r  2 / 2

e 0

dr |r r*  

M r 1   02 sin 2 



M   02 2  1  sin   r  2 

.

r 1 0 2 sin 2  r  2 / 2

e 0

dr |r r*  

M r 1   02 sin 2 



(19b)

M   02 2  1  sin   r  2 

Development of the Generalized Nonlinear Schrödinger Equation …

61

3. THE DERIVATION OF THE GENERAL ANTIDIFFUSION EQUATION FOR A SLOWLY EVOLVING PROCESS OF INITIAL GRAVITATIONAL CONDENSATION OF A ROTATING SPHEROIDAL BODY In our previous work [1] we obtain the antidiffusion equation of initial gravitational condensation of immovable (or slowly rotating) spheroidal body whose isostera is a sphere. Now let us derive the general differential equation for a process of spheroidal body forming when its isostera is evolving from the sphere to a spheroid. In this connection, using the cylindrical frame of reference (h,  , z ) we consider the mass density function (10a) of a rotating



spheroidal body (with uniform angular velocity Ω ) in a vicinity of relative mechanical equilibrium. Let us calculate the derivatives of  with respect to the spatial coordinates h and z as well as the parameters

 and  0 supposing these are

to be slowly changing functions with the time, i.e.,    (t ) and  0   0 (t ) :

 3    (1   02 )  [ h 2 (1 02 )  z 2 ] / 2    (1   02 )h2  z 2  [ h 2 (1 02 )  z 2 ] / 2  M   e  M    (1   02 )  ( )e   2  2  2 2  2    [3   (h2 (1   02 )  z 2 )]; 2 1/ 2

3/ 2

(20a) 2 2 2 2 2 2   h   0 (2 0 )e  [ h (1 0 ) z ] / 2   0 (1   02 ) { (2 0 )}  e  [ h (1 0 ) z ] / 2   0 2

2



2 0   [1  h 2 (1   02 )]; 2 1 0 2 (20b)

2 2 2 1     1   2h     h    (1   02 ) 2 0 {  (h 2e [ h (1 0 ) z ] / 2 )}   (1   02 )  [2   (1   02 )h 2 ]; h h  h  h h (20c)

62

Alexander M. Krot 2 2 2 2 2 2 2     2  0 (1   02 ) {z  e  [ h (1 0 ) z ] / 2 }  0 (1   02 )e  [ h (1 0 ) z ] / 2 (1  z 2 )    [1  z 2 ]; z z

2 z

  [ h 2 (1 02 )  z 2 ] / 2 2  2  [ h 2 (1 02 )  z 2 ] / 2   ( 1   ) {   z  e }    ( 1   )e (1  z 2 )    [1  z 2 ]; 0 0 0 0 2 z z

2

 2    2h    2z   [3   (h 2 (1   02 )  z 2 )]  2 02[1 

(20d)

 (1   02 )h 2 ]. 2 (20e)

So, taking into account of Eq. (20a) we can see that

[3   (h 2 (1   02 )  z 2 )]  2

 

(21a)

and according to Eq. (20b) we find:

[1 

 2

(1   02 )h 2 ]  

1   02  .  2 0  0

(21b)

With regard for (21a), (21b) Eq. (20e) becomes:

 2   2 2

    0 (1   02 )   0 .

(22)

Now we suppose that an evolution of the mass density of rotating spheroidal body with the time can be expressed by a composite function of

   (t ) and  0   0 (t ) . In the first place, we obtain the main antidiffusion equation in the special case of the fixed parameter

 0 [1, 4]:

  d  1 d  2 2     2        G(t )    , t  dt  2 dt 

(23a)

On the other hand, the following equation is true under consideration of the fixed parameter  :

Development of the Generalized Nonlinear Schrödinger Equation …

d   d 0  1       0 2 t  0 dt   0 (1   0 ) dt

 2      .

63

(23b)

To study a temporal evolution of solution of the equation (22) we need investigate the functional dependence

 0   0 ( ) occurring when    c

 c   (tc ) and t c is a moment of rotation origin. However, it is impossible directly by virtue of the initial statement about independence  on where

the coordinates h,  , z . Consequently, the total derivative of the mass density function  (with respect to the time) can be represented by the following relation:

d  d  d 0 .     dt  dt  0 dt

(24)

To find  /  0 let us use Eq. (22) at the fixed parameter  whence the desired partial derivative is equal to  1  2  . 2  0  0 (1   0 )

(25)

Analogously, if the parameter  0 is fixed then the main antidiffusion

(relative to  ) equation [1, 4] follows directly from Eq. (22):  1   2  2  .  2

(26)

Substitution (25) and (26) in Eq. (24) leads to the following general equation of antidiffusion with regard to a deformation of spheroidal body as a result of its rotation:

d ~   G(t ) 2  , dt

(27a)

64

Alexander M. Krot

~

where G(t ) is an antidiffusion function, i.e., the generalized gravitational compression function (GCF), taking into account a flattening process (aside from antidiffusion condensation) of rotating ellipsoid-like cosmogonical body: ~ G(t ) 

d d 1   0 . 2 2 (t ) dt  0 (1   0 ) dt 1 2



(27b)

~

In the case of finite value of  and 0 the antidiffusion function G(t ) can increase unlimitedly when   0 (at the so-called initial antidiffusion condensation) and when

 0  0 (at the initial flattening). Therefore, the

antidiffusion condensation moment and the flattening moment can be the same, but possibly, they can be inconsistent in generally. Probably, the flattening becomes when the gravitational field arises in spheroidal body, i.e., in the case if  (t ) exceeds its threshold value

 c [1, 4, 33].

When the parameter  0 becomes finite one  0  0 then a sphere-like cosmogonical body begins to deform (to be flattened) that implies a bifurcation on the diagram of dynamical states of this body. As noticed by Jeans [8 p.188, p.190, p.191], “on continually varying some parameter (say  0 , is allowed slowly to vary1) we pass through a whole series of continuous configurations of equilibrium, which form what Poincaré has called a “linear series.” …Every point on linear series is a configuration of equilibrium; a question which is of the utmost importance in cosmogonical problem is whether this equilibrium is stable or unstable. …Thus we see that there is an exchange of stabilities at the point of bifurcation.” In this connection, if we suppose that    (t ) is a variable of onedimensional state-space of spheroidal body then

 0 can be considered as a

control parameter [37]. Really, “the conditions of secular stability assume a somewhat different form for a mass rotating freely in space. In this case the rate of rotation is not constant, but changes as the moment of inertia of the mass changes … Secular stability is lost at a ‘turning point’ or ‘point of bifurcation’” [8 p. 199-201]. Then Jeans clarified [8 p. 207, p. 209] that “if Ω  0 ,…so that the configuration must be spherical. If Ω is small, although 1

The author’s remark.

Development of the Generalized Nonlinear Schrödinger Equation …

65

not actually zero, a spherical surface does not satisfy the condition, the term 1 2 2  ( x  y 2 ) destroying the spherical symmetry. In this case, as we shall 2 see almost immediately, the configuration is that of an oblate spheroid of small ellipticity… Two linear series of equilibrium configuration, which are spheroidal and ellipsoidal respectively. The configuration which form the first series are commonly known as Maclaurin’s spheroid; those which form the second as Jacobi’s ellipsoids….” So, according to Eq. (27a) a variation of form of spheroidal body is caused by the dissipation, i.e., by the gravitational energy changing due to the interior energy of particles of gaseous cloud. According to Eq. (10a) the flattening process cannot decrease the antidiffusion condensation in the axial direction, but it can reduce the antidiffusion condensation in the plane of rotation of spheroidal body. In particular, the obtained relation (26) reminds, in form, namely the antidiffusion equation because an initial density disturbance does not decrease but increases. Denoting through GCF G(t ) 

1 2

2



d dt

(28)

with usage (26)-(27b) we can write a pure antidiffusion equation of initial gravitational condensation of immovable (or slowly rotating) spheroidal body in the form [1–5]:    G(t ) 2  . t

(29)

According to (6d), (11)  r   Mr  , so that analogous equation of initial gravitational condensation of gaseous substance is also true for the distribution function Φ of an immovable (of slowly rotating) spheroidal body [1–5]: Φ   G(t ) 2 Φ . t

(30)

So, if Φ  Φ( x, y, z, t ) be a function of probability density for location of particles in an immovable isolated molecular gaseous cloud in a time moment

66

Alexander M. Krot

t  t0 then, as mentioned above due to initial oscillating interactions of particles, a sphere-like cosmogonical body (11) can be formed in this gaseous cloud. The process of initial gravitational condensation of a spheroidal body in a vicinity of mechanical equilibrium is described by the differential equations (29), (30) under assumption that the parameter   0 is a slowly changing function of time starting from some moment t 0 , i.e.,

   t  at t  t0 is a

positively defined monotonically increasing time function. In this case G(t )  0 according to Eq. (28). As follows from Eq. (30) directly, the immovable spheroidal body can be considered as a model of cosmic infinitely  distributed gas-dust substance in the case when the velocity v  0 . Thus, the derived equation (30) as well as Jeans’ equation (4) describes the different stages of evolution of a gaseous cloud. Consequently, there exists a more common evolutionary equation which generalizes both of them [1, 4]:

Φ  Φ  v  Φ    g  G(t ) 2 Φ  0 . t v

(31)



Obviously, if v  0 then Eq. (31) becomes the initial evolution equation  (30), on the contrary, if v  0 then it can be transformed to the Jeans’



equation (4) because G(t ) 2 Φ  v  Φ

Φ    g as a rule. v

4. THE ANTIDIFFUSION VELOCITY INTO A ROTATING COSMOGONICAL BODY AND THE CHARACTERIZING NUMBER K AS CONTROL PARAMETER OF ITS DYNAMICAL STATES We shall use Eq. (27a) of the slow-flowing gravitational compression of rotating cosmogonical body in order that rewrite it taking into account of the

~

generalized GCF G (t ) does not depend on the spatial variables:





 ~  div G(t ) grad   0 . t

(32)

Development of the Generalized Nonlinear Schrödinger Equation …

67

The relation (32) reminds completely the continuity equation expressing the law of conservation of mass in a nonrelativistic system [38]:

   div j  0 , t

(33)



where j is a continuum flow density. In this connection, the value in round brackets of Eq. (32) has the sense of a mass flow density (like a conductive



flow) j arising at the slow-flowing gravitational compression of spheroidal body [1, 4]:

 ~ j  G(t ) grad  .

(34)

The conductive (owing to diffusion or thermal conductivity) flows in dissipative systems were investigated by I. Prigogine, G. Nicolis, P. Glansdorff (see, for example, [37, 39, 40]). As it follows from Eq. (34) directly, there exists an antidiffusion mass flow density in a slowly compressible gravitating spheroidal body [1, 2]. Let us introduce a conductive velocity for the antidiffusion mass flow density or, simply say, the antidiffusion velocity (unlike of the ordinary  hydrodynamic velocity v ) for a rotating spheroidal body [2, 4]:

 ~  ~ (  /  0 ) ~ u  G(t )  G(t )  G(t ) grad ln(  /  0 ) .   / 0

(35)



Obviously, the antidiffusion velocity u of antidiffusion mass flow density satisfies the well-known continuity equation:    div(  u)  0 . t

(36)

Using this continuity equation (36) we can calculate the partial derivative of the antidiffusion velocity (35) with respect to the time:

68

Alexander M. Krot

~ ~   1   dG(t )  1   ~ 1    u dG(t ) ~  grad ln(  /  0 )  G(t ) grad      ~ u   G(t )  (div (  u))  t dt dt  G(t )    t    ~ ~ d ln G(t )  ~        2 d ln G(t )  ~  u  G(t )  u  u   G(t ) grad( div u)  grad( u )  dt u . dt   

(37)



Taking into account Eq. (35) we can see that rotu  0 , so that the familiar formulas of vector analysis [38, 41] become:

   grad u 2  2u  u ;

(38)

  2 u  grad( div u) .

(39)

Substituting Eqs. (38), (39) in Eq. (37) we obtain:

 ~   d ln G(t )  u ~ 2   G(t ) u  2u   u  u. t dt

(40)

Taking into account the formula (38) again, Eq. (40) can be written as follows:

 ~    d ln G(t )  u   u   u   grad( u 2 / 2)  G(t ) 2 u  u t dt .

(41)

The obtained equation (41) is similar to Navier–Stokes’ equation of motion of a viscous liquid [38, 41] under conditions that a gas-dust substance of spheroidal body is isolated from influence of external fields and

~ ~ G(t )  G s  const .

Now let us estimate the antidiffusion velocity (35) of particles inside a rotating ellipsoid-like cloud [4] taking into account its mass density formula (10a):



  



 

 ~ ~ u  G(t ) grad ln  (h, z) /  0   G(t ) grad ln 1   02   h 2 1   02  z 2 / 2      ~   G(t ) 1   02 h  eh  z  ez  u h  eh  u z  ez , (42a)







Development of the Generalized Nonlinear Schrödinger Equation …



where eh and

69

 ez are the basis vectors of cylindrical frame of reference, u h

and u z are the radial h -projection and the axial z -projection of antidiffusion velocity:





~ u h   G(t ) 1   02 h ;

(42b)

~ u z   G(t )z .

(42c)



Taking into account that rotu  0 for the antidiffusion velocity defined by Eq. (42a) we can confirm that Eqs. (38), (39) are true. We can also see that the  antidiffusion velocity u is expressed by the enough simple relations (42a-c) in the general case of a rotating and gravitating ellipsoid-like cosmogonical body.  Along with the antidiffusion velocity u there exists an ordinary  hydrodynamic velocity v (or a convective velocity in the sense of Prigogine  [38– 41]). In principle, the hydrodynamic velocity v of mass flow arises as a result of powerful gravitational contraction of a spheroidal body on the next (field) stages of its evolution. The growing magnitude of gravitational field



strength a induces the significant (i.e., observable) value of hydrodynamic  velocity v of mass flows moving into spheroidal body. This means that the value of antidiffusion velocity (35) becomes much less than the value of hydrodynamic velocity, i.e.,

  u  v .

(43)

Under this condition (43), a common (hydrodynamic and antidiffusion) mass flow density inside a spheroidal body satisfies the hydrodynamic equation of continuity [38, 43]:

   div (  v)  0 t .

(44)

Taking into account Eq. (44) we can also calculate the partial derivative of the antidiffusion velocity (35) with respect to the time in accord with the condition (43):

70

Alexander M. Krot

~ ~   1   dG (t )  1   ~ 1    u dG (t )  grad ln(  /  0 )  G (t ) grad    ~ u   G (t ) (div (  v))  t dt dt  G (t )    t    ~ ~ d ln G (t )  ~         d ln G (t )  ~  u  G (t ) v v u.   G (t ) grad( div v)  grad( vu)  dt  dt  

(45) As known from a fluid-like description [38, 41], the complete time  derivative of the common (hydrodynamic plus antidiffusion) velocity v u inside a spheroidal body defines the common acceleration (or gravitational field strength of spheroidal body) including the partial time-derivatives and convective derivatives [2, 4, 5]:

     d (v u)  v   u   a   (v ) v  u   u . dt t t

(46a)

Taking into account Eq. (41) as well as Eq. (38), the complete acceleration (46a) can be represented in the form [2, 4, 5]:    ~ d ln G(t )  .  d (v u)  v     ~ 2  a   (v ) v u   u  G(t ) u  u dt t dt

(46b)

Let us note since the mass density of rotating ellipsoid-like spheroidal body is directly proportional to the probability volume density function  according to Eq. (9), then antidiffusion velocity (35) (or (42a)) can be defined by the probability volume density function:

 ~ Φ ~ u  G(t )  G(t ) grad lnΦ Φ .

(47)

Obviously, the antidiffusion velocity (47) of probability volume flow density also satisfies Eqs. (37), (41), (42a), (45)-(46a,b). Using Eqs. (45), (46b) we can carry out an analysis of dynamical states of a rotating spheroidal body by introducing the scales of physical values

    ~ T , L, V , U , F , G s and the respective dimensionless variables  ,  , v , u , f , g

as follows:

Development of the Generalized Nonlinear Schrödinger Equation …

 ~        ~ t  T ; r  L ; v  V v ; u  Uu; a  Ff ; G(t )  G s g (t ) . ~

71 (48)

~

where G(t )  G s  const under the condition of mechanical equilibrium state. By substituting Eqs. (48) in Eqs. (45), (46b) we obtain:

 ~  VU   U d ln G(t )  U u ~ V  G(t ) 2 grad( div v )  grad( v u )  u; T  L T dt L

(49)

 ~  V2   U2 2  U d ln G(t )  V v ~ U  Ff  (v  ) v  grad( u / 2)  G(t ) 2 grad( divu )  u T  L L T dt L .

(50) Similarly to [41], dividing Eq. (50) by V 2 / L and Eq. (49) by VU / L we derive the following dimensionless equations: ~  Gs u   d ln g (t )  ; 1 Sh   grad( div v )  grad( v u )  Sh u   K Re dt

Sh

(51)

~  G K v 1   d ln g (t )      f  (v  )v  K 2 grad( u 2 / 2)  s  g (t ) grad( divu )  Sh  K u  Fr  Re dt ,

(52) where Sh  L / VT is the Strouhal number, Fr  V 2 / FL is the Froude number, Re  VL / is the Reynolds number ( is a kinematic coefficient of viscosity of flow of particles [41]), K  U / V is a new number of similarity.





The new number of similarity is a measure of the values u versus v prevailing:

  K u /v .

(53)

When this similarity number exceeds unity ( K  1 ) then the antidiffusion contraction of a rotating spheroidal body occurs exclusively, i.e.,



the value of hydrodynamic velocity is negligible ( v  0) because a

72

Alexander M. Krot

gravitational field is absent practically. If the similarity number becomes close  to unity ( K  1 ) then the hydrodynamic velocity v of mass flow arises as a result of a gravitational contraction of a spheroidal body on the field stage of its evolution. As mentioned relative to Eq. (43), the value of antidiffusion velocity (35) becomes much less than the value of hydrodynamic velocity

  u  v when K  1 . This means that the growing magnitude of powerful  gravitational field strength a induces the significant value of hydrodynamic  velocity v of mass flows moving into a rotating spheroidal body. Thus, like the Mach number M [41] the new number of similarity K is a control parameter of dynamical states of a forming spheroidal body. In particular, in the special case K  1 the dimensionless Eqs. (51), (52) are reduced to one dimensionless equation of the kind:

 2  Gs 1 u Kgrad( u / 2)   g (t ) grad( divu )  Sh  ,  Re 

(54)

which corresponds the following equation:

 2  u grad( u / 2)  G(t ) grad( div u)  . t

(55)

Except the antidiffusion solution, the equation (55) has a wave solution in



the vicinity of equilibrium state when G s  const and u  1 :   2   u  u 0 e  i kr k G s t .

(56)

In the quasi-equilibrium gravitational condensation state with a

~

~

~

periodically varying GCF G(t )  G s  G(t )t the wave solutions (56) are generated, moreover, they induce specific additional periodic forces [30, 31] and respective spatial oscillations in the different domains of a forming spheroidal body.

Development of the Generalized Nonlinear Schrödinger Equation …

73

5. THE DERIVATION OF THE GENERALIZED NONLINEAR SCHRÖDINGER EQUATION IN THE STATISTICAL THEORY OF ROTATING COSMOGONICAL BODIES As shown in Section 3 of this paper, initially the probability density to observe an oscillating particle satisfies the antidiffusion equation (27a) or (30), i.e., considerations in Section 2 and Section 3 point to an initial quasiequilibrium gravitational condensation occurring in a forming spheroidal body. On the other hand, according to Section 4 a sharp increase of the antidiffusion velocity (35) when  0  0 can lead to the coherent displacement of particles inside a spheroidal body and, as a consequence, to a

resonance increase of the parameter of gravitational condensation  (t ) [2]. This means that nonlinear phenomena arise owing to self-organization processes [40] into a rotating spheroidal body under its formation. These nonlinear phenomena induce nonlinear autowaves [42] satisfying a nonlinear undulatory Schrödinger-like equation [5]. Now let us note the well-known linear Schrödinger equation [43] as well as its generalization within framework of Nottale’s scale relativity [17–22] can

~

~

~

be obtained in the special cases of constant G(t )  G s when G s   / 2m0

~

and G s   M / 2 respectively [5]. Unlike [5] this paper studies the general

~

case of GCF G (t ) . That is why we must return to the derived (in Section 4) equations for calculating the partial derivatives of antidiffusion and hydrodynamic velocity with the aim to obtain a nonlinear generalized Schrödinger equation for a rotating spheroidal body. So, now let us consider again Eqs. (45), (46b). Taking into account the simple formulas (38), (39), (47), these equations can be rewritten in the form:

 ~ u    d ln G(t )  ~  G(t ) grad( div v)  grad( v u)  u; t dt

(57a)

 ~ v    2  d ln G(t )  ~  a  (v ) v grad( u / 2)  G(t ) grad( div u)  u . (57b) t dt

74

Alexander M. Krot

Let us investigate some special solution of Eqs. (57a, b) in the case that the acceleration (or gravitational field strength) comes from a gravitational field potential of a spheroidal body, i.e.,

 a   grad  g ,

(58)



under the assumption that the hydrodynamic velocity v is a gradient of a statistical action  which is a potential of velocity [38, 41]:

 ~ v  2G(t ) grad  .

(59)

~

In the special case of a constant G (t ) as  / 2m0 Eq. (59) becomes the

  grad  . In this connection, rotv  0 , i.e., m0   2  (v ) v  grad( v / 2) . Since u is also the gradient due to Eq. (47) as well   as a and v according to Eqs. (58), (59), so that Eqs. (57a, b) become the 

Nelson’s formula [23]: v 

following: grad

~ ~ (G(t ) ln ) d ln G(t ) ~    ~  G(t ) grad( div v)  grad( v u)  { }G(t ) grad ln  ; (60a) t dt

~ ~ (2G(t ))    d ln G(t ) ~ ~   grad  g  grad( v2 / 2)  grad( u 2 / 2)  G(t ) grad( div u)  { }G(t ) grad ln  t dt ~    d ln G(t ) ~ ~ (60b)  g  grad( v2 / 2)  grad( u 2 / 2)  G(t ) grad( div u)  { }G(t ) grad ln  dt grad

Integrating these Eqs. (60a, b) and taking into account a simplification

~ ~ ~ {d ln G(t )/dt}  G(t )  d G(t )/dt , we can find that

~ ~ (G(t ) ln )    d G(t ) ~  G(t )div v v u  ln  ; t dt

(61a)

Development of the Generalized Nonlinear Schrödinger Equation …

  ~ ~ (2G(t ))  d G(t ) v2 u 2 ~   g    G(t )div u  ln  . t 2 2 dt

75

(61b)

Let us carry out a change of dependent variable:

1 ln  ; 2

(62a)

  e i  ,

(62b)



where  is defined by Eq. (59), i  (62a, b) directly

 1 . Obviously, as it follows from Eqs.

    ei  ,

(63)

      as usually. According to the first change (62a) as 2

so that

well as Eq. (59) it is not difficult to see that

~ ~ (2G(t ) ) d G(t ) ~2 ~2 2  2G (t )   4G (t )    2  ; (64a) t dt ~ ~ (2G(t )) d G(t ) ~ ~ ~   g  2G 2 (t )() 2  2G 2 (t )() 2  2G 2 (t ) 2   2  t dt . (64b) Let us rewrite these two Eqs. (64a, b) as one. To this end, after multiplication of the second Eq.(64b) on imaginary unit and then addition both of Eqs. (64a, b), we find

~ d G(t )  ~ ~2 ~ 2 2 2 2G(t )(  i )   i  g  i 2G (t )(   i  )  i 2G(t )(  i )  2(1  i)  t dt ~ d G(t ) ~ ~ (65)  i )   i  g  i 2G 2 (t )( 2   i  2 )  i 2G(t )(  i ) 2  2(1  i)  dt















76

Alexander M. Krot Taking into account the second change (62b) we can see that

  i   ln ; 2  ln   ln    ln  ; (  i )   ln    / ; 2

 2 (  i )   2  /   () 2 /  2 , so that Eq. (65) takes the form: ~ d G(t )  ~ ~ 2 2 2 2G(t ) ln    i  g  i 2G (t )  (1  i) ln  . t  dt





(66)

After some transformations and simplifications Eq. (66) can be represented as follows: ~ ~ d G(t ) d G(t ) ~  ~ i 2G(t )   g   2G 2 (t ) 2   2 i(1  i)  ln   2 i  ln  , (67) t dt dt

whence we can obtain the nonlinear time-dependent generalized Schrödinger equation describing formation of a rotating and gravitating ellipsoid-like cosmogonical body:

~ d G(t ) ~ Ψ ~ ln Ψ  argΨΨ i 2G(t )   2G 2 (t ) 2   g Ψ 2 t dt .





~

(68)

~

Let us note that GCF is G(t )  G s  const only in the relative mechanical (virial) equilibrium states of rotating spheroidal body [1, 3, 5], so the generalized nonlinear time-dependent Schrödinger equation (68) becomes linear one in these special cases: for example, a macroscopic analog of the time-dependent linear Schrödinger equation [43] is a particular case of Eq.

~

(68) if G (t ) satisfies the Nelson basic assumption [23] as well as the timedependent general Schrödinger equation in the form of Nottale [17–22] is a special case of Eq. (68). Thus, the derived generalized nonlinear time-dependent Schrödinger equation (68) describes both the mentioned state of virial relative mechanical equilibrium (or quasi-equilibrium with a slowly varying GCF) [2, 4, 5] and the gravitational instability state leading to formation of a cosmogonical body. Let us consider different dynamical states of a gravitating spheroidal body as well as the respective forms of the generalized nonlinear timedependent Schrödinger equation (68). Indeed, the derived equation (68)

Development of the Generalized Nonlinear Schrödinger Equation …

77

describes not only the mentioned state of virial relative mechanical

~

~

equilibrium [2, 4, 5] when GCF G(t )  G s  const  R and Ψ  R or

Ψ C : ~ Ψ  ~ 2 2 1  iG s   Gs   g  Ψ t  2 

(69a)

and the quasi-equilibrium gravitational condensation state [2, 4, 5] with a

~

~

~

periodically varying GCF increment when G(t )  G s  G(t )t  R and

Ψ  R or Ψ  C : ~ d G(t ) 1  ~ Ψ  ~ 2 iG(t )   G (t ) 2   g  Ψ ln Ψ  Ψ t 2  dt 

(69b)

but also the initial equilibrium gravitational condensation state [1, 4, 33] occurring in a forming gas-dust protoplanetary cloud:

i

Ψ ~  G s  2 Ψ t

(69c)

as well as the soliton disturbances state arising in a spheroidal body under formation:

~ Ψ  ~ 1 d ln G(t ) 2  2 i   G(t )  Ψ Ψ t  2 dt  and the gravitational instability states when GCF

  e

i arg 

(69d)

~ G(t )  C

and

C :

~ d G(t ) 1  ~ Ψ  ~ 2 2 ln Ψ  argΨΨ , (69e) iG(t )   G (t )   g  Ψ t 2  dt 

78

Alexander M. Krot

including the increase of gravitational compression of spheroidal body providing a formation of core of cosmogonical body if 0  arg Ψ  2 (the case of unlimited gravitational compression leading to a collapse occurs when arg Ψ  arg Ψ 2n, n  Z ). Let us note that according to the relation (63) the probability density function    satisfies the antidiffusion equation of the type (27a) while this wave function  satisfies the generalized nonlinear time-dependent

~

Schrödinger equation (68). However, in the case of a constant G (t ) in Eq. (27a) the equivalence between Eq. (27a) and Eq. (68) becomes possible because the derived equation (68) goes over the well-known linear timedependent Schrödinger equation (69c). It should be also mentioned that like the cubic nonlinear Schrödinger equation describing an evolution of an envelope of electromagnetic wave packet propagating in weakly nonlinear dispersible media [44] we can suppose that the cubic nonlinear Schrödinger equation (69d) describes an evolution of the envelope of Jeans’ substantive wave packet propagating in a nonlinear dispersible medium in accord with the Jeans’ criterion (1).

6. A FREQUENCY INTERPRETATION OF THE GRAVITATIONAL POTENTIAL AND THE GRAVITATIONAL STRENGTH OF A ROTATING SPHEROIDAL BODY Preliminarily we suppose that in the case of an initial gravitational instability (the quasi-equilibrium gravitational condensation under condition of



unobservable velocities of particles [2, 4, 5]) the hydrodynamic velocity v of  moving particles is absent practically, i.e., v  0 , as well as its partial and convective derivatives are equal to zero, too:

  v   (v ) v  0 . t

(70)

Taking into account Eqs. (35), (70) as well as the simplified formulas of vector analysis (38), (39) the relation (46b) takes the form:

Development of the Generalized Nonlinear Schrödinger Equation …

79

~ ~  u 2 ~       ~  d ln G(t )    d lnG(t ) ~ a  u   u  G(t ) 2 u  u     G(t ) divu   G(t ) ln   dt dt   0    2  u 2 ~     ~ (71)    G(t ) divu  G(t ) ln     0  2 Using estimations of the radial h -projection (42b) and the axial z projection (42c) of antidiffusion velocity (42a) of particles as well as Eqs. (10a), an acceleration (or initial gravitational field strength) induced by the antidiffusion process inside an ellipsoid-like cloud is calculated by the formula [4]:

   ~ ~ a  [u 2h / 2  u 2z / 2  G(t )  divu h  eh  u z  e z   G(t ) ln  /  0 ] 





2 1~ ~   grad[ G 2 (t ) 2 h 2 1   02   z 2  G 2 (t ) 21   02   1 2 1 ~ (72)  G(t ) (h 2 (1   02 )  z 2 )  2 ln 1   02 ] . 2



According to Eq. (72), when a gravitational field becomes its strength a

 g directly [34]:

can be calculated by the gravitational potential

 a   grad  g , therefore 

 g  G 2 (t ) 2 (t ) 1   02  h 2  z 2  1~ 2

2





4(1   02 )  2    (t ) 



 2 ln 1   02  1 ~  G(t ) (t )(1   02 )h 2  z 2  . 2  (t )   As seen, the arising gravitational potential spheroidal

body



is

the

sum

 g  G 2 (t ) 2 (t ) 1   02  h 2  z 2  1~ 2



2

of

(73a)

 g of a forming an

regular

part

4(1   )  2   and a fluctuation part  (t )  2 0

80

Alexander M. Krot

1 ~ 2



 g  G(t ) (t )(1   02 )h 2  z 2  





2 ln 1   02  ~ ~  if G(t )  G(t ) , so that  (t ) 

Eq. (73a) takes the form:



 g   g  G 2 (t ) 2 (t ) 1   02  h 2  z 2  1~ 2

2



4(1   02 )  2  .  (t ) 

(73b)

Using Eq. (72) we can obtain the induced acceleration of an initial gravitational field:













2    ~   ~ a  G 2 (t ) 2 (t ) 1   02 h  eh  z  ez  G(t ) (t ) 1   02 h  eh  z  ez (74)

According to Newton’s second law [34] the equation of motion of a  particle under action of specific force a into a forming spheroidal body is the following:

  d 2 ( h  eh  z  e z )  ~ ~   (t ) 1   02 [G 2 (t ) (t ) 1   02  G(t )]h  eh  2 dt (75)  ~2 ~   (t )[G (t ) (t )  G(t )]z  e z  0.









Since Eq. (75) is a sum of two harmonic oscillator equations then the inducible  acceleration a leads to the oscillating movement of particles. According to Eq. (75) we can obtain the radial h -projection and the axial z -projection of this vector equation of motion of particle under action of a specific force (74) into an ellipsoid-like cosmogonical body:



d 2h ~ 2  [G (t ) 2 (t ) 1   02 dt 2



2





~  G(t ) (t ) 1   02 ]h  0 ;

d 2z ~ 2 ~  [G (t ) 2 (t )  G(t ) (t )]z  0 . 2 dt

(76a)

(76b)

It follows from Eq. (76a) the main circular frequency of the radial oscillations is expressed by the formula:

Development of the Generalized Nonlinear Schrödinger Equation …









2 ~ ~ ωh (t )  G 2 (t ) 2 (t ) 1   02  G(t ) (t ) 1   02 ,

81

(77)

so that at the stage of formation of an ellipsoid-like cosmogonical body the following representation for ωh (t ) in accord with the formula (77) is true:

ω 2h (t )  ωh (t )  ( h ) 2 (t ) , 2

(78)

Where





(79a)





(79b)

2 2 ~ ωh (t )  G 2 (t ) 2 (t ) 1   02 ;

~ ( h ) 2 (t )  G(t ) (t ) 1   02

and  h , generally speaking, is a generalized circular frequency of the

~

~

radial oscillations since G(t ) can be a negative value ( G(t )  0 ). This representation (78) is equivalent to the mentioned expansion of gravitational potential (73a) of forming spheroidal body as the sum of the regular part and the fluctuation part

 g

 g . Consequently, according to Eq. (78) the regular 

2

part of ω 2h can be a squared angular velocity of rotation Ω 2 (t )  ωh (t ) in the equatorial ( x, y)  plane whereas its fluctuation part is a squared generalized circular frequency ( h ) of fluctuations. Indeed, by substituting 2

Eqs. (79a, b) in Eq. (74) we obtain:

ah  [ωh (t )  ( h ) 2 (t )] h   Ω 2 (t ) h  ( h ) 2 (t ) h   f c  f a h 2

,

(80)

where f c is a specific (per mass unit) centrifugal force and f a h is a h projection of specific additional periodic force of Alfvén–Arrhenius [30, 31]. It follows from Eq. (76b) the main circular frequency of the axial oscillations is expressed by the formula:

82

Alexander M. Krot

~ ~ ω z (t )  G 2 (t ) 2 (t )  G(t ) (t ) ,

(81)

so that according to Eq. (81)

ω 2z (t )  ωz (t )  ( z ) 2 (t ) , 2

(82)

where

and

2 ~ ωz (t )  G 2 (t ) 2 (t ) ;

(83a)

~ ( z ) 2 (t )  G(t ) (t )

(83b)

 z is a generalized circular frequency of the axial oscillations. 



So, the gravitational acceleration a (or the specific force of gravity f g ) is



balanced by the vector sum of specific centrifugal force f c and specific



additional periodic force f a of Alfvén–Arrhenius [30, 31]:

   a   fc  fa ,

(84)

where

     f c  Ω 2 (t )r  [Ω(t )  [r  Ω(t )]] ;

(85a)

   f a  ( ) 2 (t )r   Re A(t )e i (t )t Ω 2 (t )r ,

(85b)

besides A(t )  1 . Now let us estimate the frequency of the radial and axial oscillations of a rotating and gravitating ellipsoid-like cosmogonical body under the condition

~

~

of stabilization of the generalized GCF G(t )  G s  const . In this case,

Development of the Generalized Nonlinear Schrödinger Equation …

83

according to Eq. (72) the induced gravitational field strength of an ellipsoidlike cloud is equal [4]:

     ~ ~ a   grad[ u 2 / 2  G s  divu]   grad[u 2h / 2  u 2z / 2  G s  divu h  eh  u z  ez ] 









 





2 2     ~ ~ ~   G 2s  2 1   02 h  eh  z  ez  G 2s  grad 2 1   02  1   G 2s  2 1   02 h  eh  z  ez



(86) According to Eqs. (77), (79a), (81), (83a), (86) in the particular case of relative mechanical equilibrium when

~ ~ G(t )  G s  const the circular

frequencies of the radial  h and the axial oscillations

 z inside an ellipsoid-

like cosmogonical body are described by the formulas respectively [4]:





~ ωh  ωh  G s  1   02 ;

(87a)

~ ω z  ωz  G s  .

(87b)

According to (87a) and (87b) we can see that in the case of relative mechanical equilibrium of an ellipsoid-like cloud the following inequality is true:

 z  h ,

(88)

that fully confirms analogous conclusion of Alfvén and Arrhenius [30, 31]. On the other words, when a gravitational field in an ellipsoid-like cosmogonical body becomes stable, an interference of the orthogonal radial and the axial oscillations leads to the rotation of core of this spheroidal body. Moreover, the interference of these orbital oscillations may be nonuniform at different latitudes of core of ellipsoid-like cosmogonical body modeled a star, so that we can say about a constant angular velocity of rotation  0 only in the equatorial plane of a star. In particular, the Sun’s sidereal rotation period is TSun  25.05 days at the equator, TSun  25.38 days at the 16 latitude and

TSun  34.4

days

at

the

poles

[45],

i.e.,

 0 Sun  2.90308  10 6 s 1 ,

6 1 16 Sun  2.86533  10 6 s 1 and  90 Sun  2.11401  10 s respectively.

84

Alexander M. Krot When a forming ellipsoid-like cosmogonical body reaches the relative

~

~

mechanical equilibrium state ( G(t )  G s  const ) at t  t s (that means the equality of gravitational and centrifugal potentials  g   c ) then the equivalence of Eq. (19a) and Eq. (73b) takes place whence

1   02 1 ~2 2 . G s  (t s )  2 0 (t s ) 2 2 2 1   02  1



(89)



where t s is a stabilization time moment,  0 (t s )  M ( (t s ) / 2 ) 3 / 2 is a central mass density. As follows from Eq. (89)



~ G 2s  2 [2 1   02



2





 1]  4M ( / 2 ) 3 / 2 1   02 .

(90)

Taking into account the definitions (87a) and (87b) we can find that





2 ωh  ωz  4M ( / 2 ) 3 / 2 1   02 . 2

2

Frequently

(91)

 0  1 therefore ωz  ωh  Ω s in Eq. (91), so that

M Ω  3

2 3

2 s



,

(92)

whence we can obtain the following analog of Kepler 3rd law:

Ω 2s Rs3  M

(93)

under supposition that a characteristic orbital radius is equal

Rs 

3

3  /2



.

(94)

Development of the Generalized Nonlinear Schrödinger Equation …

85

Really, as mentioned relative to Eq. (89) the equality of gravitational and centrifugal potentials means the decay of a spheroidal cloud at distances

r  Rs where Rs  1.554988 /  . Taking into account that the Keplerian angular velocity of the motion is

ΩK 

M R3

(95)

with the period TK  2 / Ω K [31], R is a radius of orbit, the expression (95) through Eq. (94) can be represented as follows:

Ω K  Ω s ( Rs / R) 3 / 2 ,

(96)

where  s is a main circular frequency of oscillations. The separation process of a spheroidal cloud leads to formation of its inner zone I (the core) and its remote zone II (the exterior shell). Respectively we consider the inner gravitational potential (19a) as well as the exterior gravitational potential (19b). The evolution of the core and the exterior shell describes the processes of formation of a central cosmogonical body (a protostar) and hulls (a disk with embryos of forming protoplanets). Let us note that owing to the fluctuation part

 g of gravitational

potential (73a) of a forming spheroidal body in its quasi-equilibrium state there exist the radial and the axial oscillations of orbital motion of particles under action of a specific additional periodic force in the remote zone, too. Really, following Alfvén and Arrhenius [30, 31] the circular orbit of moving particles can be modified by both the radial and the axial oscillations but for a slowly rotating spheroidal body these radial and axial oscillations are degenerate in the sense that ω h  ω z  Ω K (see our explanation before Eq. (92)), i.e.,

ω 2h  ω 2z  2 Ω K , 2

that confirms the Alfvén–Arrhenius’s equation completely.

(97)

86

Alexander M. Krot

Thus, by analogy with the mentioned expansion of gravitational potential (73a) we find that the regular part of the main circular frequency of oscillations ω(t ) is an angular velocity Ω(t ) of rotation whereas its fluctuation part is a generalized circular frequency  (t ) of fluctuations. As consequence, the inducible gravitational acceleration (84) is compensated by the sum of a



specific centrifugal force f c and a specific additional periodic force of Alfvén–



Arrhenius f a [30, 31]. This result points to a possibility of presence of statistical oscillations of motion in planetary orbits, i.e., oscillations of their major semi-axes a and orbital angular velocities Ω of rotation of planets around stars. Really, such conclusion is confirmed by existing the Alfvén– Arrhenius radial and axial orbital oscillations of bodies [31]. Concretely, this

~

paper shows that the temporal deviation of the generalized GCF G(t ) of a rotating spheroidal body under the condition of its mechanical quasiequilibrium leads to becoming Alfvén–Arrhenius additional periodic force modifying forms of circular orbits to slightly elliptical orbits of moving bodies. Let us note that the temporal deviation of the generalized GCF is

~

determined by an oscillation behavior of its derivative G(t ) that implies the

~

special case when G(t )  0 and, therefore, ( )  0 , i.e., according to Eqs. 2

 (83b), (85b) the additional periodic force f a becomes oriented opposite to the 

gravitational force f g . It means that the principle of an anchoring mechanism is realized in any exoplanetary system.

CONCLUSION The obtained result relative to the generalized nonlinear time-dependent Schrödinger equation (68) has been suggested in accordance with similar conclusions of Ord [46, 47] and El Naschie [25] that the Schrödinger equation could be universal. In other words, it may have a large domain of applications, but with interpretations different from that of standard quantum mechanics. In this connection, it should be mentioned that the derived nonlinear generalized Schrödinger equation is a macroscopic equation (68) unlike the quantum

Development of the Generalized Nonlinear Schrödinger Equation …

87

mechanical Schrödinger equation [43]. According to the question relative to a connection with quantum mechanics, we emphasize that the quantum mechanical Schrödinger equation describes a microscopic behavior of particle. Due to Casimir’s effect the oscillation behavior of particles (before origin of the gravitational field in a molecular cloud) is described by the time-dependent Schrödinger equation for the quantum-mechanical harmonic oscillator [43, 48]:

i

 2  k  Ψx, t      2  x 2  Ψx, t , t 2   2 m0

k  m0  2 ,

where its solution is a wave function x, t  , m0 is a mass of particle, its angular frequency of oscillations, x  q1  q2  q3 , besides 2

2

2

2

(98)



is

qi is

a displacement of particle from an equilibrium position. To construct a realistic physical picture when a wave is localized in a finite region of the space, the concept of a wave packet is introduced (in which amplitudes are localized in some spatial domain). Thus, by the wave packet one means a superposition of a sufficient number of wave functions of different frequencies and amplitudes: 

Ψx, t    an n x, t  .

(99)

n 0

As a result, we can describe a particle (moving under influence of spontaneous harmonic forces from local cores inside a molecular cloud) not as a “point mass” but as a “wave packet.” It has been reported in [48], a particle described by the three-dimensional harmonic oscillator is characterized also as an oscillating expectation value of three-dimensional Gaussian wave packet and an oscillating width of this packet as well. In the one-dimensional case (

x  q1 ), an evolution of the probability density to observe a particle described by a quantum mechanical oscillator with the initial expectation values of

x0 and momentum p 0  0 , width of the initial wave packet  x0 with time t is also characterized by a Gaussian wave packet [48]: position

88

Alexander M. Krot

wx, t    x, t   2

1

2  x t 

e



 x  x0 cost 2 2 x 2 t 



2   2 x0 2 x20 x  x0 cos  t  1    exp  4 , 2 4 2 2 4 x40 cos 2 t   04 sin 2  t  4 x0 cos  t   0 sin  t 

(100) with the oscillating expectation value x 0 t   x 0 cos  t and the oscillating width  x t   4 x4 cos2 t   04 sin 2  t 0

2 x  and  0

0

  / m0 .

Taking into account that oscillations of three-dimensional Gaussian wave packet are independent (they are also orthogonal to one another), their resulting oscillation has an elliptical trajectory of motion. This means that the shape of an oscillating particle is described by a changing with the time ellipsoid just as the trajectory of motion of this particle in the space is elliptical [48]. During a slowly evolving process of initial gravitational condensation of a forming spheroidal body from an infinitely distributed substance (a molecular cloud) the parameter of gravitational condensation

   (t )

increases with

the time t that leads to a growth of the potential gravitational energy [4, 6]:

Eg (t )  where  and

  1  (r , t ) g (r , t )dV  2V

g

,

(101)

are mass density (10a) and gravitational potential (19a, b) of

a rotating spheroidal body respectively. When an essential growth of the potential gravitational energy (101) occurs then nonlinear phenomena arise owing to self-organization processes [40, 49] of interactions of oscillating particles into a spheroidal body under its formation. These nonlinear phenomena induce nonlinear waves satisfying a nonlinear undulatory Schrödinger-like equation (68), in particular Eq. (69d) (in which except a



wave function r, t  there is a temporal function G(t ) , i.e., the generalized GCF). The generalized GCF (27b) is a measure of interactions of oscillating

~

Development of the Generalized Nonlinear Schrödinger Equation …

89

particles into a forming spheroidal body. Really, in the case of a constant

~ ~ G(t )  G s in Eq. (68) the derived nonlinear Schrödinger equation (69a)

becomes similar to the time-dependent Schrödinger equation (98) if we might



~

~

assume formally that r  x , G(t )  G s   / 2m0 and

 g  m0 2x2 / 2 .

Moreover, it is known that an evolution of an envelope of electromagnetic wave packet propagating in weakly nonlinear dispersible media is described by the cubic nonlinear Schrödinger equation [44]:

i

A 2  [   2   A ] A t ,

(102)



where A  Ar , t  is an amplitude of the envelope of wave packet and  ,  are parameters, so that we can suppose that the nonlinear time-dependent generalized Schrödinger equation (69d) describes an evolution of the envelope of Jeans’ substantive wave packet propagating in a nonlinear dispersible ~

1 d ln G(t ) . ~ , medium if we might assume formally that A   ,   G (t )   2

dt

In spite of the derived Schrödinger-like equation (68) is a macroscopic equation while the quantum mechanical Schrödinger equation (98) is a microscopic equation, they are associated among themselves like the macroscopic hydrodynamic equation of Euler and the microscopic kinetic equation of Boltzmann. Indeed, it is well-known that Euler’s macroscopic equation can be derived by means of Boltzmann’s microscopic equation [49]. In this connection, the derived macroscopic nonlinear equation (68) can aid to obtain a respective microscopic nonlinear equation describing an evolution of nonlinear interactions of wave functions of quantum-mechanical oscillators (oscillating particles).

REFERENCES [1] [2]

Krot, A. M. A statistical approach to investigate the formation of the solar system. Chaos, Solitons & Fractals. 2009, 41, 1481–1500. Krot, A. M. A quantum mechanical approach to description of initial gravitational interactions based on the statistical theory of spheroidal bodies. Nonlinear Sci. Lett. A. 2010, 1, 329–369.

90 [3]

[4] [5]

[6] [7] [8] [9] [10]

[11] [12]

[13] [14] [15] [16] [17]

Alexander M. Krot Krot, A. M. A model of forming planets and distribution of planetary distances and orbits in the Solar system based on the statistical theory of spheroidal bodies. In Solar System: Structure, Formation and Exploration; de Rossi M.; Ed; Nova Science Publishers: New York, 2012, Ch. 9, pp. 201–264. Krot, A. M. A Statistical Theory of Formation of Gravitating Cosmogonical Bodies; Bel. Navuka: Minsk, 2012; 448 pp. [in Russian]. Krot, A. M. A nonlinear Schrödinger-like equation in the statistical theory of formation of cosmological bodies. In Chaos and Complexity Research Compendium; Orsucci F. and Sala N.; Eds.; Chaos and Complexity; Nova Science Publishers: New York, 2013, Vol.3, Ch. 7, pp. 93-112. Krot, A. M. On the universal stellar law for extrasolar systems. Planet. Space Sci. 2014, 101C, 12–26. Jeans, J. Problems of Astronomy and Stellar Dynamics; University Press: Cambridge, 1919. Jeans, J. Astronomy and Cosmogony; University Press: Cambridge, 1929. Schmidt, O. Yu. Origin of Earth and Planets; Acad. of Sci. USSR Press: Moscow, 1962 [in Russian]. Safronov, V. S. Evolution of Protoplanetary Cloud and Formation of Earth and Planets; Nauka: Moscow, 1969 [reprinted by NASA Tech. Transl.: Washington, D. C., 1972; F-677]. Goldreich, P.; Ward, W. R. The formation of planetesimals. Astrophys. J. 1973, 183, 1051–1062. Lifschitz, E. M. On the gravitational stability of the expanding Universe. Zh. Eksp. Teor. Fiz. 1946, 16, 587–602 [reprinted by J. Phys. 1946, 10, 116]. Bonnor, W. B. Jeans’ formula for gravitational instability. Mon. Not. R. Astron. Soc. 1957, 117, 104. Einstein, A. Kosmologiche Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsber. d. Berl. Akad. 1917, 1, 142. Weinberg, S. The cosmological constant problem. Rev. Mod. Phys. 1989, 61, 1–23. Parenago, P. P. Lectures on Stellar Astronomy; Gosizdat: Moscow, 1954 [in Russian]. Nottale, L. Scale relativity, fractal space-time, and quantum mechanics. Chaos, Solitons & Fractals. 1994, 4, 361–388.

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[18] Nottale, L. Scale-relativity: from quantum mechanics to chaotic dynamics. Chaos, Solitons & Fractals.1995, 6, 399–410. [19] Nottale, L. Scale-relativity and quantization of extra-solar planetary systems. Astron. Astrophys. 1996, 315, L09–L12. [20] Nottale, L.; Schumacher G.; Gay, J. Scale relativity and quantization of the solar system. Astron. Astrophys. 1997, 322, 1018–1025. [21] Nottale, L. Scale-relativity and quantization of the universe: I. Theoretical framework. Astron. Astrophys. 1997, 327, 867–889. [22] Nottale, L.; Schumacher, G.; Lefèvre, E. T. Scale-relativity and quantization of exoplanet orbital semi-major axes. Astron. Astrophys. 2000, 361, 379–387. [23] Nelson, E. Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev. 1966, 150, 1079–1085. [24] Nelson, E. Quantum Fluctuations; Princeton University Press: Princeton, NJ, 1985. [25] Quantum Mechanics, Diffusion and Chaotic Fractals; El Naschie, M. S., Rössler, E., Prigogine, I., Eds.; Pergamon Press: Oxford, 1995. [26] Nottale, L.; El Naschie, M. S.; Athel, S.; Ord, G. Fractal space–time and Cantorian geometry in quantum mechanics. Chaos, Solitons & Fractals. 1996, 7 [a special issue]. [27] Ord, G. N. Classical particles and the Dirac equation with an electromagnetic force. Chaos, Solitons & Fractals. 1997, 8, 727–741. [28] Sidharth, E. G., The Chaotic Universe; Nova Science Publishers: New York, 2001. [29] Chernin, A. D. Cosmic vacuum. Usp. Fiz. Nauk. 2001, 171, 1153–1175. [30] Alfvén, H.; Arrhenius, G. Structure and evolutionary history of the Solar system. I. Astrophys. Space Sci. 1970, 8, 338–421. [31] Alfvén, H.; Arrhenius, G. Evolution of the Solar System; Sci. and tech. inform. office of NASA: Washington, 1976. [32] Landau, L. D.; Lifschitz, E. M. Statistical Physics; Addison–Wesley Publishing Co.: Reading, MA; 1955; Part 1. [33] Krot, A. M. On the principal difficulties and ways to their solution in the theory of gravitational condensation of infinitely distributed dust substance. In Observing our Changing Earth; Sideris, M. G.; Ed.; Springer: Berlin/Heidelberg, 2009; Vol. 133, pp. 283–292. [34] Landau, L. D.; Lifschitz, E. M. Classical Theory of Fields; Addison– Wesley Publishing Co.: Reading, MA, 1951. [35] Gurevich, L. E.; Lebedinsky, A. I. On the planet formation.–I. Gravitational condensation; – II. Planetary distance law and rotation of

92

[36]

[37] [38] [39] [40]

[41] [42]

[43] [44] [45] [46] [47] [48] [49]

Alexander M. Krot planets; – III. Structure of initial cloud and separation of planets by outer and inner ones. Izv. Akad. Nauk SSSR, Ser. Fiz. 1950, 14, 765– 775; 776–789; 790–799 [in Russian]. Vityazev, A. V.; Pechernikova, G. V.; Safronov, V. S. Terrestrial Planets: Origin and Early Evolution; Nauka: Moscow, 1990 [in Russian]. Nicolis, G.; Prigogine, I. Exploring Complexity: An Introduction; W. H. Freeman and Co.: New York, 1989. Landau, L. D.; Lifschitz, E. M. Fluid Mechanics; Pergamon: Oxford, 1959. Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stability and Fluctuations; London, 1971. Nicolis, G.; Prigogine, I. Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuation; John Willey and Sons: New York, 1977. Loytsyanskiy, L. G. Mechanics of Fluid and Gas; Nauka: Moscow, 1973 [in Russian]. Krot, A. M. Self-organization processes in a slow-flowing gravitational compressible cosmological body. In Topics on Chaotic Systems; Skiadas, C. H., Dimotikalis, I., Skiadas, C.; Eds; World Scientific: Singapore, New Jersey etc., 2009, pp. 190–208. Schrödinger, E. An undulatory theory of the mechanics of atoms and molecules. Phys. Rev. 1926, 28, 1049–1070. Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C. Solitons and Nonlinear Wave Equations; Academic Press: London, 1984. Sun, http://en.wikipedia.org/wiki/Sun/; 2016. Ord, G. N. Fractal space-time and the statistical mechanics of random walks. Chaos, Solitons & Fractals. 1996, 7, 821–843. Ord, G. N.; Deakin, A. S. Random walks, continuum limits, and Schrödinger’s equation. Phys. Rev. 1996, A54, 3772–3778. Brandt, S.; Dahmen, H. D. The Picture Book of Quantum Mechanics; Springer-Verlag: New York, 1994. Ebeling, W. Origin of Structures at Irreversible Processes: An Introduction in the Theory of Dissipative Structures; Rostock, 1977.

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BIOGRAPHICAL SKETCH Alexander M. Krot Affiliation: Professor, DSc Education: Radio-physicist Business Address: 6, Surganov Str., Minsk, 220012 Belarus Research and Professional Experience: complex dynamical systems, statistical theory of gravitating body formation, digital signal processing Professional Appointments: Head of Laboratory of Self-Organization System Modeling at the United Institute of Informatics Problems of National Academy of Sciences of Belarus Honors: Recipient of Komsomol’s Republic awards in the area of science and engineering (1988 and 1990), A.S. Popov award of Radio-engineering Society (1991), award of National Academy of Sciences of Belarus (1993), grant of International Science Foundation (1993), grant of President of the Republic of Belarus (2007), award of Ministry of Industry of the Republic of Belarus (2010). Publications: 1. Krot, A. M. A Statistical Theory of Formation of Gravitating Cosmogonical Bodies; Bel. Navuka: Minsk, 2012; 448 pp. [the monograph]. 2. Krot, A. M. A nonlinear Schrödinger-like equation in the statistical theory of formation of cosmological bodies. In Chaos and Complexity Research Compendium; Orsucci F. and Sala N.; Eds.; Chaos and Complexity; Nova Science Publishers: New York, 2013, Vol. 3, Ch. 7, pp. 93-112. 3. Krot, A. M. On the universal stellar law for extrasolar systems. Planet. Space Sci. 2014, vol. 101C, pp. 12–26 (Elsevier Ltd.). Professor, DSc Alexander M. Krot is a Head of the Laboratory of SelfOrganization System Modeling at the United Institute of Informatics Problems

94

Alexander M. Krot

of the National Academy of Sciences of Belarus. He graduated from the Belarusian State University, Department of Radiophysics in 1982. He received PhD degree in 1985, then degree of Doctor of Sciences (DSc) in 1991 (by the decision of the Higher Attestation Commission of USSR in Moscow) and Professor degree in 1997 (by the decision of the Higher Attestation Committee of Republic of Belarus in Minsk). Since 1993 Dr. A.M. Krot is a Head of the Laboratory at the United Institute of Informatics Problems (the former Institute of Engineering Cybernetics) of the National Academy of Sciences of Belarus. Now he is also a Professor at the Department of Mathematical Physics of the Belarusian State University in Minsk. His current research interests are the analytical theory and computational modeling of self-organization processes and phenomena in complex systems (gas-dust protoplanetary media, aerohydrodynamic viscous flows, nervous fibres, neural network structures etc.), statistical theory of planetary (stellar) system forming, theory of nonlinear analysis of attractors of complex systems, digital signal processing etc. He has published 280 scientific works including 3 monographs (A.M. Krot “Discrete models of dynamical systems based on polynomial algebra”: Minsk, Navuka i tekhnika, 1990, 312 pp.; A.M. Krot, H.B. Minervina “Fast algorithms and programs for digital spectral processing of signals and images”: Minsk, Navuka i tekhnika, 1995, 407 pp. and A.M. Krot “A statistical theory of formation of gravitating cosmogonical bodies,” Bel. Navuka, Minsk, 2012, 448 pp.) and more than 80 articles in refereed journals. Since 1995 till now Professor Alexander M. Krot is a subject of biographical records in “Who is Who in the World” and “Who is Who in Science and Engineering,” New Providence, USA.

In: Complex Systems Editor: Rebecca Martinez

ISBN: 978-1-53610-860-6 © 2017 Nova Science Publishers, Inc.

Chapter 4

THE APPLICATION OF NEURAL NETWORK MODELING IN ORGANIZING A HIERARCHICAL TEACHING SYSTEM BASED ON MENTORSHIP A. Dashkina and D. Tarkhov Peter the Great Saint Petersburg Polytechnic University, Saint Petersburg, Russian Federation

ABSTRACT In this chapter we consider the application of the hierarchical teaching systems and mentorship. We illustrate that they have many advantages over the conventional system of organizing the learning process within a group of learners with different levels of knowledge. The process of interaction between a teacher and students is complex and nonlinear. If the knowledge is transferred not only from a teacher to learners, but also from one student to another, the system can be referred to as a hierarchical one. Neural networks prove to be an appropriate tool for creating models of such systems, so we apply them here. We have conducted a number of practical experiments which involved application of mentorship and the hierarchical teaching systems within some groups of learners. The results of the experiments were processed by neural networks. It allowed us to create a sociodynamic model of the learning process and to forecast and maximize its further results. At the end of this

96

A. Dashkina and D. Tarkhov research we formulated recommendations.

the

conclusions

and

gave

practical

One of the primary goals that educators face is optimizing the teaching and learning process. It is particularly critical given the tendency towards computer-assisted learning, virtual learning communities, and a higher percentage of assignments which are done autonomously. Another reason for seeking the most efficient ways of organizing the students’ cognitive process is the growing size of university students’ groups. Switching to computer-assisted language learning is a prerequisite for new working paradigms and organizational settings based on learners’ collaboration and group support (Ganesan, Edmonds and Spectror, 2003, p.99). Students doing a foreign-language course on the Internet need interactivity and networking, which can be achieved by collaboration with other learners. Learners’ mutual cooperation is also necessary when they are supposed to do different types of assignments in a face-to-face learning situation. A poorly organized teaching and learning process in big groups can be particularly counterproductive in the context of foreign language learning because the students do not have enough time in class to practice their foreignlanguage skills. The duration of the teacher’s contact with each individual student is limited by the length of the class (90 minutes), which is why it is necessary to split the group of learners into a few subgroups so that the students with entry-level foreign-language skills can get instructions and assistance from the students with a high level of foreign language knowledge (who can be referred to as mentors). Such a technique of organizing the students’ cognitive process is a hierarchical system, in which the teacher acts as the manager delegating authority to the mentors. In our previous research we explored such a system in numerical experiments by training neural networks, and it proved quite efficient (D. A. Tarkhov, A. I. Dashkina, 2015, p. 20). We also considered organizational and methodical issues of introducing a hierarchical system in communicating knowledge into the educational process in the context of forming a scientific school (Group Monograph, 2015, p. 67). In this experiment we are going to do research into the optimal ways of dividing the students into pairs. Much research has been done into different ways of organizing groups of learners. For instance, a small group consisting of one student with a rudimentary level of knowledge, one with a medium level and one with a high level appeared to be very efficient in acquiring foreign-language skills (Polat

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Ye. S, p. 334, 2012). In the article we consider pairs of students (not small groups) in order to ascertain whether these pairs should be formed on the basis of the learners’ preferences or on the basis of the difference in the participants’ level of knowledge. The experiment was conducted at St. Petersburg Polytechnic University and it involved two groups of linguistic majors (first-year and second-year students) and two groups of second-year and third-year students specializing in economics and management. The purpose of the experiment was to find out whether friendly relationships between the students working in pairs are conducive to more efficient learning. At the beginning of the experiment the students had to do a placement test in general English. We assessed the students’ command of English and identified their initial level of knowledge before the experiment was conducted. The groups involved in the experiment were divided into pairs after the test. Half of these pairs were formed on the basis of the students’ personal choices. These preferences were not based on the students’ command of English, so in some of these pairs the participants had the same level of knowledge, whereas the members of other pairs were at completely different levels. The other halves of the groups involved in the experiment were divided into pairs by the teacher, regardless of the students’ personal preferences. In accordance with the results of the placement test, one student in each of these pairs had mastery of English, while the other student’s knowledge was rudimentary. In these pairs the students had not established any personal relationships before the beginning of the experiment. We hypothesized that the pairs formed on the basis of the students’ personal preferences would show better results at the end of the experiment because friendly relationships between the students would intensify their cognitive process. Each of the groups involved in the experiment was supposed to learn a certain amount of material in accordance with their curriculum. The linguistic majors in both of the groups were supposed to read fiction (short stories). The first-year students read classical horror stories, and the second-year students read classical love stories. Both groups had to learn the words from the texts and perform literary analysis, which allowed them to consolidate the vocabulary. The group of the second-year students specializing in economics and management had a very low level of knowledge. They were supposed to work with a course in General English called “Language leader” (preintermediate). The group of third-year students specializing in economics and management had a high level of foreign language knowledge. They had to

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work with an English course in business English called “Business Advantage” (upper-intermediate). We have already mentioned that the students had to perform both classroom and home assignments in pairs, formed either by the teacher, or on the basis of the students’ preferences. All the pairs in the groups specializing in linguistics had to give each other vocabulary dictations, do vocabulary exercises and perform literary analysis together. The second-year students specializing in economics and management had to read topic-orientated texts (about environmental protection, international organizations and sports) in pairs, do the assignments related to these texts, give each other vocabulary presentations and prepare presentations on the basis of the topic-orientated vocabulary. The third-year students specializing in economics and management also had to check each other’s vocabulary knowledge, read some texts from the course book, do vocabulary exercises and prepare relevant dialogues or presentations on the basis of the ”case study” sections. All the students participating in the experiment had to do the home assignments by sending each other messages on the Internet. These e-mail messages were supposed to include some informal correspondence in English (about the studies, the weather, cultural events, sports, etc.) as well as the information related to the home assignment. The printouts of the messages were submitted to the teacher. The students with a high level of knowledge who worked in the pairs formed on the basis of the teacher’s choice were supposed to be mentors for the students with a lower level of knowledge. It meant that they were supposed to provide their partners with necessary explanations; correct their mistakes and check their home assignments. In the “friendship-based” pairs all the assignments were evaluated on the basis of peer assessment, which is defined as “the carefully organized, periodic analysis of oral and written work by other students and comparison with models using specific criteria” (D. M. Brinton, M. A. Snow, M. Wesche, 2006, p. 192). In our experiment we focused mainly on learning the key words from the portion of material that they worked with in the course of this study, because it was easy to measure and assess in quantitative terms how well the students had expanded vocabulary. At the end of the experiment each group had to do a multiple-choice lexical test, which consisted of twenty points. In the first fifteen points students had to fill out the gaps with a suitable word; in the last five points they had to choose the words corresponding to the definitions. Here

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is an extract from the test conducted in the group of the first-year linguistic majors, who read and analyzed classical horror stories. Fill in the blanks: 1. The clock on the distant church ___ twelve. A. stroke B. struck C. punched 2. The hunter caught sight of the ___ . A. quarry B. loom C. glimpse 3. Dr Hardcastle held his ___ forefinger upon the trigger. A. wretched B. clenched C. crooked Choose the right word for the following definitions: 1. A very deep crack in rock or ice. A. shaft B. burrow

C. chasm

2. To move with great difficulty and be in danger of falling. A. flounder B. boulder C. stoop We prepared separate tests for each of the four groups which took part in the experiment. The tests were based on the limited amount of lexical material, but they were at the same level as the initial placement tests for each of the groups involved in the experiment. This enabled us to determine how well they had acquired the vocabulary. We used a ten-point scale when we evaluated the results of the final tests. This meant that if all the twenty points were correct, the student scored ten and with each mistake their total score was 0.5 lower. On the basis of the final test we made a separate table for each group. We pointed out which of the pairs were formed by the teacher regardless of the students’ personal preferences (with the mentors italisized in the table), and which of them were formed on the basis of the students’ personal choices. Here is a fragment of such a table. Pairs formed by the teacher: №1+№2 Pairs formed on the basis of the students’ personal choices: №9+№10

100 № 1. 2. 9. 10.

A. Dashkina and D. Tarkhov Name, surname Tatiana Antonova Elvira Andreyeva Darina Antsiferova Natalia Doronina

Placement test 4 6 6 4

Final test 7.5 9.5 6.5 5

Progress 3.5 3.5 0.5 1

For each student we indicated the results of the placement and final tests and the progress they made within the framework of the experiment. Since the groups had a different level of foreign language knowledge, and they were offered the tests in accordance with their level of knowledge, the results for each group were processed separately. The samples of the test results were ranked in order to obtain a sustained and visual result. The student’s rank could change in accordance with his or her results. The resulting samples were smoothed by a two-element neural network function F ( x)  c1th[a1 ( x  xc )]  c2th[a2 ( x  xc )] . The coefficients 1

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functional

 F  х   F  i

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Figure 1. The results of the placement and the final tests (grey and black respectively) for group 2-6.

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It is clearly visible that the results of the final test are much better than those of the placement one. It indicates that the teaching and learning process was organized well. The moderated dependencies are virtually lineal, which means that the test was suitable for the group of students under consideration.

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Figure 2. The results of the placement and the final tests (grey and black respectively) for the experimental subgroup in group 2-6.

Here it can be observed that the results of the final test are much better than those of the placement one and the difference is much higher than in the group taken in its entirety. In addition to that, the results of the placement test indicate a great difference in the command of English in each pair. In the final test the difference was less considerable. 7 6 5 4 3 2 1

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Figure 3. The results of the placement and the final tests (grey and black respectively) for the control subgroup in group 2-6.

From this figure it is quite obvious that there was very little difference in the results of the placement and the final test, which indicates that the teaching and learning process was not very efficient.

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Figure 4. The difference in the results of the placement and the final tests for group 2-6.

In Figure 4 we can observe that the results are incoherent, although for most of the group the progress is visible (a significant part of the graph is above the horizontal axis). Besides, there is a sharp bend in the left-hand section of the graph. It indicates that some students’ results were not characteristic of the group as a whole. We ascertained that these students’ attitude towards learning was not responsible enough: they skipped several classes and failed to do their homework.

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Figure 5. The difference in the results of the placement and the final tests for the experimental and control subgroups in group 2-6 (grey and black respectively).

In Figure 5 it can be seen that the experimental subgroup’s results were much higher than the ones in the control subgroup. The results in the experimental subgroup were fairly coherent, which indicates that there was a marked similarity in the efficiency of different students’ learning process. The performance of the control subgroup was much poorer and the results were incoherent, although the assignments were the same for the whole group. This means that the performance was influenced by some factors not linked with the teaching and learning process.

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Now it is necessary to test the assumption that the difference in the final results can be explained by the disparity in the levels of the student who acts as a mentor and the one who is instructed by the latter.

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Figure 6. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair (for group 2-6).

In Figure 6 there is no obvious correlation, but it is possible to observe that the greatest efficiency can be achieved if the average difference in the students’ level of knowledge is about 2 points. Further this fact can be taken into consideration in the process of forming pairs after the placement test.

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Figure 7. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair for the experimental and control subgroups respectively (group 2-6).

The neural network smoothing was not carried out for these dependencies because the number of points was insufficient. Figure 7 illustrates that the above mentioned conclusion about the optimal difference in the level is only characteristic of the experimental subgroup, whereas for the control subgroup

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the result is higher when the difference in the performance of the students in the pair is maximum.

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Figure 8. The results of the placement and the final tests for group 3-a (grey and black respectively).

Figure 8 illustrates that the results of the final test are much higher than those of the placement test, which indicates that the teaching and learning process was organized efficiently. The smoothed correlation is virtually lineal, which means that the test was quite appropriate for this group of students. The results of the final test are approaching to saturation. It means that the students with mastery of English should have been given more challenging assignments. 10

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Figure 9. The results of the placement and the final tests (grey and black respectively) for the experimental subgroup in group 3-a.

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In Figure 9 it is visible that the results of the final test are much higher than those of the placement test compared with the group as a whole. In addition, the results of the placement test reveal a significant difference in the performance of the students with mastery of English and those with rudimentary knowledge. The results of the final test indicate that the number of the students with poor command of English decreased because they acquired knowledge in the course of the experiment. 9

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Figure 10. The results of the placement and the final tests (grey and black respectively) for the control subgroup in group 3-a.

As we can see in Figure 10, the results of the final test for the majority of the subgroup were higher than those of the placement test. Such results imply that the teaching and learning process was quite efficient. Two of the students produced results different from the primary trend because they were influenced by individual factors characteristic of them. 4

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Figure 11. The difference in the results of the placement and the final tests for group 3-a.

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This graph in Figure 11 reveals significant scatter of the results, although the efficiency is noticeable for the majority of the group (most of the graph is above the horizontal axis). Besides, the left-hand part of the graph indicates that one of the student’s results were not in line with the rest of the group.

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Figure 12. The difference in the results of the placement and the final tests for the experimental and control subgroups in group 3-a (grey and black respectively).

Figure 12 allows us to come to the conclusion that the results in the experimental subgroup were much better than those in the control one. The scatter in each subgroup was not significant (if we do not take into consideration the results produced by one of the students). It illustrates that different students within this group have similar efficiency of the learning process. The reason for the poor results produced by the above mentioned student was frequent absences from class.

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Figure 13. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair (for group 3-a).

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In Figure 13 there is no marked dependence, although it is evident that the greatest efficiency is achieved when there is maximum difference in the students’ command of English.

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Figure 14. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair for the experimental and control subgroups (grey and black respectively) (group 3-a).

The neural network smoothing for these dependencies was not carried out because the number of points was insufficient. The graph in Figure 14 illustrates that the difference between the experimental and control groups can be caused by the difference in the initial foreign language competence of the students working in the pair.

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Figure 15. The results of the placement and the final tests for group 2-d (grey and black respectively).

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Figure 15 illustrates that the results of the final test are much higher than those of the placement test, which indicates that the teaching and learning process was organized efficiently. 8

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Figure 16. The results of the placement and the final tests (grey and black respectively) for the experimental subgroup in group 2-d.

In Figure 16 we can observe that the results of the final test were much higher than those of the placement test. Besides, according to the results of the placement test, the group was markedly divided into the students with a high level of knowledge and those whose foreign language skills were rudimentary. The final test reveals that by the end of the experiment this division had virtually disappeared.

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Figure 17. The results of the placement and the final tests (grey and black respectively) for the control subgroup in group 2-d.

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Figure 17 illustrates that the teaching and learning process was not efficient enough since the results of the placement and the final tests were virtually the same. The only tangible result was a decrease in the scatter in the students’ level of knowledge. 4

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Figure 18. The difference in the results of the placement and the final tests for group 2-d.

Figure 18 illustrates a noticeable scatter of the results; although for most of the group the efficiency is visible (the major part of the graph is above the horizontal axis). In addition to this, the bend in the middle shows that the group falls into two subgroups with different distribution of the results. 4

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Figure 19. The difference in the results of the placement and the final tests for the experimental and control subgroups in group 2-d (grey and black respectively).

In Figure 19 it can be seen that the experimental subgroup’s results were much higher than the ones in the control subgroup. In the experimental group

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the divergence was much more considerable than in the control one, although the assignments were the same for the whole group. Now we will test the assumption that there is a correlation between the difference in the final results and the divergence of the mentor’s and the mentee’s levels of knowledge. 4

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Figure 20. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair (for group 2-d).

Figure 20 indicates that the greatest efficiency can be obtained if there is a maximum divergence of the students’ levels. 4

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Figure 21. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair for the experimental and control subgroups respectively (group 2-d).

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For these dependencies the neural network smoothing was not carried out because the number of points was insufficient. Figure 21 illustrates that the divergence between the experimental and control groups may result from the initial difference in the command of English between the students’ working in each pair.

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Figure 22. The results of the placement and the final tests (grey and black respectively) for group 1-4.

It is can be observed in Figure 22 that the results of the final test were much better than those of the placement one. It illustrates that the teaching and learning process was organized well. The moderated dependencies are virtually lineal, which means that the test was suitable for this particular group of students. 10

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Figure 23. The results of the placement and the final tests (grey and black respectively) for the experimental subgroup in group 1-4.

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In Figure 23 we can see that the results of the final test were much higher than those of the placement test compared with the group as a whole. In addition, the results of the placement test show a significant difference between the students with mastery of English and those with rudimentary knowledge. The results of the final test illustrate that the number of the students with poor command of English had dropped dramatically by the end of the experiment. 6.5

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Figure 24. The results of the placement and the final tests (grey and black respectively) for the control subgroup in group 1-4.

Figure 24 illustrates that the results of the final and the placement tests are almost similar, which indicates that the learning and teaching process in this subgroup was not efficient.

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Figure 25. The difference in the results of the placement and the final tests for group 1-4.

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In Figure 25 we can observe that the results were incoherent, although for most of the students the progress was conspicuous (a significant part of the graph lies above the horizontal axis).

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Figure 26. The difference in the results of the placement and the final tests for the experimental and control subgroups in group 1-4 (grey and black respectively).

In Figure 26 it is obvious that the results in the experimental subgroup were much better than those in the control one. The scatter in the students’ level of knowledge for the experimental group is not significant, which indicates that the efficiency of the different students’ learning process was noticeably similar. Now it is necessary to ascertain whether the difference in the final results can be explained by the difference in the knowledge of the student who acted as a mentor and the one who was instructed by the latter. 3.5 3.0 2.5 2.0 1.5 1.0 0.5

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Figure 27. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair (for group 1-4).

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Figure 27 shows that the greatest efficiency can be achieved if there is a maximum divergence of the students’ levels. 3.5 3.0

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Figure 28. The dependence of the difference between the placement and final tests’ results for a student (who works in a pair) with a low level of knowledge on the difference between the placement test’s results for the students in the same pair for the experimental and control subgroups (grey and black respectively) (group 1-4).

Neural network smoothing was not carried out for these dependencies due to the paucity of the points. Figure 28 shows that the difference between the experimental and control groups could have been caused by the difference in the initial foreign language competence of the students working in each pair. In order to measure results of the test more precisely, we applied mathematical models. The system of Kolmogorov differential equations for Markovian process with a discrete set of states was proposed in the capacity of such models. Let us describe this model in the simplest case, when the system state is assigned by vector n with integral-valued coordinates {ni}. Let P (n,t) be a probability of being in state n at time point t. We denominate as nji a vector, which differs from n by increase of coordinate j and decrease of coordinate i by 1. Then ni+ is a vector with coordinate i increased by 1, and ni– is a vector with coordinate i decreased by 1. Then “Main Equation” will look as follows:

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are intensities of

transitions from the current state to the state with an increased or decreased coordinate i (Blagoveshchenskaya E. A., Dashkina A. I., Lazovskaya T. V., Ryabukhina V. V., Tarkhov D. A., 2016). We will take group 3-a as an example of applying this model. For this group the average placement test score was 6.28 and the average final test score was 7.67. We will take a pair of numbers, each of which equals 0 if the results of the student’s test are worse than the average, and equals 1 if the results of the student’s test are better than the average in order to evaluate the condition of the pair. Let us assume that the first number refers to the better student in the pair. Then all the pairs in the experimental subgroup were in initial state 10. As a result of learning, three of them passed into state 11, whereas two of them remained in state 10. For the control subgroup the dynamics was divergent. One pair was initially in state 11 and remained in it. One pair passed from state 00 into 01, one pair was initially in state 10 and remained in it, and one pair was initially in state 10 and passed into state 00. Let us assume that the same tendency will remain the same further. In this case the most probable result of the next stage of work without restructuring the groups will be the following condition of the experimental subgroup: four pairs will be in state 11 and one pair will be in state 10. If we proceed from the intrinsic assumption that as a result of the learning process the level of knowledge in the group exponentially goes to 10, we can suppose that after the next stage of work the average will be 8.54, which means that nine out of ten students in the experimental group will score 9 or more. Further, when more data are collected, it will be possible to develop a more precise model of this kind.

CONCLUSION 1. In all the groups the pairs purposefully formed by the teacher delivered a good overall performance. 2. In all the groups the results are much better in the pairs formed by the teacher, than in those that were formed on the basis of the students’ preferences. Perhaps the poorer performance in the latter can be attributed to a minor difference in the level of the students’ competence. It is recommendable to form pairs for collaborative

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REFERENCES Blagoveshchenskaya E. A., Dashkina A. I., Lazovskaya T. V., Ryabukhina V. V., Tarkhov D. A. Neural Network Methods for Construction of Sociodynamic Models Hierarchy//Springer International Publishing Switzerland 2016 L. Cheng et al. (Eds.): ISNN 2016, LNCS 9719, pp. 513–520. Content-based Second Language Instruction. Donna M. Brinton, Marguerite Ann Snow, Marjorie Wesche, the University of Michigan Press, 2006, 283 pages. Ganesan, Edmonds and Spector. The changing nature of instructional design.//Christine Steeples and Chris Jones. Networked Learning: perspectives and issues.//Centre for studies in Advanced Learning Technology (CSALT), Department of Educational Research, Lancaster University, Lancaster, 2003, (second printing with corrections) – 341 pages. Hierarchical system of education as the means of forming scientific schools (pages 67-77). //Levin V. I, Dashkina A. I., Semyonov A. P., Seryogin N. N., Tarkhov D. A. Group Monograph “Education in modern Russia,” Vol. 5, edited by V. I. Levin, Penza, 2015. Polat Ye. S. Modern pedagogic technologies (pages 330-337)//Methods of foreign language teaching: traditions and modernity, edited by Miroliubov A. A., Obninsk, 2012, 464 pages. Tarkhov, D. A., Dashkina, A. I. On hierarchical system of teaching (pages 2023)/“Neirokompiutery” (Neurocomputers) №9, 2015, the Materials of the 13th All-Russian Conference “Neurocomputers and their Application,” Moscow, 2015. Weidlich W (2000) Sociodynamics. A systematic approach to mathematical modelling in the social sciences. Harwood Academic, Amsterdam.

In: Complex Systems Editor: Rebecca Martinez

ISBN: 978-1-53610-860-6 © 2017 Nova Science Publishers, Inc.

Chapter 5

MODELLING ORGANISATION NETWORKS COLLABORATING ON HEALTH AND ENVIRONMENT WITHIN ASEAN P. Mazzega* and C. Lajaunie 1

Geosciences Environment Toulouse, CNRS, University of Toulouse, Toulouse, France 2 Ceric - DICE - International, Comparative and European Law, CNRS, Aix-Marseille University, Aix en Provence, France

ABSTRACT The emergence of infectious diseases is related to environmental factors such as biodiversity loss, land use and land cover changes, and regional impacts of climate change. The threat of worldwide pandemics led to the development of prevention and mitigation strategies implemented via public policies and national/international legal instruments. These measures involve networks of organisations engaged in a wide range of quite disparate activities. We present two methods to evaluate the collaborative potential of a network of 16 organisations and identify measures to promote their coordination: 1) the first method uses Galois lattices to identify groups of organisations and issues forming a nexus able to tackle with the environment and health challenges. It also allows pointing out the divide *

Corresponding Author email: [email protected].

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P. Mazzega and C. Lajaunie between some sub-issues that should be considered together in order to develop an integrated approach of these problems as recommended for example by the One Health initiative; 2) the second method, inspired by the functioning of networks of cerebral cortex areas for the realisation of high-level cognitive functions, analyses the graphs induced by mutual information functions between organisations. Here we evaluate these functions based on the composition of the governing boards (altogether involving 91 organisations) and partnerships (altogether involving 263 organisations) of these organisations. The approach gives the opportunity to assess a priori the effects induced by a change in the profile of the collaborating organisations. The contributions of these two methods are then discussed and compared to other approaches developed for the analysis of social intelligence or socio-cognitive artificial systems. Both methods are illustrated by the analysis of a network of several organisations involved in the management of health and environmental issues in Southeast Asia (hot spot for emerging infectious diseases and for biodiversity). Taken as a whole, our findings show that regional governance in the health-environment sector is polycentric and entangled, and provide guidance for improving governance on the basis of the competences and collaborations of participating organisations.

Keywords: Organisations, networks, environment, health, information, lattice, concepts, One Health, Southeast Asia

1. INTRODUCTION Global changes such as climate or land use changes, lead to the depletion of natural resources, biodiversity erosion, alter food security or health, and have serious impacts on the environment and thus affect sustainability. While issues are ever more pressing, initiatives to address them – programs, legal or political institutions (Phommasack et al., 2013), sociotechnical systems for the monitoring of natural resources and the state of the environment (SERVIRMekong, 2015), assessment and conservation of ecosystem services (Brander and Eppink, 2015), knowledge production and dissemination towards policymakers (Hirsch, 2006) – are growing without really overcoming the issues our societies are facing. One of the factors of the inefficiency of the schemes established to address current environmental challenges reside in the difficulty for the organisations to set up suitable collective actions in order to reach precisely targeted goals, following efficient operational methods at the most appropriate scale of

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intervention. The cognitive basis explaining this difficulty is manifesting in variety of forms: uncertainty and incompleteness of our knowledge of socioenvironmental dynamics; multiplicity and overlapping or even confrontation of the goals underpinning collective action; diversity or divergence of values or principles driving the actors (public or private organisations, governmental or non-governmental, international or local); segregation and empowerment of a political sphere; supremacy of a specific sector regarding public policies; legal regime fragmentation and limited efficiency of legal measures; low interdisciplinary porosity within the academic and scientific sphere, to cite only few symptoms of this chronic disease1. To respond to those challenges, the environmental governance – or the way we see it – has changed from a centralised and national vision to a global and decentralised one (Wälti, 2011). However, to our knowledge, only a few tools allow us to seize the essential empirical outline in relation to a specific environmental issue. For instance, within the perspective of an institutional strengthening of a research partnership on emerging infectious diseases related to environmental changes in Southeast Asia, we have to identify the organisations involved at the regional level. This study area relies on various disciplines (biological and environmental or information and education sciences) and concerns crosscutting issues (human, animal and ecosystem health; land use and land cover changes; climate changes; water cycles; etc.). Obviously, organisations involved as well as legal and political frameworks at the regional and national level should be taken into consideration. Research on the web based on keywords allows identifying the organisations, their roles and missions, the composition of their governing board or the list of their partners (Section 2). Then, we examine the way organisations cooperate – via a formal structuration in a network or a network of networks – and determine their respective structural role within this changing and entangled architecture. In Section 3, we develop an approach based on Galois lattices (and the Formal Concept Analysis) which highlights partial overlapping and merging between themes linked to “environmenthealth” area and organisations involved.

1

We deliberately associate this concept with environmental governance, the World Health Organisation defining the notion of “chronic diseases” as follows: “a disease which has one or more of the following characteristics: is permanent; leaves residual disability; is caused by unreversible pathological alternation; requires special training of the patient for rehabilitation; or may be expected to require a long period of supervision, observation or care.” See WHO, 2004, p.14.

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In Section 4, we pursue the exploitation of a model of organisational network based on mutual information functions considering the potential of co-operation induced by the board’s composition and the partnerships of each member organisation. We discuss our approach in Section 5 and detail some of its specificities relatively to other modelling approaches of complex institutions and their functioning in relation to common resources management (natural resources, knowledge, and services). If this discussion strengthens the relevance of the notion of polycentric governance, it undermines the notion of multi-level governance: indeed our empiric approach underlines the entangled aspect of this sort of governance. The main conclusions of this study are presented in Section 6, and some technical information is gathered in annex (Section 7).

2. MINING THE ORGANISATIONS’ PROFILES The first step of our study consisted in the identification of the main organisations working on health or environment within ASEAN countries thanks to a research on the internet using specific keywords (Lajaunie and Mazzega, 2016a). The governance on health and environment forms an open system so there is no unique or strict criterium to delimit the whole group of organisations to take into account or on the contrary those to put aside. The choice of criteria is at the discretion of the researcher studying the cooperation between organisations on health and environment (and therefore, the results of the analysis will be validated within the framework of those criteria). In this study aiming at presenting the modelling methodology developed for our research, we focus on a set of 13 organisations at the regional level taking an active part in the area of health and environment to which we add the three main international organisations concerned (Table 1). Data about those organisations are obtained from their respective websites listed in annex (Section 7, Table A1). We underline that entities hereafter designated as “organisations” can present different structures: national organisations, regional organisations having offices in several countries within the region (TROPMED), organisation networks (like SEAMEO), international network having components into ASEAN countries (CORDS), multi-regional organisations (like RFEH: Regional Forum on Environment and Health in Southeast and East Asian countries, or AP-BON: Asia-Pacific Biodiversity Observation Network), international organisations collaborating in various ways. The

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finding of the multiplicity of structuration of organisation leads us to take into account the notion of levels of governance as research question, position that will be confirmed by our further analyses (see Sec.4). The notion of governance covering as well the “policy content” (Zürn, 2011; Stone, 2012) we will also consider the regional political framework “ASEAN 2025” (ASEAN, 2015) promoted by ASEAN during the official launch of the regional economic community2 in 2015. Table 1. List of organisations considered in this study (see Table A1 in the Appendix for the sources of information on governing boards and partnerships)

2

01

Acronym SEAMEO

Upper Level -

02

BIOTROP

SEAMEO

03

RECFON

SEAMEO

04

SEARCA

SEAMEO

05

TROPMED

SEAMEO

06

CORDS

-

07

APEIR

CORDS

08 09

MBDS GEOBON

CORDS -

10

APBON

GEOBON

11

ARAHIS

-

12

RFEH

ASEAN +

13

ACB FAO

ASEAN -

OIE WHO

-

Organisation Full Name Southeast Asian Ministers of Education Organisation Southeast Asian Regional Centre for Tropical Biology Seameo Regional Center for Food and Nutrition Southeast Asian Regional Center for Graduate Study and Research in Agriculture Seameo Regional Centre for Tropical Medicine Connecting Organisations for Regional Disease Surveillance Asia Partnership on Emerging Infectious Diseases Research Mekong Basin Disease Surveillance Group on Earth Observations Biodiversity Observation Network Asia-Pacific Biodiversity Observation Network ASEAN Regional Animal Health Information System Regional Forum on Environment and Health in Southeast and East Asian Countries ASEAN Centre for Biodiversity Food and Agriculture Organisation of the United Nations World Organisation for Animal Health World Health Organisation

See http://asean.org/asean-economic-community/(accessed 19 Nov., 2016).

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Most of the keywords used to identify the organisations are closely linked to themes listed in Table 2. The presentation of roles, missions and competences of organisations on different websites are not standardised. Some organisations are presenting them exhaustively while other organisations are less prolific. Organisations sometimes use the same template to display their information. (e.g., organisations depending on SEAMEO). The update of the publish content depends on the good will and on the reactivity of the organisations as well as on their finance. Nevertheless, the website’s content is the main source of information public and easily accessible. This information might be complemented when necessary by the analysis of public documents and reports produced by those organisations. In all cases, even for the political framework of ASEAN 2025, the information used for the research is of textual form: we apply simple tools of terminological search or text mining (Feldman and Sanger, 2007; Clark et al., 2010; in the context of legal studies also see Lajaunie and Mazzega, 2016c; Wagh, 2013). The information collected on the themes related to the activities of the organisations studied is presented in Table 2, using an incidence matrix: a cross in a cell indicates that an organisation (in row) is active on the considered theme (in column). One organisation generally works on several themes linked to health and environment. Reciprocally, one theme corresponds to the activity of different organisations. All the themes chosen are considered through the perspective “health-environment” (surveillance, risks assessment or policy design). The analysis of the Galois lattice coming from this matrix allows the identification of clusters of themes and organisations (that we call “nexus organisations/issues”) and their ranking in Section 3. On the relevant websites, we collected the composition of the governing board of each of the 13 regional organisations (designated by Org. 𝑥𝑘 ) and the list of their partners (Org. 𝑦𝑗 ). This data constitute the basis of the model allowing the assessment of the potential of various types of collaboration of the organisations as exposed in Section 5. Regarding the governing board of an organisation 𝑥𝑘 , we keep the broadest affiliation of its members for our analysis (Org. 𝑦𝑗 ): for instance we mention the ministry of health of one specific country but we do not precise the department or office concerned; or one university without detailing the faculty. The choice of this resolution level is justified by the fact that members of one organisation 𝑦𝑗 (ex. ministry, university) have various opportunities to meet each other within internal boards taking decisions about the inner life of this 𝑦𝑗 and to share information about the themes treated during the meetings of the boards of organisations

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𝑥𝑘1 and 𝑥𝑘2 they belong to. This capacity to share information—which might be an element of the strategy of Org. 𝑦𝑗 – is specially emphasised when the organisations 𝑥𝑘1 et 𝑥𝑘2 are themselves part of another organisation 𝑥𝑘3 of higher level (for instance a network 𝑥𝑘3 of which 𝑥𝑘1 et 𝑥𝑘2 are members). We consider the organisation 𝑥𝑘1 as a member of its own governing board: otherwise the link with the board of another organisation—let us say 𝑥𝑘2 (e.g., APBON)—having one of its members affiliated to 𝑥𝑘1 (ex. ACB) would stay unnoticed. The notion of partnership is fuzzy and does not follow any standard. For instance, MBDS and APEIR are two networks, members of CORDS network: nevertheless if APEIR lists CORDS as one of its partners, MBDS does not mention it. It can concern long term collaborations (technical, scientific, training co-operation), common projects on a certain time period, funding, or indistinct links resulting for the common organisation of events. As above, we kept the broadest regional or national affiliation for a partner. For instance, in the case of multinational or transnational companies, we have chosen the national (or regional when necessary) designation of the branch (e.g., Company X Thailand) over the parent company (Company X). The analysis of partnerships–resulting from a strategy of the organisations- is interesting as it shows the internationalisation of co-operation and the international or regional interest for health-environment issues. Of course, the partnership framework constantly evolves over time.

3. THE ORGANISATIONS/ISSUES LATTICE Which clusters of organisations address common themes on healthenvironment? Which clusters of themes are treated together or conversely appear separately? At which level and through which organisations the themes of human health, animal health and ecosystem health are treated jointly, according to the One Health approach recommendations (WCS, 2004)? Is the theme of biodiversity, which appears to be linked to the emerging and reemerging of infectious diseases (Morand, 2011; Lajaunie et al., 2015), connected with the theme of health? Does the broad field covering together public policies design, capacity building, environmental monitoring and risks management present enough porosity through the organisations involved to resolutely take into account the thematic in health-environment? Which subthemes related to the environment are already linked to health via the activities of the organisations?

Table 2. Incidence matrix linking the organisations (and the ASEAN 2025 policy framework) and the HealthEnvironment related issues. The labels of issues are as follows: HH = human health; AH = animal health; EH = ecosystem health; FS = food security; BD = biodiversity; LU = land use land cover; WR = water resources; CC = climate change; MS = monitoring or survey; RA = risk assessment or risk analysis; DK = data and knowledge management; PD = policy design or implementation; TC = training or capacity building

01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17

Acronym SEAMEO BIOTROP RECFON SEARCA TROPMED CORDS APEIR MBDS GEOBON APBON ARAHIS RFEH ACB FAO OIE WHO ASEAN 2025

HH x x x x x x x

AH

EH

x

x

FS

LU

x x x

x x

BD

x

WR

CC

MS

RA

x x x x x x x x

x

x x

x

x

x x x x

x x x x

x x x x x

x x x

x

x x

x x

x

x

x

x

x

x

x

x x

x x x x

x x x x x x

DK x x x x x x x x x x x x x x x x

PD x x x x x x x x

TC x x x x x x x x

x x x x x x

x x x x x x

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3.1. The Formal Context of Organisations and Issues Elements of responses are provided by constructing and analysing the Galois lattice associated with the Organisations / Issues incidence matrix (Table 2; Sec.2). We will borrow some of the notions of the Formal Concept Analysis (FCA for short), which has greatly contributed to the theoretical development of this approach (Wolff, 1993; Ganter and Wille, 1999) and to its application in particular to many problems arising from the Social Sciences (Freeman and White, 1993), Biological Sciences (Poelmans et al., 2010) or Information Science and Artificial Intelligence (Cimiano et al., 2005; Škopljanac-Mačina and Blašković; 2014, Codocedo and Napoli, 2015). Our objective here is not to introduce a summary of the mathematical aspects of this approach - the previous references describing them in detail, including the latest theoretical developments1 - but rather to show its relevance for the analysis of the functioning of organisational networks and entangled governance. The organisations (as well as the ASEAN 2025 text) presented in rows of Table 2 can be designated as objects, and the issues (columns) are then attributes. A cross at the intersection of the row of the object 𝑥𝑘 and the column of the attribute 𝑎𝑗 means that the object has this attribute and we shall write this binary relation ∈ 𝑅: in our context it means that the organisation 𝑥𝑘 develops an activity linked to the theme 𝑎𝑗 . The formal context consists of the triplet composed of the set of objects (organisations) X={𝑥1 , 𝑥, … 𝑥𝐾 }, the set of attributes (issues) A={𝑎1 , 𝑎2 , … 𝑎𝐽 } and the binary relation R between them. The formation of formal concepts (in the sense of the FCA) uses the two operators, noted ↑ and ↓, defined as follows: for each subset of organisations 𝑋 ⊆ 𝑿 and sub-set of issues 𝐴 ⊆ 𝑨, we have: 𝑋 ↑ = {𝑎𝑗 ∈ 𝑨| 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑥𝑘 ∈ 𝑋, < 𝑥𝑘 , 𝑎𝑗 >∈ 𝑅}

(1)

𝐴↓ = {𝑥𝑘 ∈ 𝑿| 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑎𝑗 ∈ 𝐴, < 𝑥𝑘 , 𝑎𝑗 >∈ 𝑅}

(2)

𝑋 ↑ is the set of all the themes covered by all the organisations in subset 𝑋. 𝐴↓ is the set of all organisations that work on all the themes of subset 𝐴. A formal concept is a pair of subsets < 𝑋, 𝐴 > such that 𝑋 ↑ = 𝐴 and 𝐴↓ = 𝑋. 1

In particular, the International Conferences on Formal Concept Analysis publish proceedings on a yearly basis.

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In terms of socio-environmental governance, we propose to speak of “nexus organisations-issues” (NOI for short). A formal NOI < 𝑋, 𝐴 > is then a pair comprising a) the subset 𝑋 of the only organisations which develop activities on all the themes of subset 𝐴, and b) the subset 𝐴 of the only activities on which work all organisations of 𝑋. 𝐴 is the extension of the nexus, 𝑋 its intention. In addition to identifying these formal NOIs, this approach is of major interest for governance analysis: formal NOIs are organised hierarchically. Indeed the nexus < 𝑋1 , 𝐴1 > is hierarchically less than or equal to the nexus < 𝑋2 , 𝐴2 > if and only if the following two conditions are realized:  

𝐴1 ⊆ 𝐴2 : the set of issues 𝐴1 is included in or equal to the set of issues 𝐴2; 𝑋2 ⊆ 𝑋1 : the set of organisations 𝑋2 is included in or equal to the set of organisations 𝑋1;

and we shall write < 𝑋1 , 𝐴1 >≤< 𝑋2 , 𝐴2 >. This relationship introduces a partial order, < 𝑋2 , 𝐴2 > being a super-NOI for < 𝑋1 , 𝐴1 > (and conversely < 𝑋1 , 𝐴1 > is a sub-NOI for < 𝑋2 , 𝐴2 >). The collection of all formal NOIs equipped with the partial order relation ≤ allows representing the formal context in the form of a lattice, as we shall see.

3.2. Navigating the Organisations/Issues Lattice We use the free software Concept Explorer 1.2 (Yevtushenko, 2000) to represent the lattice graph (Figure 1) corresponding to the formal context2 of Table 2. The lattice contains 45 formal concepts (nodes on the graph) distributed over 8 hierarchical levels (and therefore a height of 7) and 98 links. Figure 1 is based on a representation with reduced labeling: attaching the label of an organisation 𝑥𝑘 to some NOI (or node; e.g., 12 RFEH) means that 𝑥𝑘 occurs in intents of all NOIs reachable by descending paths from this concept to the bottom node; attaching the label of an issue 𝑎𝑗 to some NOI (e.g., BD for biodiversity) means that 𝑎𝑗 lays in extents of all concepts reachable by ascending paths in the lattice from this concept to the top node. The supremum (top node of the lattice graph) is labelled with the sole FAO which indeed is 2

In fact to facilitate the reading of the results we use here as context the transpose of the matrix: the columns of the Table 2 are used as rows and the rows as columns.

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involved in all issues related to health and environment. Near the bottom of the graph, ARAHIS is engaged in the least number of issues considered in the formal context we built, say animal health and data and knowledge management (as can be easily checked from Table 2).

Figure 1. Reduced labelling representation (see text) of the lattice corresponding to the transposed formal context of Table 2. Each formal concept or NOI (nexus organisations/issues) is a node. Wherever an organisation (resp. issue) is attached to a node, an upper grey (resp. lower black) semi-circle is drawn.

The joint of the “policy design and implementation,” “training and capacity building” plus “data and knowledge management” is covering all the other sub-issues and all the organisations in this context. Among the issues, “land use and land cover” (LULC) is covering no other issue but is taken into account by SEARCA and FAO and is acknowledged by the ASEAN 2025 policy framework. Some formal NOIs are also somewhat latent: the higher

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level NOI (node with a X on Figure 1) connecting both the WHO and the regional forum RFEH (Regional forum on environment and health in Southeast and East Asian countries) is also involving the FAO, the ASEAN 2025 policy frame, and in terms of issues, the “human health,” “climate change,” “risk assessment,” “policy design and implementation” and “training and capacity building” (but it does not include “ecosystem health” or “animal health”). This NOI indicates the minimum formal list of themes and organisations that should be gathered, for instance, to strengthen collaborations between WHO and the RFEH. This example shows that it is worth navigating the lattice in order to explore the visible and more latent shared interests of organisations, as well as the lack or weakness of coordination on some issues. Let us further illustrate this point with two other questions of interest. Numerous scientific works show the importance of interactions between animal and human health (e.g., Mackenzie et al., 2001). More than 70% of emerging diseases in human are zoonoses (Jones et al., 2008), Southeast Asia being a hotspot of emergence of infectious diseases. The sub-lattices corresponding to formal NOI of respective labels “human health” and “animal health” are represented in Figure 2A and 2B respectively. Within the context of our study, human health is a sub-domain of public policies and capacity building. Nonetheless, it encompasses ecosystem health which concerns 11 organisations of our cluster. Animal health is related to the broader theme of knowledge and data management. The organisations encompassing the two types of activities are international organisations– FAO, and CORDS network – as well as BIOTROP and APEIR. WHO in particular is not directly developing activities on animal health, and OIE is not yet concerned with human health. Finally, the overlapping of NOI taken into account both animal and human health remains very limited. This limitation is in contradiction with the recommendations within the One Health (Hall and Coghlan, 2011; Zinstaag, 2012) and EcoHealth framework (e.g., Zinsstag et al., 2005; Brown, 2007). Moreover, emergence and distribution of infectious diseases are linked to biodiversity erosion (Morand et al., 2014). This erosion results from converging factors such as habitat fragmentation and degradation due to land use and land cover changes (Patz et al., 2004), hydrological regime modifications at various spatial and time scales (Gordon et al., 2007), environmental disruption induced by climate change (IPCC, 2014). However, in the organisational context we decided to analyse, the only overlapping between the NOIs having human health as label and the NOI concerned by

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biodiversity issues are materialised via FAO, BIOTROP and the ASEAN 2025 policy framework (see Figure 1). The themes of climate change, land use and water resources changes appear as sub-themes of biodiversity but they remain separated from sub-lattices linked to human health (as well as animal and ecosystem health). The apparition of the One Health concept within the political and legal sphere which intends to integrate policies related to human, animal and ecosystem health3 challenges are not benefiting from a supportive context if we considered the organisations we study. Even though international organisations such as FAO and CORDS and to a lesser extent WHO and OIE cover a wide spectrum of issues, they are not able by themselves to take into account all the interactions between environment and health. The implication of national and regional organisations is essential. To better assess the potential of collaboration between those organisations, we analyse now the network of regional organisations considered as a socio-cognitive system.

Figure 2. (Continued). 3

For its part, the HiAP (Health in All Policies) approach promotes health in all policies, including environmental policies. – Dora et al., 2013.

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Figure 2. Sub-lattices labelled by “Human Health” (A) and “Animal Health” (B) respectively. Note that the “Human Health” sub-lattices covers the “Environmental Health” sub-lattice.

4. ORGANISATION NETWORKS AS COGNITIVE SYSTEMS The way Hayek (1945) formulates the issue of the construction of a rational empirical economic order finds an equivalent in the manner to build a governance of the challenges in environment-health (and more generally in the main current environmental dilemmas). Indeed, if all the knowledge and useful data were available, they would not be accessible to a “single mind” able to explore all the possible options and chose one or more options leading to an optimal governance4. Knowledge and data are distributed between a myriad of

4

“The problem is thus in no way solved if we can show that all the facts, if they were known to a single mind (as we hypothetically assume them to be give, to the observing economist), would uniquely determine the solution; instead we must show how a solution is produced

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actors, spread among numerous socio-technical systems so that is the capacity to collaborate which control the quality of the governance. Moreover, interdisciplinary5 research is now focusing on the notion of “single mind” itself. In the highest cognitive functions, the material support of spirit–the brain- considered at a certain scale of organisation, is itself a cluster of cortex regions functionally specialised and interconnected (in relation with artificial consciousness, see Reggia, 2013). Using the analogy “cortex regions/neuronal connectivity” versus “organisations/informational connectivity (cf. Lajaunie and Mazzega, 2016b6) we transposed and adapted a model of emergence of consciousness (Tononi, 2004; Tononi, 2008) to the governance of a network of organisations built to address challenges linked to environment and health.

4.1. Organisation Networks as Information Integration Systems (IIS) The model of information integration system (hereafter IIS) is based on the assessment of mutual information functions between pairs of organisations considered among a cluster 𝑆 of organisations. We proceed as follows: to each organisation 𝑥𝑘 we associate a textual corpus 𝐶𝑘 containing 𝑁𝑘 segments of text 𝑡𝑛𝑘 . In the present study, 𝐶𝑘 contains the list of affiliations of the members of the governing board or the list of names of partner organisations (see Section 2). The concatenation of all corpora constitutes the corpus 𝐶𝑆 of the set 𝑆 of considered organisations. The a posteriori probability 𝑃[𝑡𝑛𝑘 (𝐶𝑚 )] of occurrence of a term 𝑡𝑛𝑘 in any corpus 𝐶𝑚 is usually calculated by dividing the number of occurrences of this term in 𝐶𝑚 by the number of its occurrence in the global corpus 𝐶𝑆 . The same term can appear in many corpora (and will induce an informative link between corresponding organisations). The joint 𝑙 (𝐶 )] 𝑙 probability 𝑃[𝑡𝑛𝑘 (𝐶𝑚 ); 𝑡𝑛′ of co-occurrence of two terms 𝑡𝑛𝑘 and 𝑡𝑛′ in 𝑚 the corpus 𝐶𝑚 is evaluated in the same way. The elementary information 𝑘𝑙 𝑘 𝑙 𝑒𝑛𝑛 ′ (𝑚) between terms 𝑡𝑛 and 𝑡𝑛′ on corpus 𝐶𝑚 is then given by:

by the interactions of people each of whom possesses only partial knowledge” in Hayek (1945), p. 530. 5 From a neurophysiological point of view see e. g. Sperry R., 1984, or the famous book by O. Sacks (1998). 6 In this article we situate this problematic in the context of the analysis of collective action—cf. Oslon, 1965, and in the debate about the possibility of the emergence of thinking institutions discussed in particular by M. Douglas, 1986.

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P. Mazzega and C. Lajaunie 𝑙 (𝐶 )] 𝑃[𝑡 𝑘 (𝐶𝑚 );𝑡𝑛′ 𝑚 } 𝑙 )]𝑃[𝑡 𝑛 𝑚 𝑛′ (𝐶𝑚 )]

𝑘𝑙 𝑛 𝑘 𝑙 𝑒𝑛𝑛 ′ (𝑚)=𝑃[𝑡𝑛 (𝐶𝑚 ); 𝑡𝑛′ (𝐶𝑚 )]ln{ 𝑃[𝑡 𝑘 (𝐶

(3)

The average mutual information between organisations 𝑥𝑘 and 𝑥𝑙 in the pool S is obtained by summing up the elementary information pieces over all pairs of terms and all corpora: 𝑁

𝑁

𝑘𝑙 𝑘 𝑙 𝐼𝐴𝑀𝐼 [𝑥𝑘 , 𝑥𝑙 ]𝐶𝑠 =(𝑁𝑘 × 𝑁𝑙 )−1 ∑𝑀+ 𝑚=1 ∑𝑛=1 ∑𝑛′ =1 𝑒𝑛𝑛′ (𝑚)

(4)

It is important to notice that the intensity of the informational link between two organisations (eq.4) depends on the composition of the cluster of organisations within which this link is considered. Thus, as we have shown (Lajaunie and Mazzega, 2016a, b) if a new organisation come into the cluster, all the informational links between organisations considered by pairs are modified7. It is also possible to add other corpora to the global corpus 𝐶𝑆 —for instance, the text of a convention, an agreement or a public policy (like ASEAN 2025)—and to evaluate their impact on the relative informational positioning of organisations. Finally, the affiliation of an organisation to the set 𝑆 induces a self-information if it shares elements with at least another organisation. So, for instance, the self-informational content of the governing board of 𝑥𝑘 appears when some of its members are affiliated to organisations which are also members of other boards of organisations from 𝑆.

4.2. IIS Induced by Organisations’ Boards The corpus associated to each organisation is constituted of affiliations of members of governing board. The set of boards of the 13 regional organisations (Table 1) gathers members from 91 distinct organisations. There are 140 multiple occurrences of affiliations in the corpus 𝐶𝑆 . Figure 3 shows the non-oriented graph of organisations which links are weighted by the value of the function 𝐼𝐴𝑀𝐼 . This network represents an aspect of the regional governance in health-environment. These values, whether they concern closeness centrality degrees or betweenness centrality degrees put APEIR and RFEH in first position then MBDS, RECFON et TROPMED in second 7

This property coincides with an experience that is commonly possible when one is a member of a social network (virtual or concrete): the interpersonal links change when the configuration of the network is modified by the arrival of new members or the departure of older members.

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position. These high centrality degrees are mainly explained by the fact that these board members affiliations have the higher rates of co-occurrence in other board compositions (RFEH: 29 co-occurrences; MBDS: 17, RECFON: 23, TROPMED: 24). However this does not explain the closeness and betweenness first position of APEIR with only 12 co-occurrences of its board members affiliations in other boards. But APEIR is linked with other organisations having high scores of co-occurrences: with RFEH, TROPMED, RECFON and MBDS through the Ministries of Health of Cambodia and of Lao PDR, and once again with TROPMED and RECFON through the University of Indonesia. Note also that the high density of the graph (0.455) tends to level the values of degrees of centrality to similar scores.

Figure 3. Networks induced by the memberships of the boards of the 13 organisations considered. The links are weighted by the value of the mutual information (multiplied by 1000). The histogram of the AMI values is represented in the bottom left corner (with the AMI maximum likelihood in the interval [0, 75]).

The value of information functions has no direct interpretation but their comparison between pairs of organisations indicates the relative intensity of their links. The histogram of values of AMI functions is right skewed (Figure

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3). The highest values are mainly concerning MBDS, RECFON and TROPMED as well as RFEH and ACB to a lesser extent8. The high mutual information between MBDS and ACB being nor expected nor intuitive, deserves an explanation. Indeed there is zero overlap of the composition of the two governing boards. But all MBDS (say 𝑥𝑘 in eq.4) board members are also in the board of the regional forum RFEH; and in the same way all the ACB (say 𝑥𝑙 in eq.4) board members are also in the RFEH board (say corpus 𝐶𝑚 en eqs.3 and 4). Therefore the strong informational link between MBDS and ACB is mediated by the board of RFEH. A similar situation occurs with MBDS and RECFON which is very favourable to link the issues of diseases survey with food and nutrition at the regional scale. SEAMEO had a decisive role in the constitution of a subnetwork of organisation implied in regional missions of education. Nevertheless, the composition of its board implies ministries of education of ASEAN countries, ministries poorly represented in the boards of other boards of organisations. It leads to a quite paradoxical situation as the integration of SEAMEO is the result of the sole affiliation of its secretariat to boards of few organisations.The identification of the best actions to conduct in order to improve the governance of issues related to environment-health can also follow another strategy: we can identify the weakest informational links between sets of organisations. The exploration of all the possible partitions constituted from a set of 13 organisations would be very computer time-consuming. Thus we limit our study to the 4095 bipartitions of components 𝑆𝜅 and 𝑆𝜋 and look for the minimum expressed by: Φ𝐼𝐼𝑆 [𝑆] = 𝑚𝑖𝑛[𝑆𝜅 ,𝑆𝜋 ]∈Π2(𝑆) {𝐼𝐴𝑀𝐼 [𝑆𝜅 , 𝑆𝜋 ]𝐶𝑆 }

(5)

Π2 (𝑆) being the set of all bipartitions of 𝑆. The histogram of all the values of mutual information functions between two 𝑆𝜅 and 𝑆𝜋 of bipartitions of 𝑆 is presented Figure 4b, with the bipartition which minimises (respectively maximise) this information in Figure 4c (respectively Figure 4a).

8

The top ten scores of 𝐼𝐴𝑀𝐼 are as follows: (MBDS, MBDS)=1435; (MBDS, ACB)=1359; (TROPMED, MBDS)=1282; (RECFON, RECFON)=1262; (RECFON, MBDS)=1256; (MBDS, RFEH)=1237; (TROPMED, TROPMED)=1219; (RECFON, TROPMED)=1205; (TROPMED, RFEH)=951; (TROPMED, ACB)=936.

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Figure 4. A) Bipartition maximising mutual information between its components 𝑆 and 𝑆 (see text); B) Histogram of the values of the mutual information between components 𝑆 and 𝑆 of the 4095 bipartitions of all 13 organisations; C) Bipartition minimising mutual information between its components 𝑆 and 𝑆 .

A better integration of the challenges in health-environment could benefit from a recomposition of the boards of SEAMEO and GEOBON as shown in Figure 4c. A simple way to proceed would be for instance to invite members affiliated to ministries of health or environment of ASEAN countries (and as such members of the board of RFEH notably) into SEAMEO’s board, thus joining ministries of education. Of course, the impact of this scenario on the

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informational structuration of this network could be easily simulated9, as well as other scenarios taking into account the real opportunities to strengthen regional institutional collaborations. We can notice the strong position of the board of MBDS considered in comparison to the composition of all other boards (Figure 4a). Thus, it is possible to consider the composition of boards at the scale of networks of organisations as an instrument for strengthening governance and facilitating collective management of issues requiring a strong integration of knowledge and skills. The same conclusion can be drawn when considering the partnerships built by these organisations.

4.3. IIS induced by Organisations’ Partnerships The governance is also expressed through the partnerships between organisations. In order to analyse this fact, we associate to each organisation the corpus constituted by the list of affiliations of each of its partners (see Sec.2). The global corpus 𝐶𝑆 gathers 263 distinct organisations (including those of Table 1). There are also 114 occurrences of partners in several lists (partners of several organisations that we are studying). This number, higher than the co-occurrences of board members, explains the higher level of mutual information values (compare histograms in Figures 5 and 3). Some organisations show an important number of partners such as SEARCA with its 124 partners. However with few overlapping with the partners of other organisations of the networks, the levels of information remain similar to the others as we can see on the graph, Figure 5. The high density (0.558) of the network induced by the partnerships tends to homogenise centralities (closeness, betweenness) of organisations thus its interpretation is not very significant. A better understanding of the structuring of partnerships is obtained by analysing the distribution of mutual information. The organisations with the greatest number of partners in common with other organisations in the regional network are in decreasing order: APEIR with 20 co-occurrences of partners (among its 32 partners), RECFON with 14 co-occurrences of partners (among 32 partners), then MBDS with 12 co-occurrences (among 10 partners10) and ACB with 10 co-occurrences (among 21 partners). However,

9

This involves modifying the corresponding corpus and estimating the new values of the information functions. 10 The same partner may appear in several other partner lists.

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the 10 pairs with the highest information values11 mainly exhibit TROPMED (19 partners) and ARAHIS (5 partners, including ASEAN and the international organisation OIE). In other words, the importance of their partnership results mainly from mediation by third-party organisations (as we saw in the boards of MBDS and ACB in Sec.5.1).

Figure 5. Networks induced by the partnerships of the 13 organisations considered. The links are weighted by the value of the mutual information. The histogram of the AMI values is represented in the bottom left corner (with the AMI maximum likelihood in the interval [1000, 1250]).

The excellent positioning of the ARAHIS’ partnership in regional healthenvironment governance is confirmed by the analysis of the informative performances of the bipartitions (Figure 6). Indeed, the most favourable bipartition to the development of collaborations through partnerships places ARAHIS’ partnership as interlocutors in front of all the other partnerships (Figure 6a). 11

The top-10 𝐼𝐴𝑀𝐼 values are as follows: (TROPMED, ARAHIS) = 2592; (ARAHIS, RFEH) = 2205; (RECFON, TROPMED) = 2184; (TROPMED, MBDS) = 2125; (TROPMED, RFEH) = 2118; (MBDS, ARAHIS) = 2097; (TROPMED, APEIR) = 2034; (APEIR, ARAHIS)=2002; (ARAHIS, ARAHIS) = 1978; (CORDS, ARAHIS) = 1884.

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Figure 6. A) Bipartition maximising mutual information between its components 𝑆 and 𝑆 (see text); B) Histogram of the values of the mutual information between components 𝑆 and 𝑆 of the 4095 bipartitions of all 13 organisations; C) Bipartition minimising mutual information between its components 𝑆 and 𝑆 .

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This fact is quite remarkable since ARAHIS is basically an information system. On the contrary, the group of partners of GEOBON, APBON and ACB (organisations working on the study and monitoring of biodiversity) remains isolated from other partnerships more oriented towards health (human, animal or ecosystem health) or education, as shown by the least efficient bipartition in Figure 6c. This observation reinforces the findings based previously on totally independent evidences (via the analysis of the lattice of the nexus organisations-issues, Sec.3.2): the theme of biodiversity is not integrated with those of health via the thematic implications of the organisations considered here, and their existing partnerships do not promote such integration. Let us end this section with a general remark: the networks presented in Figure 3 and Figure 5 synthesize a large amount of information, because if the complete network of boards (or partnerships) were to be represented, it would have 91 (resp. 263) organisations (or nodes).

5. DISCUSSION Other formal models of network governance or of their collective functioning are being developed. If their census exceeds the ambition of this study, it is interesting to open a comparative reflection with some of them.

5.1. Comparison with Other Approaches Noriega et al., (2014) develop a characterisation of artificial sociocognitive systems12 to model their structure, functioning and evolution. The idealised generic system is decomposed in the “world subsystem” (including mainly the human or software agents), the institutional system and the set of technological artefacts (Christiaanse et al., 2014). These three large components are interacting following relations that can be organised in a typology and classically requires the specification of integrity conditions. The characteristics of these systems13 proposed by these authors as well as the

12

Those systems involve many rational participants in various kinds of social coordination mediated by technological artefacts like for example in crowd-based systems or electronic markets. 13 E.g., social space and agents; perceivable interactions; system openness; action coordination.

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identification of the elements required for their modelling14 constitute a framework of generic analysis with a high level of abstraction. Our study differs mainly in two ways: a) we do deal properly with a socio-cognitive system in which all the interactions between agents pass through the mediation of technical devices. The life of the governing boards and the activities of partnerships depend above all on interpersonal relations more or less influenced by the institutional frameworks on which they depend15; b) based on information published by the organisations themselves, our approach is resolutely empirical. It is also the empirical nature of our approach that distinguishes it in the first place from very stimulating recent works on holonic institutions (Calabrese et al., 2010). A. Koestler’s (1969) holon is a representational model inspired from biology that considers a living organism as a multi-levelled hierarchy of semi-autonomous sub-wholes, structured into sub-wholes of a lower order, etc. (see also Calabrese, 2010). Each such semi-autonomous subwhole is a holon. In the following of the axiomatization by Pitt et al., (2011) of 6 of the 8 principles proposed by E. Ostrom (1991) as required to ensure a sustainable self-management of common pool resources (CPRs; see also Ostrom, 1999), Diaconescu and Pitt (2015) propose an approach by holonomic institutions of the 8th principle of nested enterprises which concerns the layered or encapsulated CPRs, with local CPRs at the base level. This analysis scheme is adapted to the representation of groups of organisations structured in “formal” and hierarchical networks such as SEAMEO or GEOBON (Tab.1). Some nexus organisations / issues that we have presented could also be represented as holons, in particular by assigning to each organisation the issues that are specific to it and assigning common outcomes to the higher organisation level as higher common goals (see Figure 2 in Diaconescu and Pitt, 2015).

5.2. The entanglement of Governance However, empirical observation shows that the reticular structuring of these organisations is entangled, which seems to defeat the approach by 14

15

E.g., ontology; formalisation of events, actions and activities; regulatory system; agents typology; system dynamics, etc. We also analysed the collaborative potential of organisations evaluated from their roles and missions (Lajaunie and Mazzega, 2016a). But their involvement in a network results primarily from political strategies and opportunities.

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holonic institutions. Indeed, in the most general case of the collective functioning of organisations, the configuration of the arrangements changes (participating organisations, modes of interaction, chosen objectives, incentives, etc.) and is adapted to each objective pursued. For example, the NOI labelled “Human Health” (Figure 1) is a component of the NOI labelled “BIOTROP” (via “APEIR/CORDS”) but also of the NOIs labelled “RFEH” and “RECFON” respectively, neither of which is a component of the NOI “Human Health.” Organisations and issues belong simultaneously to several entities (NOI or “concepts”) superior in the double hierarchy: the issues dealt with by the organisations are entangled and not simply nested. Each organisation simultaneously pursues several objectives with possibly for each of them different partners. Reciprocally within a network, an organisation may pursue only some of the objectives pursued by the network as a whole. In fact from the start, it remains difficult to assign a “level” to the profile of each organisation. As we have noted (Sec.2), many of them are already composed of several offices in different countries, or are networks, or networks of networks, or are developing and structuring their activities following a multiregional geopolitical strategy (such as RFEH, APABON). While the notion of “level” is useful for developing and simplifying discourse, it poses the risk of imposing a vision that is often unrelated to the real practices of organisations or institutions. In summary, the governance in health-environment that we analyse here is decentralised, polycentric (Andersson and Ostrom, 2008), and entangled. Decentralised and polycentric insofar as each organisation has relative autonomy, develops more or less local activities and makes individual decisions or as part of a collective decision-making arena. It is entangled in several ways (and the following list is not exhaustive): a) each organisation participates in the management of several issues, and each issue is supported by several organisations; b) each governing board involves members of several organisations, each of which may be a member of several boards; c) an organisation has collaborative relationships with several partner organisations, and each partner generally collaborates with several organisations; d) Each organisation pursues several missions on the basis of a set of competencies, with each mission or competence being “available” in several organisations. These characteristics of environmental-health governance are probably common to most (if not all) systems developed or emerging to address contemporary socio-environmental issues. The approaches we have presented here allow us to analyse and interpret these structures on the basis of empirical data produced by the involved organisations themselves.

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CONCLUSION We have developed two tools for analysing networks of organisations collaborating in health-environment in Southeast Asia. The first, based on Galois lattices and the use of some notions of Formal Concept Analysis (FCA), makes it possible to identify various organisations/issues nexus (a structure that corresponds to the notion of a “formal concept” of the FCA), say clusters of organisations all working on the same subset of themes or issues. An inspection of the lattice shows in particular that the themes of human health, animal health and ecosystem health are not yet really articulated into the joint competences of regional or international organisations. Similarly, the theme of biodiversity is connected to changes in land use or water resources, but the link is not established with health issues through the implications of organisations16. The method also identifies some cross-fertilisation of themes in health-environment and organisations’ competences that, although latent, could be opportunities to promote regional governance on these issues. The second approach is to establish a network between organisations, based on mutual information functions. The values of these functions are obtained by mining text corpora associated with each organisation. These corpora provide information about the composition of the governing boards, or the list of their partners (they may also describe the roles and competences of each organisation). Collaboration between organisations is considered a highlevel cognitive activity, with the whole network functioning as an integrative information system. The analysis of this network highlights the interrelations between pairs of organisations that are most conducive to their collaboration or, on the contrary, the weakest links. It identifies at the network level the informational links that should be strengthened between organisational subgroups and provides guidance to overcome these barriers to efficient collaborations. Overall, the approaches we are developing are a contribution to the analysis of governance as an “interdisciplinary and transformative concept” (Zumbansen, 2008). They constitute operational tools for conducting analyses of various aspects of the governance of socio-environmental themes, considered from the point of view of the characteristics of the organisations involved, and preserving the empirical properties of a polycentric and entangled governance. 16

For its part, the theme of climate change is considered more in the field of policy design and implementation than in relation to other environmental or health topics.

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ACKNOWLEDGMENTS The Ecology and Environment Institute of the National Center for Scientific Research (InEE CNRS, France) supports the International Multidisciplinary Thematic Network “Biodiversity, Health and Societies in Southeast Asia,” Thailand (PI: S. Morand, CNRS / CIRAD) to which this study contributes.

APPENDIX Table A1: List of organisations considered in this study. Only the compositions of the governing boards and list of partners of the first 13 organisations are analysed on the basis of the information published on the web pages indicated in the last column (B: link to the boards—P: link to the partnerships; all sites have being accessed between November 11 and 19, 2016)

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INDEX A animal health, 121, 123, 124, 127, 128, 130, 142, 148 annihilation, 20, 21, 27 antidiffusion velocity, 50, 66, 67, 68, 69, 70, 72, 73, 79 ASEAN, v, 117, 120, 121, 122, 124, 125, 127, 129, 132, 134, 135, 137, 144 Asia, x, 118, 119, 120, 121, 128, 142, 143, 144, 145, 146 Asian countries, 120, 128 assessment, 98, 118, 122, 124, 128, 131, 147

B biodiversity, ix, 117, 118, 123, 124, 126, 128, 139, 142, 144, 146 biodiversity erosion, 118, 128 board members, 133, 134, 136 brain, 131, 147

C capacity building, 123, 124, 127, 128 causal interpretation, 11, 12 causal relationship, 17 centrality degree, 132

cerebral cortex, x, 118 challenges, ix, 28, 117, 118, 129, 130, 135 Chinese government, 48 classical electrodynamics, 30 command of English, 97, 101, 105, 107, 111, 112 communication, 44 computational modeling, 94 computational theory, 40 concepts, 7, 118, 126, 141 cortex, x, 118, 131 covering, 7, 14, 15, 16, 17, 28, 40, 121, 123, 127 critical value, 50 curriculum, 97

D Department of Education, 116 differential equations, 66, 114 diffusion process, 53 Dirac equation, 91 distribution function, 51, 52, 54, 65 divergence, 7, 110, 111, 114, 119 diversity, viii, 43, 46, 119 dynamical systems, 93, 94

150

Index

E economics, 45, 97, 98, 144 ecosystem health, 119, 123, 124, 128, 129, 139, 142 education, 116, 119, 134, 135, 139 educational process, 96 educators, 96 environment, v, ix, 7, 8, 9, 117, 118, 119, 120, 121, 122, 123, 124, 127, 128, 129, 130, 132, 134, 135, 137, 141, 142, 143, 145, 146 environmental change, 119 environmental factors, ix, 117 environmental issues, x, 118, 141 environmental protection, 98 equality, 52, 84, 85

F final test, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115 foreign language, 96, 98, 100, 107, 108, 114, 116 formal concept, 119, 125, 126, 127, 142, 144 formal context, 125, 126, 127

G Galois lattice, ix, 117, 119, 122, 125, 142, 145 generalized nonlinear Schrödinger equation, v, vii, viii, 49, 50, 53, 73 gravitational field, 50, 51, 53, 54, 58, 60, 64, 69, 70, 72, 74, 79, 80, 83, 87 gravitational force, ix, 50, 51, 57, 86

H health, v, ix, 117, 118, 119, 120, 121, 122, 123, 124, 127, 128, 129, 130, 132, 134,

135, 137, 139, 141, 142, 143, 144, 145, 146, 147, 148 health care, 148 Hilbert space, 4, 11, 12, 14, 15, 18, 19, 21, 22, 23, 26, 27, 30, 33, 34, 36, 39

I infectious diseases, ix, 117, 118, 119, 121, 123, 128, 146 information, iv, vii, 1, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 22, 23, 24, 25, 29, 30, 31, 32, 33, 36, 38, 44, 47, 98, 118, 119, 120, 121, 122, 125, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 142, 143, 144, 146, 147, 148 information exchange, 31 information retrieval, 144 information technology, 44 institutions, 118, 120, 131, 140, 141, 144 instructional design, 116 integration, 59, 131, 134, 135, 139, 148

L land cover changes, ix, 117, 119, 128 language skills, 96, 97, 108 lattice(s), ix, 18, 30, 31, 32, 117, 118, 119, 123, 126, 127, 128, 130, 139, 142, 145 learning process, ix, 95, 96, 101, 102, 104, 105, 106, 108, 109, 111, 113, 115 level of knowledge, 96, 97, 98, 100, 103, 106, 107, 108, 109, 110, 113, 114, 115

M Maxwell equations, 27 measurement, 4, 5, 6, 9, 13, 25, 26, 28, 40 mentorship, vii, ix, 95 methodology, 28, 120, 145 molecular clouds, 50 mutual information function, x, 118, 120, 131, 134, 142

151

Index

N networks, v, ix, 44, 48, 95, 96, 117, 118, 119, 120, 123, 125, 130, 131, 133, 136, 137, 139, 140, 141, 142, 146 neural network, vii, ix, 94, 95, 96, 100, 103, 107, 111, 147

quantum fluctuations, 53 quantum mechanics, vii, 1, 3, 4, 6, 7, 9, 11, 13, 17, 19, 24, 25, 27, 29, 40, 86, 90, 91 quantum realm, 6 quasi-equilibrium, viii, 50, 72, 73, 76, 77, 78, 85, 86

R O OIE, 121, 124, 128, 129, 137 one dimension, 12, 72 One Health, ix, 118, 123, 128, 129, 145, 148 orbital oscillations, 50, 83, 86 Organisations, 118, 120, 121, 122, 123, 125, 126, 132, 136, 141, 144

P placement test, 97, 99, 101, 103, 104, 105, 106, 107, 108, 110, 112, 113, 114, 115 Poisson equation, 51, 58 policy design, 122, 124, 127, 142 principles, 2, 119, 140, 147, 148 probability, 4, 6, 11, 19, 25, 55, 65, 70, 73, 78, 87, 114, 131 probability density function, 55, 78 public policy, 132

Q quantum electrodynamics, vii, viii, 2, 28, 40 quantum field theory, 27

regulatory system, 140 resources, 45, 118, 120, 124, 129, 140, 142

S Schrödinger equation, vii, viii, 49, 50, 53, 73, 76, 78, 86, 89, 91 slow-flowing gravitational condensation, 50 special relativity, 4, 12, 24, 25 special theory of relativity, 23

T teaching and learning process, 96, 101, 102, 104, 105, 108, 109, 111 training, 96, 119, 123, 124, 127 transformation, 22, 60, 76

V virtual learning communities, 96 vocabulary, 97, 98, 99