Nonlinear Fokker-Planck equations: fundamentals and applications
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Series: | Springer series in synergetics
Springer complexity |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 404 S. graph. Darst. |
| ISBN: | 3540212647 9783642059544 |
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| 650 | 4 | |a Fokker-Planck equation | |
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Record in the Search Index
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| adam_text | TILL DANIEL FRANK NONLINEAR FOKKER-PLANCK EQUATIONS FUNDAMENTALS AND
APPLICATIONS WITH 86 FIGURES AND 18 TABLES FYJ SPRINGER 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 FUNCTIONAL
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
7 3 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 SCHLOGL S DECOMPOSITION OF
KULLBACK MEASURES . . . 100 4.8 FIRST, SECOND, AND THIRD CHOICE
THERMOSTATISTICS 107 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 I^M^-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 XII CONTENTS 6 ENTROPY
FOKKER-PLANCK EQUATIONS 213 6.1 EXISTENCE AND UNIQUENESS OF STATIONARY
SOLUTIONS* 215 6.2 ENTROPY INCREASE AND ANOMALOUS DIFFUSION 218 6.2.1
ENTROPY INCREASE 218 6.2.2 ANOMALOUS DIFFUSION 219 6.3 DRIFT- AND
DIFFUSION FORMS 222 6.4 ENTROPY AND INFORMATION MEASURES 224 6.4.1
PROPERTIES* 224 6.4.2 EXAMPLES 231 6.5 EXAMPLES AND APPLICATIONS 246
6.5.1 POROUS MEDIUM EQUATION 246 6.5.2 T 5 Q -ENTROPY FOKKER-PLANCK
EQUATION 250 6.5.3 SHARMA-MITTAL ENTROPY FOKKER-PLANCK EQUATION 261
6.5.4 FOKKER-PLANCK EQUATIONS FOR FERMIONS AND BOSONS ... 280 6.5.5
MULTIVARIATE GENERALIZATIONS 284 6.5.6 METAL ELECTRON MODEL, BLACK BODY
RADIATION MODEL, AND PLANCK S RADIATION FORMULA 288 6.5.7 POPULATION
DYNAMICS 296 7 GENERAL NONLINEAR FOKKER*PLANCK EQUATIONS 299 7.1 LINEAR
STABILITY ANALYSIS 299 7.1.1 STATIONARY SOLUTIONS 299 7.1.2 STABILITY OF
STATIONARY SOLUTIONS 300 7.1.3 ON AN ADDITIONAL STABILITY COEFFICIENT*
305 7.2 FREE ENERGY AND LYAPUNOV FUNCTIONAL ANALYSIS 306 7.2.1 MOVING
FRAME TRANSFORMATIONS 306 7.2.2 DERIVATION OF ENTROPY AND INFORMATION
MEASURES 310 7.2.3 DERIVATION OF LOCAL LYAPUNOV FUNCTIONALS 312 7.2.4
DERIVATION OF LYAPUNOV FUNCTIONALS 317 7.3 EXAMPLES 324 7.3.1 TRAVELING
WAVES 324 7.3.2 REENTRANT NOISE-INDUCED PHASE TRANSITIONS 329 7.3.3
SYSTEMS WITH MULTISTABLE VARIABILITY 332 7.4 APPLICATIONS 341 7.4.1
LANDAU FORM AND PLASMA PARTICLES 341 7.4.2 BUNCH-PARTICLE DISTRIBUTIONS
OF PARTICLE BEAMS 342 7.4.3 NOISE GENERATOR 349 7.4.4
ACCURACY-FLEXIBILITY TRADE-OFF 351 7.5 BIBLIOGRAPHIC NOTES 360 8
EPILOGUE 367 REFERENCES 371 INDEX 401
|
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| indexdate | 2024-07-09T20:07:43Z |
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| isbn | 3540212647 9783642059544 |
| language | English |
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| physical | XII, 404 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
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| publisher | Springer |
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| series2 | Springer series in synergetics Springer complexity |
| spelling | Frank, Till Daniel Verfasser aut Nonlinear Fokker-Planck equations fundamentals and applications Till Daniel Frank Berlin [u.a.] Springer 2005 XII, 404 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer series in synergetics Springer complexity Fokker-Planck, Équation de Teoria de campos larpcal Équations différentielles non linéaires Équations différentielles stochastiques Differential equations, Nonlinear Fokker-Planck equation Stochastic differential equations Fokker-Planck-Gleichung (DE-588)4126333-9 gnd rswk-swf Fokker-Planck-Gleichung (DE-588)4126333-9 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013181611&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Frank, Till Daniel Nonlinear Fokker-Planck equations fundamentals and applications Fokker-Planck, Équation de Teoria de campos larpcal Équations différentielles non linéaires Équations différentielles stochastiques Differential equations, Nonlinear Fokker-Planck equation Stochastic differential equations Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| subject_GND | (DE-588)4126333-9 |
| title | Nonlinear Fokker-Planck equations fundamentals and applications |
| title_auth | Nonlinear Fokker-Planck equations fundamentals and applications |
| title_exact_search | Nonlinear Fokker-Planck equations fundamentals and applications |
| title_full | Nonlinear Fokker-Planck equations fundamentals and applications Till Daniel Frank |
| title_fullStr | Nonlinear Fokker-Planck equations fundamentals and applications Till Daniel Frank |
| title_full_unstemmed | Nonlinear Fokker-Planck equations fundamentals and applications Till Daniel Frank |
| title_short | Nonlinear Fokker-Planck equations |
| title_sort | nonlinear fokker planck equations fundamentals and applications |
| title_sub | fundamentals and applications |
| topic | Fokker-Planck, Équation de Teoria de campos larpcal Équations différentielles non linéaires Équations différentielles stochastiques Differential equations, Nonlinear Fokker-Planck equation Stochastic differential equations Fokker-Planck-Gleichung (DE-588)4126333-9 gnd |
| topic_facet | Fokker-Planck, Équation de Teoria de campos Équations différentielles non linéaires Équations différentielles stochastiques Differential equations, Nonlinear Fokker-Planck equation Stochastic differential equations Fokker-Planck-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013181611&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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