Physics of Stochastic Processes: How Randomness Acts in Time

Physics of Stochastic Processes......Page 5
Contents......Page 7
Preface......Page 13
Part I Basic Mathematical Description......Page 21
1.1 Wiener Process, Adapted Processes and Quadratic Variation......Page 23
1.2 The Space of Square Integrable Random Variables......Page 28
1.3 The Ito Integral and the Ito Formula......Page 35
1.4 The Kolmogorov Differential Equation and the Fokker–Planck Equation......Page 43
1.5 Special Diffusion Processes......Page 47
1.6 Exercises......Page 49
2.1 Bounded Multidimensional Region......Page 51
2.2 From Chapman–Kolmogorov Equation to Fokker–Planck Description......Page 53
2.2.1 The Backward Fokker–Planck Equation......Page 55
2.2.2 Boundary Singularities......Page 57
2.2.3 The Forward Fokker–Planck Equation......Page 60
2.2.4 Boundary Relations......Page 63
2.3 Different Types of Boundaries......Page 64
2.4 Equivalent Lattice Representation of Random Walks Near the Boundary......Page 65
2.4.1 Diffusion Tensor Representations......Page 66
2.4.2 Equivalent Lattice Random Walks......Page 74
2.4.3 Properties of the Boundary Layer......Page 76
2.5 Expression for Boundary Singularities......Page 78
2.6.1 Moments of the Walker Distribution and the Generating Function......Page 81
2.6.2 Master Equation for Lattice Random Walks and its General Solution......Page 82
2.6.3 Limit of Multiple-Step Random Walks on Small Time Scales......Page 85
2.6.4 Continuum Limit and a Boundary Model......Page 88
2.7 Boundary Condition for the Backward Fokker–Planck Equation......Page 89
2.8 Boundary Condition for the Forward Fokker–Planck Equation......Page 91
2.9 Concluding Remarks......Page 92
2.10 Exercises......Page 93
Part II Physics of Stochastic Processes......Page 95
3.1 Markovian Stochastic Processes......Page 97
3.2 The Master Equation......Page 102
3.3 One-Step Processes in Finite Systems......Page 105
3.4 The First-Passage Time Problem......Page 108
3.5 The Poisson Process in Closed and Open Systems......Page 112
3.6 The Two-Level System......Page 119
3.7 The Three-Level System......Page 125
3.8 Exercises......Page 134
4.1 General Fokker–Planck Equations......Page 137
4.2 Bounded Drift–Diffusion in One Dimension......Page 139
4.3 The Escape Problem and its Solution......Page 143
4.4 Derivation of the Fokker–Planck Equation......Page 147
4.5 Fokker–Planck Dynamics in Finite State Space......Page 148
4.6 Fokker–Planck Dynamics with Coordinate-Dependent Diffusion Coefficient......Page 153
4.7 Alternative Method of Solving the Fokker–Planck Equation......Page 160
4.8 Exercises......Page 162
5.1 A System of Many Brownian Particles......Page 165
5.2 A Traditional View of the Langevin Equation......Page 171
5.3 Additive White Noise......Page 172
5.4 Spectral Analysis......Page 177
5.5 Brownian Motion in Three-Dimensional Velocity Space......Page 180
5.6 Stochastic Differential Equations......Page 186
5.7 The Standard Wiener Process......Page 188
5.9 Geometric Brownian Motion......Page 193
5.10 Exercises......Page 196
Part III Applications......Page 199
6.1 Random Walk on a Line and Diffusion: Main Results......Page 201
6.2 A Drunken Sailor as Random Walker......Page 204
6.3 Diffusion with Natural Boundaries......Page 206
6.4 Diffusion in a Finite Interval with Mixed Boundaries......Page 213
6.5 The Mirror Method and Time Lag......Page 220
6.6 Maximum Value Distribution......Page 225
6.7.1 Reflected Diffusion......Page 228
6.7.2 Diffusion in a Semi-Open System......Page 229
6.7.3 Diffusion in an Open System......Page 230
6.8 Exercises......Page 231
7.1 Drift–Diffusion Equation with Natural Boundaries......Page 233
7.2 Drift–Diffusion Problem with Absorbing and Reflecting Boundaries......Page 235
7.3 Dimensionless Drift–Diffusion Equation......Page 236
7.4 Solution in Terms of Orthogonal Eigenfunctions......Page 237
7.5 First-Passage Time Probability Density......Page 246
7.6 Cumulative Breakdown Probability......Page 248
7.7 The Limiting Case for Large Positive Values of the Control Parameter......Page 249
7.8 A Brief Survey of the Exact Solution......Page 252
7.8.1 Probability Density......Page 253
7.8.3 First Moment of the Outflow Probability Density......Page 254
7.8.4 Second Moment of the Outflow Probability Density......Page 255
7.8.5 Outflow Probability......Page 256
7.9 Relationship to the Sturm–Liouville Theory......Page 258
7.10 Alternative Method by the Backward Fokker–Planck Equation......Page 260
7.11 Roots of the Transcendental Equation......Page 269
7.12 Exercises......Page 271
8.1 Definitions and Properties......Page 273
8.2 The Ornstein–Uhlenbeck Process and its Solution......Page 274
8.3 The Ornstein–Uhlenbeck Process with Linear Potential......Page 281
8.4 The Exponential Ornstein–Uhlenbeck Process......Page 286
8.5 Outlook on Econophysics......Page 288
8.6 Exercises......Page 292
9.1 Dynamics of First-Order Phase Transitions in Finite Systems......Page 295
9.2 Condensation of Supersaturated Vapor......Page 297
9.3 The General Multi-Droplet Scenario......Page 306
9.4 Detailed Balance and Free Energy......Page 310
9.5 Relaxation to the Free Energy Minimum......Page 314
9.6 Chemical Potentials......Page 315
9.7 Exercises......Page 316
10.1 The Car-Following Theory......Page 319
10.2 The Optimal Velocity Model and its Langevin Approach......Page 322
10.3 Traffic Jam Formation on a Circular Road......Page 336
10.4 Metastability Near Phase Transitions in Traffic Flow......Page 348
10.5 Car Cluster Formation as First-Order Phase Transition......Page 352
10.6 Thermodynamics of Traffic Flow......Page 358
10.7 Exercises......Page 368
11.1 Equilibrium and Nonequilibrium Phase Transitions......Page 371
11.2 Types of Stochastic Differential Equations......Page 374
11.3 Transformation of Random Variables......Page 378
11.4 Forms of the Fokker–Planck Equation......Page 380
11.5 The Verhulst Model of Third Order......Page 381
11.7 Noise-Induced Instability in Geometric Brownian Motion......Page 384
11.8 System Dynamics with Stagnation......Page 387
11.9 Oscillator with Dynamical Traps......Page 389
11.10 Dynamics with Traps in a Chain of Oscillators......Page 392
11.11 Self-Freezing Model for Multi-Lane Traffic......Page 401
11.12 Exercises......Page 405
12.1 Hopping Models with Zero-Range Interaction......Page 407
12.2 The Zero-Range Model of Traffic Flow......Page 409
12.3 Transition Rates and Phase Separation......Page 411
12.4 Metastability......Page 415
12.5 Monte Carlo Simulations of the Hopping Model......Page 420
12.6 Fundamental Diagram of the Zero-Range Model......Page 423
12.7 Polarization Kinetics in Ferroelectrics with Fluctuations......Page 425
12.8 Exercises......Page 429
Epilog......Page 431
References......Page 433
Index......Page 443