Random Walk and Diffusion Models: An Introduction for Life and Behavioral Scientists

✍ Scribed by Wolf Schwarz

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 218
• Category: Library

✦ Synopsis

This book offers an accessible introduction to random walk and diffusion models at a level consistent with the typical background of students in the life sciences. In recent decades these models have become widely used in areas far beyond their traditional origins in physics, for example, in studies of animal behavior, ecology, sociology, sports science, population genetics, public health applications, and human decision making. Developing the main formal concepts, the book provides detailed and intuitive step-by-step explanations, and moves smoothly from simple to more complex models. Finally, in the last chapter, some successful and original applications of random walk and diffusion models in the life and behavioral sciences are illustrated in detail. The treatment of basic techniques and models is consolidated and extended throughout by a set of carefully chosen exercises.

✦ Table of Contents

Preface
Contents
1 Introduction
1.1 Random Walk and Diffusion Models in the Life and the Behavioral Sciences
1.2 Two Historic Examples
1.2.1 Galton's Board
1.2.2 The Laws of Fick and the Diffusion of Substance
1.3 Exercises
2 Discrete Random Walks
2.1 The Symmetric Simple Random Walk
2.1.1 Unrestricted Evolution: Some Basic Properties
2.2 The Simple Random Walk with Drift
2.2.1 Unrestricted Evolution: Some Basic Properties
2.2.2 The Forward and the Backward Equation
2.3 Discrete Random Walks in the Presence of Barriers
2.3.1 Absorption Probability with a Single Barrier
2.3.2 Absorption Probability with Two Barriers
2.3.3 Mean First-Passage Time to a Single Barrier
2.3.4 Mean First-Passage Time with Two Barriers
2.3.5 Conditional Mean First-Passage Time with Two Barriers
2.3.6 Generating Functions for First-Passage Problems
2.4 The Correlated Random Walk
2.4.1 Unrestricted Evolution: The Basic Transition Matrix
2.4.2 The Mean and Variance of Correlated Random Walks
2.4.3 Correlated Random Walks Between Absorbing Barriers
2.4.4 Correlated Random Walks with Randomized Initial Directional State
2.5 More General Random Walks
2.5.1 The Ehrenfest Model
2.5.2 The Fisher–Wright Model of Random Genetic Drift
2.6 Exercises
3 Small Steps at High Speed: The Diffusion Limit
3.1 Small Steps at High Speed: The Diffusion Limit
3.2 The Diffusion Equation
3.2.1 From Difference Equations to Differential Equations
3.2.2 Fick's Laws, Galton's Board, and the Diffusion Equation
3.3 An Alternative Derivation: The Moment-Generating Function
3.4 Some Elementary Properties of the Continuous Limit
3.5 Exercises
4 Diffusion with Drift: The Wiener Process
4.1 Unrestricted Evolution
4.1.1 Limit Conditions for Discrete Random Walks with Drift
4.1.2 The Forward Equation in the Presence of Drift
4.1.3 The Backward Equation for the Wiener Process
4.1.4 Explicit Form of the Transition Density
4.2 One Absorbing Barrier
4.2.1 The Forward Equation for First-Passage Times
4.2.2 The Backward Equation for First-Passage Times
4.2.3 An Alternative Derivation: The Moment-Generating Function
4.3 One Reflecting Barrier
4.3.1 Reflecting Barrier in the Absence of Drift
4.3.2 Reflecting Barrier in the Presence of Drift
4.4 Two Absorbing Barriers
4.4.1 Absorption Probability
4.4.2 Mean First-Passage Times
4.4.3 First-Passage Time Distributions
4.4.4 Moment-Generating Functions for Two Barriers
4.5 Two Reflecting Barriers
4.6 The Mixed Case: One Absorbing and One Reflecting Barrier
4.7 The Backward Process
4.8 Exercises
5 More General Diffusion Processes
5.1 Diffusion Processes with State-Dependent Drift and Variance
5.2 The Generalized Forward and Backward Diffusion Equations
5.3 Examples of More General Diffusion Processes
5.3.1 The Ornstein–Uhlenbeck (OU) Process
5.3.2 The Fisher–Wright Process
5.4 Siegert's Equation for First-Passage Times
5.4.1 Application of Siegert's Equation to the Wiener Process
5.5 The Darling and Siegert Equations
5.6 Exercises
6 Differential Equations for Probabilities and Means
6.1 Introduction
6.2 Two Absorbing Barriers: Unconditional Results
6.2.1 Boundary Conditions and Explicit Solutions
6.2.2 Mean Unconditional First-Passage Time
6.2.3 Examples
6.3 Two Absorbing Barriers: Conditional Results
6.3.1 Boundary Conditions and Solutions
6.3.2 Absorption Probabilities
6.3.3 Mean Conditional First-Passage Time
6.3.4 Boundary Conditions for the Conditional Case
6.3.5 Examples
6.4 One Absorbing and One Reflecting Barrier
6.4.1 Boundary Conditions and Solutions
6.4.2 Mean First-Passage Times
6.5 Exercises
7 Applications of Random Walk and Diffusion Models in the Life and Behavioral Sciences
7.1 Stephen J. Gould: The Spread of Excellence
7.2 Modeling First Hitting Times
7.3 Correlated Random Walks: Generalizations and Applications
7.4 Random Walk and Diffusion Models of Animal Movement
7.5 Random Walk and Diffusion Models in Sports
7.6 Random Walk and Diffusion Models of Animal Decision-Making
7.6.1 Sequential Sampling Models of Neuronal Activity in Perceptual Decision-Making
7.6.2 Random Walk Models of Discrimination and Choice Behavior in Animals
7.6.3 Random Walk Models of Group Decision-Making in Animals
7.7 Random Walk and Diffusion Models of Human Decision-Making
7.8 Some Further Applications of Random Walk and Diffusion Models in the Behavioral and Life Sciences
7.9 Exercises
References
I. Books
II. Articles
Index