Path Integrals For Stochastic Processes: An Introduction

Description

This book provides an introductory albeit solid presentation of path integration techniques as applied to the field of stochastic processes. The subject began with the work of Wiener during the 1920's, corresponding to a sum over random trajectories, anticipating by two decades Feynman's famous work on the path integral representation of quantum mechanics. However, the true trigger for the application of these techniques within nonequilibrium statistical mechanics and stochastic processes was the work of Onsager and Machlup in the early 1950's. The last quarter of the 20th century has witnessed a growing interest in this technique and its application in several branches of research, even outside physics (for instance, in economy).The aim of this book is to offer a brief but complete presentation of the path integral approach to stochastic processes. It could be used as an advanced textbook for graduate students and even ambitious undergraduates in physics. It describes how to apply these techniques for both Markov and non-Markov processes. The path expansion (or semiclassical approximation) is discussed and adapted to the stochastic context. Also, some examples of nonlinear transformations and some applications are discussed, as well as examples of rather unusual applications. An extensive bibliography is included. The book is detailed enough to capture the interest of the curious reader, and complete enough to provide a solid background to explore the research literature and start exploiting the learned material in real situations. Contents Preface 1. Stochastic Processes: A Short Tour 1.1 Stochastic Process 1.2 Master Equation 1.3 Langevin Equation 1.4 Fokker–Planck Equation 1.5 Relation Between Langevin and Fokker–Planck Equations 2. The Path Integral for a Markov Stochastic Process 2.1 The Wiener Integral 2.2 The Path Integral for a General Markov Process 2.3 The Recovering of the Fokker–Planck Equation 2.4 Path Integrals in Phase Space 2.5 Generating Functional and Correlations 3. Generalized Path Expansion Scheme I 3.1 Expansion Around the Reference Path 3.2 Fluctuations Around the Reference Path 4. Space-Time Transformation I 4.1 Introduction 4.2 Simple Example 4.3 Fluctuation Theorems from Non-equilibrium Onsager– Machlup Theory 4.4 Brownian Particle in a Time-Dependent Harmonic Potential 4.5 Work Distribution Function 5. Generalized Path Expansion Scheme II 5.1 Path Expansion: Further Aspects 5.2 Examples 5.2.1 Ornstein–Uhlenbeck Problem 5.2.2 Simplified Prey-Predator Model 6. Space-Time Transformation II 6.1 Introduction 6.2 The Diffusion Propagator 6.3 Flow Through the Infinite Barrier 6.4 Asymptotic Probability Distribution 6.5 General Localization Conditions 6.6 A Family of Analytical Solutions 6.7 Stochastic Resonance in a Monostable Non-Harmonic Time-Dependent Potential 7. Non-Markov Processes: Colored Noise Case 7.1 Introduction 7.2 Ornstein–Uhlenbeck Case 7.3 The Stationary Distribution 7.4 The Interpolating Scheme 7.4.1 Stationary Distributions 8. Non-Markov Processes: Non-Gaussian Case 8.1 Introduction 8.2 Non-Gaussian Process η 8.3 Effective Markov Approximation 9. Non-Markov Processes: Nonlinear Cases 9.1 Introduction 9.2 Nonlinear Noise 9.2.1 Polynomial Noise 9.2.2 Exponential Noise 9.3 Kramers Problem 10. Fractional Diffusion Process 10.1 Short Introduction to Fractional Brownian Motion 10.2 Fractional Brownian Motion: A Path Integral Approach 10.3 Fractional Brownian Motion: The Kinetic Equation 10.4 Fractional Brownian Motion: Some Extensions 10.4.1 Case 1 10.4.2 Case 2 10.5 Fractional Levy Motion: Path Integral Approach 10.5.1 Gaussian Test 10.5.2 Kinetic Equation 10.6 Fractional Levy Motion: Final Comments 11. Feynman–Kac Formula, the Influence Functional 11.1 Feynman–Kac formula 11.2 Influence Functional: Elimination of Irrelevant Variables 11.2.1 Example: Colored Noise 11.2.2 Example: Lotka–Volterra Model 11.3 Kramers Problem 12. Other Diffusion-Like Problems 12.1 Diffusion in Shear Flows 12.2 Diffusion Controlled Reactions 12.2.1 The Model 12.2.2 Point of View of Path Integrals 12.2.3 Results for the Reaction A + B → B 13. What was Left Out Appendix A Space-Time Transformation: Definitions and Solutions A.1 Definitions A.2 Solutions Appendix B Basics Definitions in Fractional Calculus Bibliography Index