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Handbook of stochastic methods .pdf
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Contents
1. A Historical Introduction
1.1 Motivation 1
1.2 Some Historical Examples 2
1.2.1 Brownian Motion 2
1.2.2 Langevin's Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Birth-Death Processes 8
1.4 Noise in Electronic Systems 11
1.4.1 Shot Noise 11
1.4.2 Autocorrelation Functions and Spectra 15
1.4.3 Fourier Analysis of Fluctuating Functions:
Stationary Systems 17
1.4.4 Johnson Noise and Nyquist's Theorem 18
2. Probability Concepts
2.1 Events, and Sets of Events 21
2.2 Probabilities 22
2.2.1 Probability Axioms .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2 The Meaning of P(A) 23
2.2.3 The Meaning of the Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Random Variables. . . . . . . . . . . . . .. . . . . . .. . . . . . . . .. . . . . . 24
2.3 Joint and Conditional Probabilities: Independence. . . . . . . . . . . . . . 25
2.3.1 Joint Probabilities. . .. . . . . . . . . . .. .. . . . . . . . . . . . . .. . . . . . 25
2.3.2 Conditional Probabilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.3 Relationship Between Joint Probabilities of Different Orders 26
2.3.4 Independence................ . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Mean Values and Probability Density 28
2.4.1 Determination of Probability Density by
Means of Arbitrary Functions 28
2.4.2 Sets of Probability Zero 29
2.5 Mean Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Moments, Correlations, and Covariances 30
2.5.2 The Law of Large Numbers 30
2.6 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7 Cumulant Generating Function:
Correlation Functions and Cumulants 33
2.7.1. Example: Cumulant of Order 4: «Xt X 2X 3X 4)) ••.••••.••.• 35
2.7.2 Significance of Cumulants 35
XIV Contents
2.8 Gaussian and Poissonian Probability Distributions 36
2.8.1 The Gaussian Distribution 36
2.8.2 Central Limit Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8.3 The Poisson Distribution 38
2.9 Limits of Sequences of Random Variables 39
2.9.1 Almost Certain Limit 40
2.9.2 Mean Square Limit (Limit in the Mean) 40
2.9.3 Stochastic Limit, or Limit in Probability 40
2.9.4 Limit in Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.9.5 Relationship Between Limits 41
3. Markov Processes
3.1 Stochastic Processes 42
3.2 Markov Process 43
3.2.1 Consistency - the Chapman-Kolmogorov Equation .. . . . . . 43
3.2.2 Discrete State Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.3 More General Measures 44
3.3 Continuity in Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Mathematical Definition of a Continuous Markov Process . . 46
3.4 Differential Chapman-Kolmogorov Equation 47
3.4.1 Derivation of the Differential
Chapman-KolIhogorov Equation 48
3.4.2 Status of the Differential Chapman-Kolmggorov Equation. 51
3.5 Interpretation of Conditions and Results 51
3.5.1 Jump Processes: The Master Equation.... 52
3.5.2 Diffusion Processes - the Fokker-Planck Equation 52
3.5.3 Deterministic Processes - Liouville's Equation. . . . . . . . . . . 53
3.5.4 General Processes 54
3.6 Equations for Time Development in Initial Time -
Backward Equations 55
3.7 Stationary and Homogeneous Markov Processes 56
3.7.1 Ergodic Properties 57
3.7.2 Homogeneous Processes 60
3.7.3 Approach to a Stationary Process 61
3.7.4 Autocorrelation Function for Markov Processes 64
3.8 Examples of Markov Processes 66
3.8.1 The Wiener Process 66
3.8.2 The Random Walk in One Dimension 70
3.8.3 Poisson Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.8.4 The Ornstein-Uhlenbeck Process 74
3.8.5 Random Telegraph Process 77
4. The Ito Calculus and Stochastic Differential Equations
4.1 Motivation 80
d ., <::t("'h,,<t;r- Tntpor<lt;("\n sn
Contents XV
4.2.1 Definition of the Stochastic Integral . . . . . . . . . . . . . . . . . . . . . 83
1
4.2.2 Example I W(t')dW(t') .. .... . .... . .. .. . . . . . .. .. ... ... 84
10
4.2.3 The Stratonovich Integral 86
4.2.4 Nonanticipating Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2.5 ProofthatdW(t)2=dtanddW(t)2+N=0 87
4.2.6 Properties of the Ito Stochastic Integral . . . . . . . . . . . . . . . . . . 88
4.3 Stochastic Differential Equations (SDE) . . . . . . . . . . . . . . . . . . . . . . . 92
4.3.1 Ito Stochastic Differential Equation: Definition .... . . . . . . . 93
4.3.2 Markov Property of the Solution of an
Ito~tochastic Differential Equation 95
4.3.3 Change of Variables: Ito's Formula 95
4.3.4 Connection Between Fokker-Planck Equation and
Stochastic Differential Equation 96
4.3.5 Multivariable Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3.6 Stratonovich's Stochastic Differential Equation. . . . . . . . . . . 98
4.3.7 Dependence on Initial Conditions and Parameters....... . . 101
4.4 Some Examples and Solutions 102
4.4.1 Coefficients Without x Dependence 102
4.4.2 Multiplicative Linear White Noise Process 103
4.4.3 Complex Oscillator with Noisy Frequency. . . . . . . . . . . . . . .. 104
4.4.4 Ornstein-Uhlenbeck Process 106
4.4.5 Conversion from Cartesian to Polar Coordinates. . . . . . . . . . 107
4.4.6 Multivariate Ornstein-Uhlenbeck Process 109
4.4.7 The General Single Variable Linear Equation 112
4.4.8 Multivariable Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.9 Time-Dependent Ornstein-Uhlenbeck Process 115
5. The Fokker-Planck Equation
5.1 Background 117
5.2 Fokker-Planck Equation in One Dimension. . . . . . . . . . . . . . . . . . . . 118
5.2.1 Boundary Conditions 11 8
5.2.2 Stationary Solutions for Homogeneous Fokker-Planck
Equations . . . . . . . . . . 124
5.2.3 Examples of Stationary Solutions........ . 126
5.2.4 Boundary Conditions for the Backward Fokker-Planck
Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.2.5 Eigenfunction Methods (Homogeneous Processes) 129
5.2.6 Examples 132
5.2.7 First Passage Times for Homogeneous Processes 136
5.2.8 Probability of Exit Through a Particular End of the
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 142
5.3 Fokker-Planck Equations in Several Dimensions 143
5.3.1 Change of Variables 144
5.3.2 Boundary Conditions 146
5.3.3 Stationary Solutions: Potential Conditions 146
<; 1 d npt"ilprllhbnf'p 14R
XVI Contents
5.3.5 Consequences of Detailed Balance 150
5.3.6 Examples of Detailed Balance in Fokker-Planck Equations. 155
5.3.7 Eigenfunction Methods in Many Variables -
Homogeneous Processes 165
5.4 First Exit Time from a Region (Homogeneous Processes) 170
5.4.1 Solutions of Mean Exit Time Problems 171
5.4.2 Distribution of Exit Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6. Approximation Methods for Diffusion Processes
6.1 Small Noise Perturbation Theories 177
6.2 Small Noise Expansions for Stochastic Differential Equations 180
6.2.1 Validity of the Expansion.. ... .. . ... .... . . . 182
6.2.2 Stationary Solutions (Homogeneous Processes) 183
6.2.3 Mean, Variance, and Time Correlation Function 184
6.2.4 Failure of Small Noise Perturbation Theories 185
6.3 Small Noise Expansion of the Fokker-Planck Equation 187
6.3.1 Equations for Moments and Autocorrelation Functions 189
6.3.2 Example.... . 192
6.3.3 Asymptotic Method for Stationary Distributions 194
6.4 Adiabatic Elimination of Fast Variables 195
6.4.1 Abstract Formulation in Terms of Operators
and Projectors. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 198
6.4.2 Solution Using Laplace Transform . . . . . . . . . . . . . . . . . . . . . . 200
6.4.3 Short-Time Behavidur 203
6.4.4 Boundary Conditions ;v. . . . . . . . . . . . . . . 205
6.4.5 Systematic Perturbative Analysis 206
6.5 White Noise Process as a Limit of Nonwhite Process 210
6.5.1 Generality of the Result 215
6.5.2 More General Fluctuation Equations 215
6.5.3 Time Nonhomogeneous Systems. . . . . . . . . . . . . . . . . . . . . . . . 216
6.5.4 Effect of Time Dependence inL1 •••••••••••••••••••••••• 217
6.6 Adiabatic Elimination of Fast Variables: The General Case. . . . . .. 218
6.6.1 Example: Elimination of Short-Lived
Chemical Intermediates 218
6.6.2 Adiabatic Elimination in Haken's Model. . . . . . . . . . . . . . . .. 223
6.6.3 Adiabatic Elimination of Fast Variables:
A Nonlinear Case 227
6.6.4 An Example with Arbitrary Nonlinear Coupling 232
7. Master Equations and Jump Processes
7.1 Birth-Death Master Equations - One Variable............ 236
7.1.1 Stationary Solutions 236
7.1.2 Example: Chemical Reaction X .,. A 238
7.1.3 A Chemical Bistable System 241
7.2 Approximation of Master Equations by Fokker-Planck Equations 246
7.2.1 Jump Process Approximation of a Diffusion Process 246
Contents
7.2.2 The Kramers-Moyal Expansion .
7.2.3 Van Kampen's System Size Expansion .
7.2.4 Kurtz's Theorem .
7.2.5 Critical Fluctuations .
7.3 Boundary Conditions for Birth-Death Processes .
7.4 Mean First Passage Times .
7.4.1 Probability of Absorption .
7.4.2 Comparison with Fokker-Planck Equation .
7.5 Birth-Death Systems with Many Variables .
7.5.1 Stationary Solutions when Detailed Balance Holds .
7.5.2 Stationary Solutions Without Detailed Balance
(Kirchoff's Solution) .
7.5.3 System Size Expansion and Related Expansions .
7.6 Some Examples .
7.6.1 X + A .,. 2X .
y k
7.6.2 X;:± Y;:± A .
k y
7.6.3 Prey-Predator System .
7.6.4 Generating Function Equations .
7.7 The Poisson Representation .
7.7.1 Kinds of Poisson Representations .
7.7.2 Real Poisson Representations .
7.7.3 Complex Poisson Representations .
7.7.4 The Positive Poisson Representation .
7.7.5 Time Correlation Functions .
7.7.6 Trimolecular Reaction .
7.7.7. Third-Order Noise .
8. Spatially Distributed Systems
8.1 Background .
8.1.1 Functional Fokker-Planck Equations .
8.2 Multivariate Master Equation Description .
8.2.1 Diffusion .
8.2.2 Continuum Form of Diffusion Master Equation .
8.2.3 Reactions and Diffusion Combined .
8.2.4 Poisson Representation Methods .
8.3 Spatial and Temporal Correlation Structures .
8 3
. kl . .1 Reaction X;:± Y .
k2 k
8.3.2 Reactions B + X ~ C, A + X kt 2X .
8.3.3 A Nonlinear Model with a Second-Order Phase Transition ..
8.4 Connection Between Local and Global Descriptions .
8.4.1 Explicit Adiabatic Elimination of Inhomogeneous Modes .
8.5 Phase-Space Master Equation .
8.5.1 Treatment of Flow .
8.5.2 Flow as a Birth-Death Process .
8.5.3 Inclusion of Collisions - the Boltzmann Master Equation ..
8.5.4 Collisions and Flow Together .
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XVIII Contents
9. Bistability, Metastability, and Escape Problems
9.1 Diffusion in a Double-Well Potential (One Variable) 342
9.1.1 Behaviour for D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 343
9.1.2 Behaviour ifD is Very Small 343
9.1.3 Exit Time 345
9.1.4 Splitting Probability 345
9.1.5 Decay from an lJnstable State. . . . . . . .. . . . . . . .. . . . . . . . . 347
9.2 Equilibration of Populations in Each Well. . . . . . . . . . . . . . . . . . . . 348
9.2.1 Kramers'Method 349
9.2.2 Example: Reversible Denaturation of Chymotrypsinogen .. 352
9.2.3 Bistability with Birth-Death Master Equations
(One Variable) 354
9.3 Bistability in Multivariable Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 357
9.3.1 Distribution of Exit Points..... . 357
9.3.2 Asymptotic Analysis of Mean Exit Time 362
9.3.3 Kramers' Method in Several Dimensions 363
9.3.4 Example: Brownian Motion in a Double Potential. . . .. . . . 366
10. Quantum Mechanical Markov Processes
10.1 Quantum Mechanics of the Harmonic Oscillator. . . . . . . . . . . . . . 373
10.1.1 Interaction with an External Field. . .. . . . . . . . . . . . . . . . . 375
10.1.2 Properties of Coherent States 376
10.2 Density Matrix and Prob~ilities 380
10.2.1 Von Neumann's Equation 382
10.2.2 Glauber-Sudarshan P-Representation ..~ t. . . . . . . . . . . . . 382
10.2.3 Operator Correspondences 383
10.2.4 Application to the Driven Harmonic Oscillator. . . . . . . . . 384
10.2.5 Quantum Characteristic Function. . . . . . . . . . . . . . . . . . . . 386
10.3 Quantum Markov Processes. . . . .. . . . . . . . . . . . . . . . . .. . . . . . . . 388
10.3.1 Heat Bath................. 388
10.3.2 Correlations of Smooth Functions of Bath Operators 389
10.3.3 Quantum Master Equation for a System Interacting
with a Heat Bath 390
10.4 Examples and Applications of Quantum Markov Processes 395
10.4.1 Harmonic Oscillator. . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . 395
10.4.2 The Driven Two-Level Atom. . . . . . . .. . . . . . . . . . .. . . . . 399
10.5 Time Correlation Functions in Quantum Markov Processes 402
10.5.1 Quantum Regression Theorem 404
10.5.2 Application to Harmonic Oscillator
in the P-Representation 405
10.5.3 Time Correlations for Two-Level Atom. . . . . .. . . . . . . . . 408
10.6 Generalised P-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 408
10.6.1 Definition of Generalised P-Representation 409
10.6.2 Existence Theorems 411
10.6.3 Relation to Poisson Representation 413
10.6.4 Operator Identities 414
Contents XIX
10.7 Application of Generalised P- Representations
to Time-Development Equations , . . . . 415
10.7.1 ComplexP-Representation 416
10.7.2 PositiveP-Representation 416
10.7.3 Example......................................... 418
References 421
Bibliography 427
Symbol Index 431
.J
Author Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
Subject Index 437
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