Brownian Motion Flucuations, Dynamics, and Applications

by Mazo, Robert M.

• ISBN13: 9780198515678
• ISBN10: 0198515677
• Format: Hardcover
• Copyright: 2002-04-25
• Publisher: Clarendon Press

Summary

Brownian motion - the incessant motion of small particles suspended in a fluid - is an important topic in statistical physics and physical chemistry. This book studies its origin in molecular scale fluctuations, its description in terms of random process theory and also in terms of statistical mechanics. A number of new applications of these descriptions to physical and chemical processes, as well as statistical mechanical derivations and the mathematical background are discussed in detail. Graduate students, lecturers, and researchers in statistical physics and physical chemistry will find this an interesting and useful reference work.

Table of Contents

Historical Background	p. 1
Robert Brown	p. 1
Between Brown and Einstein	p. 3
Albert Einstein	p. 5
Marian von Smoluchowski	p. 7
Molecular Reality	p. 8
The Scope of this Book	p. 10
Probability Theory	p. 11
Probability	p. 11
Conditional Probability and Independence	p. 14
Random Variables and Probability Distributions	p. 16
Expectations and Particular Distributions	p. 18
Characteristic Function; Sums of Random Variables	p. 23
Conclusion	p. 25
Stochastic Processes	p. 26
Stochastic Processes	p. 26
Distribution Functions	p. 27
Classification of Stochastic Processes	p. 29
The Fokker-Planck Equation	p. 33
Some Special Processes	p. 35
Calculus of Stochastic Processes	p. 37
Fourier Analysis of Random Processes	p. 40
White Noise	p. 43
Conclusion	p. 45
Einstein-Smoluchowski Theory	p. 46
What is Brownian Motion?	p. 46
Smoluchowski's Theory	p. 48
Smoluchowski Theory Continued	p. 52
Einstein's Theory	p. 54
Diffusion Coefficient and Friction Constant	p. 57
The Langevin Theory	p. 59
Stochastic Differential Equations and Integrals	p. 62
The Langevin Equation Revisited	p. 62
Stochastic Differential Equations	p. 64
Which Rule Should Be Used?	p. 67
Some Examples	p. 69
Functional Integrals	p. 71
Functional Integrals	p. 71
The Wiener Integral	p. 72
Wiener Measure	p. 74
The Feynman-Kac Formula	p. 76
Feynman Path Integrals	p. 78
Evaluation of Wiener Integrals	p. 79
Applications of Functional Integrals	p. 82
Some Important Special Cases	p. 83
Several Cases of Interest	p. 83
The Free Particle	p. 83
The Distribution of Displacements	p. 85
The Harmonically Bound Particle	p. 87
A Particle in a Constant Force Field	p. 92
The Uniaxial Rotor	p. 93
An Equation for the Distribution of Displacements	p. 94
Discussion	p. 95
The Smoluchowski Equation	p. 97
The Kramers-Klein Equation	p. 97
The Smoluchowski Equation	p. 98
Elimination of Fast Variables	p. 101
The Smoluchowski Equation Continued	p. 104
Passage over Potential Barriers	p. 105
Concluding Remarks	p. 108
Random Walk	p. 111
The Random Walk	p. 111
The One-Dimensional Pearson Walk	p. 112
The Biased Random Walk	p. 114
The Persistent Walk	p. 117
Boundaries and First Passage Times	p. 120
Random Remarks on Random Walks	p. 125
Statistical Mechanics	p. 127
Molecular Distribution Functions	p. 127
The Liouville Equation	p. 129
Projection Operators--The Zwanzig Equation	p. 131
Projection Operators--The Mori Equation	p. 133
Concluding Remarks	p. 136
Stochastic Equations from a Statistical Mechanical Viewpoint	p. 138
The Langevin Equation A Heuristic View	p. 138
The Fokker-Planck Equation--A Heuristic View	p. 141
What is Wrong with these Derivations?	p. 144
Eliminating Fast Processes	p. 146
The Distribution Function	p. 153
Discussion	p. 157
Two Exactly Treatable Models	p. 159
Two Illustrative Examples	p. 159
Brownian Motion in a Dilute Gas	p. 159
Discussion	p. 162
The Particle Bound to a Lattice	p. 163
The One-Dimensional Case	p. 167
Discussion	p. 169
Brownian Motion and Noise	p. 170
Limits on Measurement	p. 170
Oscillations of a Fiber	p. 171
A Pneumatic Example	p. 174
Electrical Systems	p. 178
Discussion	p. 181
Diffusion Phenomena	p. 183
Brownian Motion in Configuration Space	p. 183
Diffusion Controlled Reactions	p. 183
The Effect of Forces	p. 187
The Coagulation of Colloids	p. 191
Taylor Diffusion	p. 192
Rotational Diffusion	p. 197
Rotational Diffusion	p. 197
Fluorescence Depolarization	p. 201
Non-Spherical Brownian Particles	p. 204
Concluding Remarks	p. 207
Polymer Solutions	p. 208
A Model for Dilute Solutions of Polymers	p. 208
Hydrodynamic Interaction	p. 210
The Equation of Motion	p. 212
Diffusion and Intrinsic Viscosity	p. 214
Historical Remarks and Additional Reading	p. 219
Interacting Brownian Particles	p. 222
Effects of Concentration	p. 222
The Fokker-Planck Equation	p. 223
The Multiparticle Smoluchowski Equation	p. 226
The Diffusion Coefficient	p. 228
The Viscosity	p. 235
Concluding Remarks	p. 238
Dynamics, Fractals, and Chaos	p. 240
Brownian Dynamics	p. 240
Brownian Paths as Fractals	p. 246
Brownian Motion and Chaos	p. 251
Concluding Remarks	p. 257
The Applicability of Stokes' Law	p. 258
Functional Calculus	p. 260
An Operator Identity	p. 263
Euler Angles	p. 264
The Oseen Tensor	p. 266
Mutual Diffusion and Self-Diffusion	p. 268
Mutual Diffusion	p. 268
Self-Diffusion	p. 269
Relation between D[subscript m] and D[subscript s]	p. 269
References	p. 271
Index	p. 285
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