An Introduction to Stochastic Processes in Physics : Containing "On the Theory of Brownian Motion" by Paul Langevin, Translated by Anthony Gythiel

Preface and Acknowledgments xi Random Variables 1(6) Random and Sure Variables 1(1) Assigning Probabilities 2(2) The Meaning of Independence 4(3) Problems: 1.1. Coin Flipping, 1.2. Independent Failure Modes 5(2) Expected Values 7(10) Moments 7(2) Mean Sum Theorem 9(1) Variance Sum Theorem 10(2) Combining Measurements 12(5) Problems: 2.1. Dice Parameters, 2.2. Perfect Linear Correlation, 2.3. Resistors in Series, 2.4. Density Fluctuations 14(3) Random Steps 17(6) Brownian Motion Described 17(1) Brownian Motion Modeled 18(1) Critique and Prospect 19(4) Problems: 3.1. Two-Dimensional Random Walk, 3.2. Random Walk with Hesitation, 3.3. Multistep Walk, 3.4. Autocorrelation, 3.5. Frequency of Heads 20(3) Continuous Random Variables 23(10) Probability Densities 23(1) Uniform, Normal, and Cauchy Densities 24(3) Moment-Generating Functions 27(6) Problems: 4.1. Single-Slit Diffraction, 4.2. Moments of a Normal, 4.3. Exponential Random Variable, 4.4. Poisson Random Variable 29(4) Normal Variable Theorems 33(8) Normal Linear Transform Theorem 33(1) Normal Sum Theorem 34(1) Jointly Normal Variables 35(1) Central Limit Theorem 36(5) Problems: 5.1. Uniform Linear Transform, 5.2. Adding Uniform Variables, 5.3. Dependent Normals 39(2) Einstein's Brownian Motion 41(12) Sure Processes 41(2) Wiener Process 43(2) Brownian Motion Revisited 45(1) Monte Carlo Simulation 46(2) Diffusion Equation 48(5) Problems: 6.1. Autocorrelated Process, 6.2. Concentration Pulse, 6.3. Brownian Motion with Drift, 6.4. Brownian Motion in a Plane 49(4) Ornstein-Uhlenbeck Processes 53(10) Langevin Equation 53(1) Solving the Langevin Equation 54(3) Simulating the O-U Process 57(2) Fluctuation-Dissipation Theorem 59(1) Johnson Noise 60(3) Problems: 7.1. Terminal Speed, 7.2. RL Circuit 62(1) Langevin's Brownian Motion 63(12) Integrating the O-U Process 63(3) Simulating Langevin's Brownian Motion 66(2) Smoluchowski Approximation 68(1) Example: Brownian Projectile 69(6) Problems: 8.1. Derivation, 8.2. X-V Correlation, 8.3. Range Variation 72(3) Other Physical Processes 75(10) Stochastic Damped Harmonic Oscillator 75(5) Stochastic Cyclotron Motion 80(5) Problems: 9.1. Smoluchowski Oscillator, 9.2. Critical Damping, 9.3. Oscillator Energy, 9.4. O-U Process Limit, 9.5. Statistical Independence 83(2) Fluctuations without Dissipation 85(12) Effusion 85(3) Elastic Scattering 88(9) Problems: 10.1. Two-Level Atoms, 10.2. Cross-Field Diffusion, 10.3. Mean Square Displacement 94(3) Appendix A: ``On the Theory of Brownian Motion,'' by Paul Langevin, translated by Anthony Gythiel 97(4) Appendix B: Kinetic Equations 101(2) Answers to Problems 103(4) References 107(2) Index 109