Handbook of Stochastic Methods

Transcription

1 Springer Series in Synergetics 13 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences von Crispin W Gardiner Neuausgabe Handbook of Stochastic Methods Gardiner schnell und portofrei erhältlich bei beck-shop.de DIE FACHBUCHHANDLUNG Springer 2004 Verlag C.H. Beck im Internet: ISBN

2 1. A Historical Introduction : :::::::::::::::::::::::: Motivation :::::::::::::::::::::::::::::: 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 ofprobability 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: hhx 1 X 2 X 3 X 4 ii : :::::: Significance of Cumulants : ::::::::::::::::: Gaussian and Poissonian Probability Distributions ::::::::: The Gaussian Distribution : ::::::::::::::::: 36

3 XII Central Limit Theorem :::::::::::::::::::: The PoissonDistribution ::::::::::::::::::: 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 acontinuous 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 ::::::::::::: t Example W(t 0 ) dw(t 0 ) :::::::::::::::::::: t 0 84

4 XIII The Stratonovich Integral : ::::::::::::::::: Nonanticipating Functions : ::::::::::::::::: Proof that dw(t) 2 = dt and dw(t) 2+N =0 : :::::::::: 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 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: ANonlinear 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 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 (Kirchoff s 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 9. Bistability, Metastability, and Escape Problems :::::::::::: Diffusion in a Double-Well Potential (One Variable) :::::::: Behaviour for D =0 : :::::::::::::::::::: Behaviour if D 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 ::::::::::::::::: Euler 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 XVII Symbol Index :::::::::::::::::::::::::::::::::: 403 Author Index :::::::::::::::::::::::::::::::::: 407 Subject Index :::::::::::::::::::::::::::::::::: 409