This volume is the third edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the translational and rotational motion of particles and spins in a potential highlighting modern applications in physics, chemistry, electrical engineering, and so on. In order to improve the presentation, to accommodate all the new developments, and to appeal to the specialized interests of the various communities involved, the book has been extensively rewritten and a very large amount of new material has been added. This has been done in order to present a comprehensive overview of the subject emphasizing via a synergetic approach that seemingly unrelated physical problems involving random noise may be described using virtually identical mathematical methods in the spirit of the founders of the subject, viz., Einstein, Langevin, Smoluchowski, Kramers, etc. The book has been written in such a way that all the material should be accessible both to an advanced researcher and a beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of scattered research papers and review articles.

Description-Table Of Contents

ch. 1. Historical background and introductory concepts. 1.1. motion. 1.2. Einstein's explanation of movement. 1.3. The Langevin equation. 1.4. Einstein's method. 1.5. Essential concepts in statistical mechanics. 1.6. Probability theory. 1.7. Application to the Langevin equation. 1.8. Wiener process. 1.9. The Fokker-Planck 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. 1.14. Applications to the theory of movement in a potential. 1.15. Rotational motion: application to dielectric relaxation. 1.16. Superparamagnetism: magnetic after-effect. 1.17. Brown's treatment of N?el relaxation. 1.18. Asymptotic expressions for the N?el relaxation. 1.19. Ferrofluids. 1.20. Depletion effect in a biased bistable potential. 1.21. Stochastic resonance. 1.22. Anomalous diffusion -- ch. 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: It? 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. 2.7. Methods of solution of differential-recurrence relations arising from the nonlinear Langevin equation. 2.8. Linear response theory. 2.9. Integral relaxation theory. 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. 2.12. Effective relaxation time. 2.13. Evaluation of the dynamic susceptibility using [symbol], [symbol] and [symbol]. 2.14. Nonlinear transient response of a particle -- ch. 3. motion of a free particle and a harmonic oscillator. 3.1. Introduction. 3.2. Ornstein-Uhlenbeck theory of motion. 3.3. Stationary solution of the Langevin equation: the Wiener-Khinchin theorem. 3.4. Application to phase diffusion in MRI. 3.5. Rotational motion of a fixed-axis rotator. 3.7. Torsional oscillator model: example of the use of the Wiener integral -- ch. 4. Rotational motion about a fixed axis in N-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. 4.5. Effect of a d.c. bias field on the orientational relaxation of a fixed-axis rotator with two equivalent sites. ; 8 ch. 5. motion in a tilted periodic potential: application to the Josephson tunneling 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 -- ch. 6. Translational 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 -- ch. 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. 7.3. rotation in a uniaxial potential. 7.4. rotation in a uniform d.c. external field. 7.5. Nonlinear transient responses in dielectric and Kerr-effect relaxation. 7.6. Nonlinear dielectric relaxation of polar molecules in a strong a.c. electric field: steady-state response. 7.7. Concluding remarks -- ch. 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. 8.3. Application to dielectric relaxation. 8.4. Kerr-effect relaxation. 8.5. Concluding remarks -- ch. 9. motion of classical spins: application to magnetization relaxation in superparamagnets. 9.1. Introduction. 9.2. Brown's model: Langevin equation approach. 9.3. Magnetization relaxation in uniaxial superparamagnets. 9.4. Reversal time of the magnetization in superparamagnets with nonaxially symmetric potentials: escape-rate theory approach. 9.5. Magnetization relaxation in superparamagnets with non-axially symmetric anisotropy: matrix continued-fraction approach. 9.6. Nonlinear a.c. stationary response of superparamagnets. 9.7. Concluding remarks -- ch. 10. Inertial effects in rotational and translational motion for a single degree of freedom. 10.1. Introduction. 10.2. Inertial effects in nonlinear dielectric response. 10.3. motion of a fixed-axis rotator in a double-well potential. 10.4. motion of a fixed-axis rotator in an asymmetric double-well potential. 10.5. motion in a tilted periodic potential. 10.6. Translational motion in a double-well potential. 10.7. Concluding remarks. ; 8 ch. 11. Inertial effects in rotational diffusion in space: application to orientational relaxation in molecular liquids and ferrofluids. 11.1. Introduction. 11.2. Inertial rotational motion of a thin rod in space. 11.3. Rotational motion of a symmetrical top. 11.4. Inertial rotational motion of a rigid dipolar rotator in a uniaxial biased potential. 11.5. Itinerant oscillator model of rotational motion in liquids. 11.6. Application of the cage model to ferrofluids -- ch. 12. Anomalous diffusion and relaxation. 12.1. Discrete- and continuous-time random walks. 12.2. Fractional diffusion equation for the continuous-time random walk model. 12.3. Solution of fractional diffusion equations. 12.4. Characteristic times of anomalous diffusion. 12.5. Inertial effects in anomalous relaxation. 12.6. Barkai and Silbey's fractional kinetic equation. 12.7. Anomalous diffusion in a periodic potential. 12.8. Fractional Langevin equation. 12.9. Concluding remarks.