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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Coffey, William, 1948- Langevin equation. River Edge, NJ : World Scientific, ©2004 (DLC) 2006276587 (OCoLC)70267251 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
William Coffey; Yu P Kalmykov; J T Waldron |

ISBN: | 9789812795090 981279509X |

OCLC Number: | 262846486 |

Description: | 1 online resource (xxiv, 678 pages) : illustrations. |

Contents: | Ch. 1. Historical background and introductory concepts. 1.1. Brownian motion. 1.2. Einstein's explanation of the Brownian movement. 1.3. The Langevin equation. 1.4. Einstein's method. 1.5. Necessary concepts of 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' escape rate theory. 1.14. Applications of the theory of Brownian movement in a potential. 1.15. Rotational Brownian motion -- application to dielectric relaxation. 1.16. Superparamagnetism -- magnetic after-effect. 1.17. Brown's treatment of Néel relaxation. 1.18. Asymptotic expressions for the Néel relaxation time. 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: Itô and Stratonovich rules. 2.4. Derivation of differential-recurrence relations from the one-dimensional Langevin equation. 2.5. Nonlinear Langevin equations in several dimensions. 2.6. Average of the multiplicative noise term in the Langevin equation for a rotator. 2.7. Methods of solution of differential-recurrence relations arising from the nonlinear Langevin equation. 2.8. Linear response theory. 2.9. Correlation time. 2.10. Linear response theory results for systems with dynamics governed by one-dimensional Fokker-Planck equations. 2.11. Smallest nonvanishing eigenvalue: the continued fraction approach. 2.12. Effective eigenvalue. 2.13. Evaluation of the dynamic susceptibility using [symbol] and [symbol]. 2.14. Nonlinear response of a Brownian particle subjected to a strong external field -- ch. 3. Brownian motion of a free particle and a harmonic oscillator. 3.1. Ornstein-Uhlenbeck theory of the Brownian motion. 3.2. Stationary solution of the Langevin equation -- the Wiener-Khinchine theorem. 3.3. Brownian motion of a harmonic oscillator. 3.4. Application to dielectric relaxation. 3.5. Torsional oscillator model: example of the use of the Wiener integral -- ch. 4. Two-dimensional rotational Brownian motion in N-fold cosine potentials. 4.1. Introduction. 4.2. Langevin equation for rotation in two dimensions. 4.3. Longitudinal and transverse effective relaxation times in the noninertial limit. 4.4. Polarisabilities and dielectric relaxation times of a fixed axis rotator with two equivalent sites. 4.5. Comparison of the longitudinal relaxation time with the results of the Kramers theory. Ch. 5. Brownian motion in a tilted cosine potential: application to the Josephson tunnelling junction. 5.1. Introduction. 5.2. Josephson junction: dynamic model. 5.3. Reduction of the averaged Langevin equation for the junction to a set of differential-recurrence relations. 5.4. DC current-voltage characteristics. 5.5. Linear response to an applied alternating current. 5.6. Effective eigenvalues for the Josephson junction. 5.7. Linear response using the effective rigenvalues. 5.8. Spectrum of the Josephson radiation -- ch. 6. Translational Brownian motion in a double-well potential. 6.1. Introduction. 6.2. Relaxation time of the position correlation function. 6.3. Comparison of characteristic times and evaluation of the position correlation function -- ch. 7. Three-dimentional rotational Brownian motion in an external potential: application to the theory of dielectric and magnetic relaxation. 7.1. Introduction. 7.2. Rotational diffusion in an external potential: the Langevin equation approach. 7.3. Gilbert's equation augmented by a random field term. 7.4. Brownian rotation in the uniaxial potential. 7.5. Brownian rotation in a uniform DC external field. 7.6. Anisotropic noninertial rotational diffusion of an asymmetric top in an external potential -- ch. 8. Rotational Brownian motion in axially symmetric potentials: matrix continued fraction solutions. 8.1. Introduction. 8.2. Application to the single axis rotator. 8.3. Rotation in three dimensions: longitudinal response. 8.4. Transverse response of uniaxial particles. 8.5. Nonlinear transient responses in dielectric and Kerr-effect relaxation. 8.6. Nonlinear dielectric relaxation of polar molecules in a strong AC electric field: steady state response. 8.7. Dielectric relaxation and rotational Brownian motion in nematic liquid crystals -- ch. 9. Rotational Brownian motion in non-axially symmetric potentials. 9.1. Introduction. 9.2. Uniaxial superparamagnetic particles in an oblique field. 9.3. Cubic anisotropy -- ch. 10. Inertial Langevin equations: application to orientational relaxation in liquids. 10.1. Introduction. 10.2. Step-on solution for noninertial rotation about a fixed axis. 10.3. Inertial rotation about a fixed axis. 10.4. Inertial rotational Brownian motion of a thin rod in space. 10.5. Rotational Brownian motion of a symmetrical top. 10.6. Itinerant oscillator model of rotational motion in liquids. 10.7. Application of the cage to ferrofluids -- ch. 11. Anomalous diffusion. 11.1. Discrete and continuous time random walks. 11.2. A fractional diffusion equation for the continuous time random walk model. 11.3. Divergence of global characteristic times in anomalous diffusion. 11.4. Inertial effects in anomalous relaxation. 11.5. Barkai and Silbey's form of the fractional Klein-Kramers equation. 11.6. Anomalous diffusion in a periodic potential. 11.7. Fractional Langevin equation. |

Series Title: | World Scientific series in contemporary chemical physics. |

Responsibility: | W.T. Coffey, Yu. P. Kalmykov, J.T. Waldron. |

### Abstract:

This volume is the second edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the Brownian motion in a potential, with emphasis on modern applications in the natural sciences, electrical engineering and so on. It has been substantially enlarged to cover in a succinct manner a number of new topics, such as anomalous diffusion, continuous time random walks, stochastic resonance etc, which are of major current interest in view of the large number of disparate physical systems exhibiting these phenomena. The book has been written in such a way that all the material should be accessible to an advanced undergraduate or beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of research papers or scattered review articles.

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