Nonlinear Fokker-Planck equations: fundamentals and applications

413 Pages·2005·2.189 MB·English

by  T.D. Frank

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Till Daniel Frank
Nonlinear
Fokker–Planck
Equations
Fundamentals and Applications
With86Figuresand18Tables
123
Dr.TillDanielFrank
Universita¨tMu¨nster
Institutfu¨rTheoretischePhysik
Wilhelm-Klemm-Strasse9
48149Mu¨nster,Germany
ISSN0172-7389
ISBN3-540-21264-7 SpringerBerlinHeidelbergNewYork
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This book is dedicated to my parents, wife and daughter.
Preface
Nonlinear Fokker–Planck equations have found applications in various fields
such as plasma physics, surface physics, astrophysics, the physics of polymer
fluids and particle beams, nonlinear hydrodynamics, theory of electronic cir-
cuitryandlaserarrays,engineering,biophysics,populationdynamics,human
movement sciences, neurophysics, psychology and marketing. In spite of the
diversity of these research fields, many phenomena addressed therein have a
fundamental physical mechanism in common. They arise due to cooperative
interactions between the subsystems of many-body systems. These cooper-
ative interactions result in a reduction of the large number of degrees of
freedom of many-body systems and, in doing so, bind the subunits of many-
body systems by means of self-organization into synergetic entities. These
synergetic many-body systems admit low dimensional descriptions in terms
of nonlinear Fokker–Planck equations that capture and uncover the essen-
tialdynamicsunderlyingtheobservedphenomena.Thephenomenathatwill
be addressed in this book range from equilibrium and nonequilibrium phase
transitions and the multistability of systems to the emergence of power law
and cut-off distributions and the distortion of Boltzmann distributions. We
will study possible asymptotic behaviors of systems such as the approach to
stationary distributions and the emergence of nonstationary traveling wave
distributions.Wewillbeconcernedwithnormalandanomalousdiffusionand
we will examine how correlation functions evolve with time in these kinds of
synergetic systems. We will discuss a Fokker–Planck approach to quantum
statistics, linear nonequilibrium thermodynamics, and generalized extensive
and nonextensive thermostatistics.
The aim of this book is to provide an introduction to the theory of non-
linear Fokker–Planck equations and to highlight what systems described by
nonlinear Fokker–Planck equations have in common. Theoretical considera-
tions and concepts will be illustrated by various examples and applications.
Due to the ramifications of the theory of nonlinear Fokker–Planck equa-
tions in various scientific fields, this book is designed for graduate students
and researchers in physics and related fields such as biology, neurophysics,
human movement sciences, and psychology. I hope that this book will make
graduate students interested in the topic of nonlinear Fokker–Planck equa-
VIII Preface
tionsandwillmakeresearchersawareoftheconnectionsbetweenthedifferent
areas in which nonlinear Fokker–Planck equations have been applied so far.
Acknowledgments
I am indebted to Professor H. Haken for inviting me to write a book on
the topic of nonlinear Fokker–Planck equations for the Springer Series of
Synergetics. I am grateful to Professor P.J. Beek and Professor R. Friedrich
forsupportingmystudiesonnonlinearFokker-Planckequations.Iwouldlike
to thank Dr. A. Daffertshofer for many helpful discussions. Finally, I wish
to thank Professor W. Beiglbo¨ck and Ms. B. Reichel-Mayer of the Springer-
Verlag for their help in preparing this manuscript.
Mu¨nster, Till Daniel Frank
September 2004
Contents
1 Introduction.............................................. 1
1.1 Fokker–Planck Equations................................ 1
1.1.1 Brownian Particles and Langevin Equations ......... 1
1.1.2 Many-Body Systems and Mean Field Theory ........ 2
1.2 Phase Transitions and Self-Organization................... 3
1.3 Stochastic Feedback .................................... 5
1.4 Applications ........................................... 7
1.4.1 Collective Phenomena............................. 7
1.4.2 Multistable Systems .............................. 7
1.4.3 Power Law and Cut-Off Distributions ............... 10
1.4.4 Free Energy Systems.............................. 12
1.4.5 Anomalous Diffusion.............................. 15
1.5 Overview.............................................. 16
2 Fundamentals............................................. 19
2.1 Stochastic Processes .................................... 19
2.2 Nonlinear Fokker–Planck Equation ....................... 20
2.2.1 Notation ........................................ 21
2.2.2 Stratonovich Form................................ 21
2.2.3 Transient Solutions ............................... 22
2.2.4 Continuity Equation .............................. 22
2.2.5 Boundary Conditions ............................. 23
2.2.6 Stationary Solutions .............................. 23
2.3 Self-Consistency Equations .............................. 24
2.4 Multistability and Basins of Attraction.................... 25
2.5 Nonlinearity Dimension ................................. 25
2.6 Classifications.......................................... 26
2.7 Derivations ............................................ 28
2.8 Numerics.............................................. 28
2.8.1 Path Integral Solutions............................ 28
2.8.2 Fourier and Moment Expansions ................... 29
2.8.3 Finite Difference Schemes ......................... 29
2.8.4 Distributed Approximating Functionals ............. 30
X Contents
3 Strongly Nonlinear Fokker–Planck Equations ............. 31
3.1 Transformation to a Linear Problem ...................... 31
3.2 What Are Strongly Nonlinear Fokker–Planck Equations? .... 33
3.3 Correlation Functions ................................... 36
3.4 Langevin Equations..................................... 36
3.4.1 Two-Layered Langevin Equations .................. 36
3.4.2 Self-Consistent Langevin Equations................. 38
3.4.3 Hierarchies and Correlation Functions............... 39
3.4.4 Numerics........................................ 39
3.5 Stationary Solutions .................................... 42
3.6 H-Theorem for Stochastic Processes....................... 43
3.7 Nonlinear Families of Markov Processes∗ .................. 46
3.7.1 Linear Versus Nonlinear Families of Markov Processes 46
3.7.2 Linear Families of Markov Processes ................ 47
3.7.3 Nonlinear Families of Markov Diffusion Processes..... 48
3.7.4 Markov Embedding............................... 50
3.7.5 Hitchhiker Processes .............................. 50
3.8 Top-Down Versus Bottom-Up Approaches∗ ................ 52
3.9 Transient Solutions and Transition Probability Densities..... 55
3.9.1 Nonequivalence of Transient Solutions
and Transition Probability Densities ................ 55
3.9.2 Gaussian Distributions∗ ........................... 57
3.9.3 Purely Random Processes ......................... 61
3.9.4 Wiener Processes................................. 62
3.9.5 Ornstein–Uhlenbeck Processes ..................... 62
3.9.6 Transient Solutions: Two Examples................. 63
3.10 Shimizu–Yamada Model – Transient Solutions.............. 66
3.11 Fluctuation–Dissipation Theorem......................... 70
4 Free Energy Fokker–Planck Equations .................... 73
4.1 Free Energy Principle ................................... 75
4.2 Maximum Entropy Principle and Relationship
between Noise Amplitude and Temperature................ 76
4.3 H-Theorem for Free Energy Fokker–Planck Equations....... 77
4.4 Boltzmann Statistics.................................... 79
4.5 Linear Nonequilibrium Thermodynamics .................. 80
4.5.1 Derivation of Free Energy Fokker–Planck Equations .. 80
4.5.2 Drift and Diffusion Coefficients..................... 84
4.5.3 Transition Probability Densities
and Langevin Equations........................... 86
4.5.4 Density Functions ................................ 87
4.5.5 Entropy Production and Conservative Force ......... 87
4.5.6 Stationary Solutions .............................. 88
4.5.7 H-Theorem for Systems with Conservative Forces
and Nontrivial Mobility Coefficients ................ 89
Contents XI
4.6 Canonical-Dissipative Systems ........................... 90
4.6.1 Linear Case...................................... 90
4.6.2 Nonlinear Case................................... 92
4.7 Boundedness of Free Energy Functionals∗.................. 96
4.7.1 Distortion Functionals ............................ 97
4.7.2 Kullback Measure and Entropy Inequality ........... 97
4.7.3 Generic Cases
and Schlo¨gl’s Decomposition of Kullback Measures ... 100
4.8 First, Second, and Third Choice Thermostatistics........... 107
5 Free Energy Fokker–Planck Equations
with Boltzmann Statistics ................................ 109
5.1 Stability Analysis....................................... 110
5.1.1 Lyapunov’s Direct Method ........................ 112
5.1.2 Linear Stability Analysis .......................... 113
5.1.3 Self-Consistency Equation Analysis ................. 115
5.1.4 Shiino’s Decomposition of Perturbations............. 118
5.1.5 Generic Cases.................................... 120
5.1.6 Higher-Dimensional Nonlinearities .................. 126
5.1.7 Multiplicative Noise .............................. 130
5.1.8 Norm for Perturbations∗ .......................... 132
5.2 Natural Boundary Conditions............................ 134
5.2.1 Shimizu–Yamada Model – Stationary Solutions ...... 135
5.2.2 Dynamical Takatsuji Model – Basins of Attraction.... 140
5.2.3 Desai–Zwanzig Model............................. 148
5.2.4 Bounded B(M1)-Model ........................... 152
5.3 Periodic Boundary Conditions ........................... 155
5.3.1 Cluster Amplitude and Cluster Phase............... 156
5.3.2 KSS Model – Cluster Amplitude Dynamics .......... 157
5.3.3 Mean Field HKB Model – Cluster Phase Dynamics ... 167
5.4 Characteristics of Bifurcations ........................... 183
5.4.1 Stability and Disorder ............................ 183
5.4.2 Emergence of Collective Behavior .................. 184
5.4.3 Multistability and Symmetry ...................... 186
5.4.4 Continuous and Discontinuous Phase Transitions ..... 187
5.5 Applications ........................................... 188
5.5.1 Ferromagnetism.................................. 188
5.5.2 Synchronization .................................. 191
5.5.3 Isotropic-nematic Phase Transitions
and Maier–Saupe Model........................... 195
5.5.4 Muscular Contraction............................. 202
5.5.5 Network Models for Group Behavior ................ 206
5.5.6 Multistable Perception-Action Systems.............. 209