The Langevin Equation

Preface to the Second Edition......Page 8
Preface to the First Edition......Page 14
Contents......Page 18
1.1 Brownian Motion......Page 26
1.2 Einstein's Explanation of the Brownian Movement......Page 31
1.3 The Langevin Equation......Page 36
1.4 Einstein's Method......Page 42
1.5 Necessary Concepts of Statistical Mechanics......Page 48
1.6 Probability Theory......Page 69
1.7 Application to the Langevin Equation......Page 85
1.8 Wiener Process......Page 88
1.9 The Fokker-Planck Equation......Page 93
1.10 Drift and Diffusion Coefficients......Page 101
1.11 Solution of the One-Dimensional Fokker-Planck Equation......Page 105
1.12 The Smoluchowski Equation......Page 108
1.13 Escape of Particles over Potential Barriers — Kramers' Escape Rate Theory......Page 110
1.14 Applications of the Theory of Brownian Movement in a Potential......Page 137
1.15 Rotational Brownian Motion — Application to Dielectric Relaxation......Page 138
1.16 Superparamagnetism — Magnetic After-Effect......Page 146
1.17 Brown's Treatment of Neel Relaxation......Page 153
1.18 Asymptotic Expressions for the Neel Relaxation Time......Page 158
1.19 Ferrofluids......Page 166
1.20 Depletion Effect in a Biased Bistable Potential......Page 168
1.21 Stochastic Resonance......Page 174
1.22 Anomalous Diffusion......Page 177
References......Page 189
2.1 Criticisms of the Langevin Equation......Page 194
2.2 Doob's Interpretation of the Langevin Equation......Page 196
2.3 Nonlinear Langevin Equation with a Multiplicative Noise Term: Ito and Stratonovich Rules......Page 197
2.4 Derivation of Differential-Recurrence Relations from the One-Dimensional Langevin Equation......Page 202
2.5 Nonlinear Langevin Equations in Several Dimensions......Page 204
2.6 Average of the Multiplicative Noise Term in the Langevin Equation for a Rotator......Page 208
2.7 Methods of Solution of Differential-Recurrence Relations Arising from the Nonlinear Langevin Equation......Page 215
2.8 Linear Response Theory......Page 226
2.9 Correlation Time......Page 232
2.10 Linear Response Theory Results for Systems with Dynamics Governed by One-Dimensional Fokker- Planck equations......Page 235
2.11 Smallest Nonvanishing Eigenvalue: The Continued Fraction Approach......Page 239
2.12 Effective Eigenvalue......Page 246
2.13 Evaluation of the Dynamic Susceptibility Using T Tef and /\1......Page 248
2.14 Nonlinear Response of a Brownian Particle Subjected to a Strong External Field......Page 251
References......Page 258
3.1 Ornstein-Uhlenbeck Theory of the Brownian Motion......Page 261
3.2 Stationary Solution of the Langevin Equation — The Wiener-Khinchine Theorem......Page 263
3.3 Brownian Motion of a Harmonic Oscillator......Page 266
3.4 Application to Dielectric Relaxation......Page 268
3.5 Torsional Oscillator Model: Example of the Use of the Wiener Integral......Page 272
References......Page 276
4.1 Introduction......Page 277
4.2 Langevin Equation for Rotation in Two Dimensions......Page 278
4.3 Longitudinal and Transverse Effective Relaxation Times in the Noninertial Limit......Page 281
4.4 Polarisabilities and Dielectric Relaxation Times of a Fixed Axis Rotator with Two Equivalent Sites......Page 286
4.5 Comparison of the Longitudinal Relaxation Time with the Results of the Kramers Theory......Page 303
References......Page 305
5.1 Introduction......Page 307
5.2 Josephson Junction: Dynamic Model......Page 308
5.3 Reduction of the Averaged Langevin Equation for the Junction to a Set of Differential-Recurrence Relations......Page 310
5.4 DC Current-Voltage Characteristics......Page 312
5.5 Linear Response to an Applied Alternating Current......Page 315
5.6 Effective Eigenvalues for the Josephson Junction......Page 318
5.7 Linear Impedance Using the Effective Eigenvalues......Page 323
5.8 Spectrum of the Josephson Radiation......Page 327
References......Page 332
6.1 Introduction......Page 334
6.2 Relaxation Time of the Position Correlation Function......Page 335
6.3 Comparison of Characteristic Times and Evaluation of the Position Correlation Function......Page 342
References......Page 348
7.1 Introduction......Page 350
7.2 Rotational Diffusion in an External Potential: The Langevin Equation Approach......Page 351
7.3 Gilbert's Equation Augmented by a Random Field Term......Page 360
7.4 Brownian Rotation in the Uniaxial Potential......Page 372
7.5 Brownian Rotation in a Uniform DC External Field......Page 392
7.6 Anisotropic Noninertial Rotational Diffusion of an Asymmetric Top in an External Potential......Page 403
References......Page 419
8.1 Introduction......Page 422
8.2 Application to the Single Axis Rotator......Page 423
8.3 Rotation in Three Dimensions: Longitudinal Response......Page 432
8.4 Transverse Response of Uniaxial Particles......Page 452
8.5 Nonlinear Transient Responses in Dielectric and Kerr- Effect Relaxation......Page 461
8.6 Nonlinear Dielectric Relaxation of Polar Molecules in a Strong AC Electric Field: Steady State Response......Page 468
8.7 Dielectric Relaxation and Rotational Brownian Motion in Nematic Liquid Crystals......Page 475
References......Page 490
9.1 Introduction......Page 493
9.2 Uniaxial Superparamagnetic Particles in an Oblique Field......Page 494
9.3 Cubic Anisotropy......Page 515
References......Page 530
10.1 Introduction......Page 532
10.2 Step-On Solution for Noninertial Rotation about a Fixed Axis......Page 533
10.3 Inertial Rotation about a Fixed Axis......Page 537
10.4 Inertial Rotational Brownian Motion of a Thin Rod in Space......Page 555
10.5 Rotational Brownian Motion of a Symmetrical Top......Page 569
10.6 Itinerant Oscillator Model of Rotational Motion in Liquids......Page 582
10.7 Application of the Cage to Ferrofluids......Page 601
Appendix A: Statistical Averages of the Hermite Polynomials of the Angular Velocity Components for Linear Molecules......Page 614
Appendix B: Averages of the Angular Velocities Components......Page 615
Appendix C: Evaluation of cos0(E) in the Low Damping Limit......Page 619
Appendix D: Sack's Continued Fraction Solution for the Sphere......Page 621
References......Page 622
11.1 Discrete and Continuous Time Random Walks......Page 625
11.2 A Fractional Diffusion Equation for the Continuous Time Random Walk Model......Page 628
11.3 Divergence of Global Characteristic Times in Anomalous Diffusion......Page 646
11.4 Inertial Effects in Anomalous Relaxation......Page 656
11.5 Barkai and Silbey' s Form of the Fractional Klein- Kramers Equation......Page 666
11.6 Anomalous Diffusion in a Periodic Potential......Page 679
11.7 Fractional Langevin Equation......Page 690
Appendix: Fractal Dimension Anomalous Exponents and Random Walks......Page 695
References......Page 697
Index......Page 700