CONTENTS -- Preface to the Tllird Edition -- Contents -- Chapter 1 Historical Background and Introductory Concepts -- 1.1. Brownian motion -- 1.2. Einstein's explanation of Brownian movement -- 1.3. The Langevin equation -- 1.3.1. Calculation of Avogadro's number -- 1.4. Einstein's Method -- 1.5. Essential concepts in Statistical Mechanics -- 1.5.1. Ensemble of systems -- 1.5.2. Phase space -- 1.5.3. Representative point -- 1.5.4. Ergodic hypothesis -- 1.5.5. Calculation of averages -- 1.5.6. Liouville equation -- 1.5.7. Reduction of the Liouville equation -- 1.5.8. Langevin equation for a system with one degree of freedom -- 1.5.9. Intuitive derivation of the Klein-Kramers equation -- 1.5.10. Conditions under which a Maxwellian distribution in the velocities may be deemed to be attained -- 1.5.11. Very-high-damping (VHD) regime -- 1.5.12. Very-low-damping (VLD) regime -- 1.6. Probability theory -- 1.6.1. Random variables and probability distributions -- 1.6.2. The Gaussian distribution -- 1.6.3. Moment-generating fimctious -- 1.6.4. Central limit theorem -- 1.6.5. Random processes -- 1.6.6. Wiener-Khinchin theorem -- 1.7. Application to the Langevin equation -- 1.8. Wiener process -- 1.8.1. Variance of the Wiener process -- 1.8.2. Wiener integrals -- 1.9. The Fokker-Planok equation -- 1.10. Drift and diffusion coefficients -- 1.11. Solution of the one-dimensional Fokker-Planck equation -- 1.12. The Smoluchowski equation -- 1.13. Escape of particles over potential barriers: Kramers' theory -- 1.13.1. Escape rate in the IHD limit -- 1.13.2. Kramers' calculation of the escape rate in the VLD limit -- 1.13.3. Range of validity of the IHD and VLD formulas -- 1.13.4. Extension of Kramers' theory to many dimensions in the IHD limit -- 1.13.5. Langer's treatment of the IHD limit -- 1.13.6. Kramers' formula as a special case of Langer's formula.

1.13.7. Kramers' turn over problem -- 1.14. Applications of the theory of Brownian movement in a potential -- 1.15. Rotational Brownian motion: application to dielectric relaxation -- 1.15.1. Breakdown of the Debye theory at high frequencies -- 1.16. Superparamagnetism: magnetic after-effect -- 1.17. Brown's treatment of Neel relaxation -- 1.18. Asymptotic expressions for the Neel relaxation time -- 1.18.1. Magnetization reversal time in a uniaxial superparamagnet: application of Kramers' method -- 1.18.2. Escape rate formulas for superparamagnets -- 1.19. Ferrofluids -- 1.20. Depletion effect in a biased bistable potential -- 1.21. Stochastic resonance -- 1.22. Anomalous diffusion -- 1.22.1. Empirical formulas for the complex dielectric permittivity -- 1.22.2. Theoretical justification for anomalous relaxation behavior -- 1.22.3. Anomalous dielectric relaxation of an assembly of dipolar molecules -- References -- Chapter 2 Langevin Equations and Methods of Solution -- 2.1. Criticisms of the Langevin equation -- 2.2. Doob's interpretation of the Langevin equation -- 2.3. Nonlinear Langevin equation with a multiplicative noise term: Ito and Stratonovich rules -- 2.4. Derivation of differential-recurrence relations from the one-dimensional Langevin equation -- 2.5. Nonlinear Langevin equation in several dimensions -- 2.6. Average of the multiplicative noise term in the Langevin equation -- 2.6.1. Multiplicative noise term for rotation in space -- 2.6.2. Multiplicative noise terms in the non-inertial limit -- 2.6.3. Explicit average of the noise-induced terms for a planar rotator -- 2.7. Methods of solution of differential-recurrence relations arising from the nonlinear Langevin equation -- 2.7.1. Matrix method -- 2.7.2. Initial conditions -- 2.7.3. Matrix continued-fraction solution of recurrence equations -- 2.8. Linear response theory.

2.9. Integral relaxation time -- 2.10. Linear response theory for systems with dynamics governed by single-variable Fokker-Planck equations -- 2.11. Smallest non-vanishing eigenvalue: continued-fraction approach -- 2.11.1 Evaluation of 1 from a scalar three-term recurrence relation -- 2.11.2 Evaluation of 1 from a matrix three-term recurrence relation -- 2.12. Effective relaxation time -- 2.13. Evaluation of the dynamic susceptibility using int, ef, and 1 -- 2.14. Nonlinear transient response of a Brownian particle -- References -- Chapter 3 Brownian Motion of a Free Particle and a Harmonic Oscillator -- 3.1. Introduction -- 3.2. Omstein-Uhlenbeck theory of Brownian motion -- 3.3. Stationary solution of the Langevin equation: the Wiener-Khinchin theorem -- 3.4. Application to phase diffusion in MRI -- 3.5. Brownian motion of a harmonic oscillator -- 3.6. Rotational Brownian motion of a fixed-axis rotator -- 3.7. Torsional oscillator model: example of the use of the Wiener integral -- References -- Chapter 4 Rotational Brownian Motion About a Fixed Axis inN-Fold Cosine Potentials -- 4.1. Introduction -- 4.2. Langevin equation for rotation about a fixed axis -- 4.3. Longitudinal and transverse effective relaxation times -- 4.4. Polarizabilities and relaxation times of a fixed-axis rotator with two equivalent sites -- 4.4.1. Matrix solution for the longitudinal response -- 4.4.2. Continued-fraction solution for the longitudinal response -- 4.4.3. Comparison of the longitudinal relaxation time with the Kramers theory -- 4.4.4. Continued-fraction solution for the transverse response -- 4.5. Effect of a d. c. bias field on the orientational relaxation of a fixed-axis rotator with two equivalent sites -- 4.5.1. Longitudinal response -- 4.5.2. Transverse response -- 4.5.3. Relaxation times -- References.

Chapter 5 Brownian Motion in a Tilted Periodic Potential: Application to the Josephson Tunnelling Junction -- 5.1. Introduction -- 5.2. Langevin equations -- 5.3. Josephson junction: dynamic model.. -- 5.4. Reduction of the averaged Langevin equation for the junction to a set of differential-recurrence relations -- 5.5. Current-voltage characteristics -- 5.6. Linear response to an applied alternating current -- 5.7. Effective eigenvalues for the Josephson junction -- 5.8. Linear impedance -- 5.9. Spectrum of the Josephson radiation -- 5.10. Nonlinear effects in d. c. and a.c. current-voltage characteristics -- 5 .11. Concluding remarks -- References -- Chapter 6 Translational Brownian Motion in a Double-Well Potential -- 6.1. Introduction -- 6.2. Characteristic times of the position correlation function -- 6.3. Converging continued fractions for the correlation functions -- 6.4. Two-mode approximation -- 6.5. Stochastic resonance -- 6.6. Concluding remarks -- References -- Chapter 7 Non-inertial Rotational Diffusion in Axially Symmetric External Potentials: Applications to Orientational Relaxation of Molecules in Fluids and Liquid Crystals -- 7.1. Introduction -- 7.2. Rotational diffusion in a potential: Langevin equation approach -- 7.2.1. Averaging the non-inertial Euler-Langevin equation -- 7.2.2. Differential-recurrence equations for spherical harmonics -- 7.2.3. Examples of differential-recurrence equations -- 7.3. Brownian rotation in a uniaxial potential -- 7.3.1. Longitudinal relaxation -- 7.3.2. Susceptibility and relaxation times: continued-fraction solution -- 7.3.3. Integral form and asymptotic expansions of the exact solution -- 7.3.4. Transverse response: continued-fraction solution -- 7.3.5. Complex susceptibilities -- 7.4. Brownian rotation in a uniform d. c. external field -- 7.4.1. Longitudinal response: continued-fraction solution.

7.4.2. Transverse response: continued-fraction solution -- 7.4.3. Comparison with experimental data -- 7.5. Nonlinear transient responses in dielectric and Kerr-effect relaxation -- 7.5.1. Nonlinear transient dielectric and Kerr -effect relaxation times -- 7.5.2. Nonlinear step-on transient response: matrix continued-fraction solution -- 7.6. Nonlinear dielectric relaxation of polar molecules in a strong a. c. electric field: steady-state response -- 7.7. Concluding remarks -- References -- Chapter 8 Anisotropic Non-inertial Rotational Diffusion in an External Potential: Application to Linear and Nonlinear Dielectric Relaxation and the Dynamic Kerr Effect -- 8.1. Introduction -- 8.2. Anisotropic non-inertial rotational diffusion of an asymmetric top in an external potential -- 8.2.1. Euler-Langevin equation in the non-inertial limit -- 8.2.2. Differential-recurrence equation for Wigner's D functions -- 8.3. Application to dielectric relaxation -- 8.3.1. Linear dielectric response of an assembly of asymmetric tops -- 8.3.2. Nonlinear response in superimposed a.c. and strong d.c. bias fields: perturbation solution -- 8.3.3. Rotational Brownian motion and dielectric relaxation in nematic liquid crystals -- 8.4. Kerr-effect relaxation -- 8.4.1. Basic relations -- 8.4.2. Dynamic Kerr effect: matrix formulation -- 8.4.3. Nonlinear dielectric and Kerr-effect transients in strong fields -- 8.5. Concluding remarks -- References -- Chapter 9 Brownian Motion of Classical Spins: Application to Magnetization Relaxation in Superparamagnets -- 9.1. Introduction -- 9.2. Brown's model: Langevin equation approach -- 9.2.1. Derivation of the differential-recurrence equation for the statistical moments from the stochastic Gilbert equation -- 9.2.2. Fokker-Planck equation approach.

9.2.3. Differential-recurrence equations for a uniaxial superparamagnet in an external magnetic field.