Handbook of Stochastic Methods

Transcription

1 C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer

2 Contents 1. A Historical Introduction Motivation I 1.2 Some Historical Examples Brownian Motion Langevin's Equation Birth-Death Processes Noise in Electronic Systems Shot Noise Autocorrelation Functions and Spectra Fourier Analysis of Fluctuating Functions: Stationary Systems Johnson Noise and Nyquist's Theorem Probability Concepts Events, and Sets of Events Probabilities Probability Axioms The Meaning of P(A) The Meaning of the Axioms Random Variables Joint and Conditional Probabilities: Independence Joint Probabilities Conditional Probabilities Relationship Between Joint Probabilities of Different Orders Independence Mean Values and Probability Density Determination of Probability Density by Means of Arbitrary Functions Sets of Probability Zero Mean Values Moments, Correlations, and Covariances The Law of Large Numbers Characteristic Function Cumulant Generating Function: Correlation Functions and Cumulants Example: Cumulant of Order 4: ((^X.X^)) Significance of Cumulants Gaussian and Poissonian Probability Distributions The Gaussian Distribution 36

3 XII Contents Central Limit Theorem The Poisson Distribution Limits of Sequences of Random Variables Almost Certain Limit Mean Square Limit (Limit in the Mean) Stochastic Limit, or Limit in Probability Limit in Distribution Relationship Between Limits Markov Processes Stochastic Processes Markov Process Consistency the Chapman-Kolmogorov Equation Discrete State Spaces More General Measures Continuity in Stochastic Processes Mathematical Definition of a Continuous Markov Process Differential Chapman-Kolmogorov Equation Derivation of the Differential Chapman-Kolmogorov Equation Status of the Differential Chapman-Kolmogorov Equation Interpretation of Conditions and Results Jump Processes: The Master Equation Diffusion Processes the Fokker-Planck Equation Deterministic Processes Liouville's Equation General Processes Equations for Time Development in Initial Time Backward Equations Stationary and Homogeneous Markov Processes Ergodic Properties Homogeneous Processes Approach to a Stationary Process Autocorrelation Function for Markov Processes Examples of Markov Processes The Wiener Process The Random Walk in One Dimension Poisson Process The Ornstein-Uhlenbeck Process Random Telegraph Process The Ito Calculus and Stochastic Differential Equations Motivation Stochastic Integration Definition of the Stochastic Integral Example j W(t')dW{t') 84

4 Contents XIII The Stratonovich Integral Nonanticipating Functions Proof that dw(tf = dt and dw(t) 2+N = O Properties of the Ito Stochastic Integral Stochastic Differential Equations (SDE) Ito Stochastic Differential Equation: Definition Markov Property of the Solution of an Ito Stochastic Differential Equation Change of Variables: Ito's Formula Connection Between Fokker-Planck Equation and Stochastic Differential Equation Multivariable Systems Stratonovich's Stochastic Differential Equation Dependence on Initial Conditions and Parameters Some Examples and Solutions Coefficients Without x Dependence Multiplicative Linear White Noise Process Complex Oscillator with Noisy Frequency Ornstein-Uhlenbeck Process Conversion from Cartesian to Polar Coordinates Multivariate Ornstein-Uhlenbeck Process The General Single Variable Linear Equation Multivariable Linear Equations Time-Dependent Ornstein-Uhlenbeck Process The Fokker-Planck Equation Background Fokker-Planck Equation in One Dimension Boundary Conditions Stationary Solutions for Homogeneous Fokker-Planck Equations Examples of Stationary Solutions Boundary Conditions for the Backward Fokker-Planck Equation Eigenfunction Methods (Homogeneous Processes) Examples First Passage Times for Homogeneous Processes Probability of Exit Through a Particular End of the Interval Fokker-Planck Equations in Several Dimensions Change of Variables Boundary Conditions Stationary Solutions: Potential Conditions Detailed Balance Consequences of Detailed Balance Examples of Detailed Balance in Fokker-Planck Equations 155

5 XIV Contents Eigenfunction Methods in Many Variables Homogeneous Processes First Exit Time from a Region (Homogeneous Processes) Solutions of Mean Exit Time Problems Distribution of Exit Points Approximation Methods for Diffusion Processes Small Noise Perturbation Theories Small Noise Expansions for Stochastic Differential Equations Validity of the Expansion Stationary Solutions (Homogeneous Processes) Mean, Variance, and Time Correlation Function Failure of Small Noise Perturbation Theories Small Noise Expansion of the Fokker-Planck Equation Equations for Moments and Autocorrelation Functions Example Asymptotic Method for Stationary Distributions Adiabatic Elimination of Fast Variables Abstract Formulation in Terms of Operators and Projectors Solution Using Laplace Transform Short-Time Behaviour Boundary Conditions Systematic Perturbative Analysis White Noise Process as a Limit of Nonwhite Process Generality of the Result More General Fluctuation Equations Time Nonhomogencous Systems Effect of Time Dependence in L, Adiabatic Elimination of Fast Variables: The General Case Example: Elimination of Short-Lived Chemical Intermediates Adiabatic Elimination in Haken's Model Adiabatic Elimination of Fast Variables: A Nonlinear Case An Example with Arbitrary Nonlinear Coupling Master Equations and Jump Processes Birth-Death Master Equations One Variable Stationary Solutions Example: Chemical Reaction X ^± A A Chemical Bistable System Approximation of Master Equations by Fokker-Planck Equations Jump Process Approximation of a Diffusion Process The Kramers-Moyal Expansion Van Kampen's System Size Expansion 250

6 Contents XV Kurtz's Theorem Critical Fluctuations Boundary Conditions for Birth-Death Processes Mean First Passage Times Probability of Absorption Comparison with Fokker-Planck Equation Birth-Death Systems with Many Variables Stationary Solutions when Detailed Balance Holds Stationary Solutions Without Detailed Balance (Kirchoffs Solution) System Size Expansion and Related Expansions Some Examples X + A^±2X X^±Y^A Prey-Predator System Generating Function Equations The Poisson Representation Kinds of Poisson Representations Real Poisson Representations Complex Poisson Representations The Positive Poisson Representation Time Correlation Functions Trimolecular Reaction Third-Order Noise Spatially Distributed Systems Background Functional Fokker-Planck Equations Multivariate Master Equation Description Diffusion Continuum Form of Diffusion Master Equation Reactions and Diffusion Combined Poisson Representation Methods Spatial and Temporal Correlation Structures Reaction X^ Y Reactions B + X C,A+X 2X A Nonlinear Model with a Second-Order Phase Transition Connection Between Local and Global Descriptions Explicit Adiabatic Elimination of Inhomogeneous Modes Phase-Space Master Equation Treatment of Flow Flow as a Birth-Death Process Inclusion of Collisions the Boltzmann Master Equation Collisions and Flow Together 339

7 XVI Contents 9. Bistability, Metastability, and Escape Problems Diffusion in a Double-Well Potential (One Variable) Behaviour for D = Behaviour if/) is Very Small Exit Time Splitting Probability Decay from an Unstable State Equilibration of Populations in Each Well Kramers' Method Example: Reversible Denaturation of Chymotrypsinogen Bistability with Birth-Death Master Equations (One Variable) Bistability in Multivariable Systems Distribution of Exit Points Asymptotic Analysis of Mean Exit Time Kramers'Method in Several Dimensions Example: Brownian Motion in a Double Potential Simulation of Stochastic Differential Equations The One Variable Taylor Expansion Euier Methods Higher Orders Multiple Stochastic Integrals The Euler Algorithm Milstein Algorithm The Meaning of Weak and Strong Convergence Stability Consistency Implicit and Semi-implicit Algorithms Vector Stochastic Differential Equations Formulae and Notation Multiple Stochastic Integrals The Vector Euler Algorithm The Vector Milstein Algorithm The Strong Vector Semi-implicit Algorithm The Weak Vector Semi-implicit Algorithm Higher Order Algorithms Stochastic Partial Differential Equations Fourier Transform Methods The Interaction Picture Method Software Resources 391 References 393 Bibliography 399

8 Contents XVII Symbol Index 403 Author Index 407 Subject Index 409