The Langevin equation : with applications to stochastic problems in physics, chemistry and electrical engineering

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The Langevin equation : with applications to stochastic problems in physics, chemistry and electrical engineering / W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron

版:

2nd ed

資料種別:

図書

出版情報:

Singapore ; New Jersey : World Scientific, 2004

形態:

xxiv, 678 p. ; 24 cm

シリーズ名:

World Scientific series in contemporary chemical physics ; v. 14 <BA19386290>

著者名:

ISBN:

9789812384621 [9812384626]

書誌ID:

BA67426261

子書誌情報

フルテキスト

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書誌詳細

注記:

Includes bibliographical references and index

主題:

統計力学

言語:

英語

目次情報:

Preface to the Second Edition

Preface to the First Edition

Contents

Historical Background and Introductory Concepts / Chapter 1：

Brownian Motion / 1.1：

Einstein's Explanation of the Brownian Movement / 1.2：

The Langevin Equation / 1.3：

Calculation of Avogadro's number / 1.3.1：

Einstein's Method / 1.4：

Necessary Concepts of Statistical Mechanics / 1.5：

Ensemble of systems / 1.5.1：

Phase space / 1.5.2：

Representative point / 1.5.3：

Ergodic hypothesis / 1.5.4：

Calculation of averages / 1.5.5：

Liouville equation / 1.5.6：

Reduction of the Liouville equation / 1.5.7：

Langevin equation for a system with one degree of freedom / 1.5.8：

Effect of a heat bath. Intuitive derivation of the Klein-Kramers equation / 1.5.9：

Conditions under which a Maxwellian distribution in the velocities may be deemed to be attained / 1.5.10：

Very high damping regime / 1.5.11：

Low damping regime / 1.5.12：

Probability Theory / 1.6：

Random variables and probability distributions / 1.6.1：

Properties of the Gaussian distribution / 1.6.2：

Moment generating functions / 1.6.3：

Central Limit Theorem / 1.6.4：

Random processes / 1.6.5：

Wiener-Khinchine theorem / 1.6.6：

Application to the Langevin Equation / 1.7：

Wiener Process / 1.8：

Variance of the Wiener process / 1.8.1：

Wiener integrals / 1.8.2：

The Fokker-Planck Equation / 1.9：

Drift and Diffusion Coefficients / 1.10：

Solution of the One-Dimensional Fokker-Planck Equation / 1.11：

The Smoluchowski Equation / 1.12：

Escape of Particles over Potential Barriers - Kramers' Escape Rate Theory / 1.13：

Escape rate in the IHD limit / 1.13.1：

Kramers' original calculation of the escape rate for very low damping / 1.13.2：

Range of validity of the IHD and VLD formulae / 1.13.3：

Extension of Kramers' theory to many dimensions in the intermediate-to-high damping limit / 1.13.4：

Langer's treatment of the IHD limit / 1.13.5：

Kramers' formula as a special case of Langer's formula / 1.13.6：

Applications of the Theory of Brownian Movement in a Potential / 1.14：

Rotational Brownian Motion - Application to Dielectric Relaxation / 1.15：

Breakdown of the Debye theory at high frequencies / 1.15.1：

Superparamagnetism - Magnetic After-Effect / 1.16：

Brown's Treatment of Neel Relaxation / 1.17：

Asymptotic Expressions for the Neel Relaxation Time / 1.18：

Application of Kramers' method to axially sym-metric potentials of the magneto-crystalline anisotropy / 1.18.1：

IHD formula for magnetic spins / 1.18.2：

Ferrofluids / 1.19：

Depletion Effect in a Biased Bistable Potential / 1.20：

Stochastic Resonance / 1.21：

Anomalous Diffusion / 1.22：

Empirical formulae for [varepsilon]([omega]) / 1.22.1：

Theoretical justification for anomalous relaxation behaviour / 1.22.2：

Anomalous dielectric relaxation of an assembly of fixed axis rotators / 1.22.3：

References

Langevin Equations and Methods of Solution / Chapter 2：

Criticisms of the Langevin Equation / 2.1：

Doob's Interpretation of the Langevin Equation / 2.2：

Nonlinear Langevin Equation with a Multiplicative Noise Term: Ito and Stratonovich Rules / 2.3：

Derivation of Differential-Recurrence Relations from the One-Dimensional Langevin Equation / 2.4：

Nonlinear Langevin Equations in Several Dimensions / 2.5：

Average of the Multiplicative Noise Term in the Langevin Equation for a Rotator / 2.6：

Multiplicative noise term for a three-dimensional rotator / 2.6.1：

Multiplicative noise terms with I taken as zero prior to averaging / 2.6.2：

Explicit average of the noise induced terms for a planar rotator / 2.6.3：

Methods of Solution of Differential-Recurrence Relations Arising from the Nonlinear Langevin Equation / 2.7：

Matrix diagonalisation method / 2.7.1：

Initial conditions / 2.7.2：

Matrix continued fraction solution of recurrence equations / 2.7.3：

Linear Response Theory / 2.8：

Correlation Time / 2.9：

Linear Response Theory Results for Systems with Dynamics Governed by One-Dimensional Fokker-Planck equations / 2.10：

Smallest Nonvanishing Eigenvalue: The Continued Fraction Approach / 2.11：

Evaluation of [lambda subscript 1] from a scalar three-term recurrence relation / 2.11.1：

Evaluation of [lambda subscript 1] from a matrix three-term recurrence relation / 2.11.2：

Effective Eigenvalue / 2.12：

Evaluation of the Dynamic Susceptibility Using [tau], [tau subscript ef], and [lambda subscript 1] / 2.13：

Nonlinear Response of a Brownian Particle Subjected to a Strong External Field / 2.14：

Analytical solutions for the relaxation time of one-dimensional systems / 2.14.1：

Nonlinear transient response in the rotational Brownian motion / 2.14.2：

Brownian Motion of a Free Particle and a Harmonic Oscillator / Chapter 3：

Ornstein-Uhlenbeck Theory of the Brownian Motion / 3.1：

Stationary Solution of the Langevin Equation - The Wiener-Khinchine Theorem / 3.2：

Brownian Motion of a Harmonic Oscillator / 3.3：

Application to Dielectric Relaxation / 3.4：

Theorem about Gaussian random variables / 3.4.1：

Torsional Oscillator Model: Example of the Use of the Wiener Integral / 3.5：

Two-Dimensional Rotational Brownian Motion in N-Fold Cosine Potentials / Chapter 4：

Introduction / 4.1：

Langevin Equation for Rotation in Two Dimensions / 4.2：

Longitudinal and Transverse Effective Relaxation Times in the Noninertial Limit / 4.3：

Polarisabilities and Dielectric Relaxation Times of a Fixed Axis Rotator with Two Equivalent Sites / 4.4：

Matrix solution / 4.4.1：

Longitudinal polarisability and relaxation times / 4.4.3：

Transverse polarisability and relaxation times / 4.4.4：

Comparison of the Longitudinal Relaxation Time with the Results of the Kramers Theory / 4.5：

Brownian Motion in a Tilted Cosine Potential: Application to the Josephson Tunnelling Junction / Chapter 5：

Josephson Junction: Dynamic Model / 5.1：

Reduction of the Averaged Langevin Equation for the Junction to a Set of Differential-Recurrence Relations / 5.3：

DC Current-Voltage Characteristics / 5.4：

Linear Response to an Applied Alternating Current / 5.5：

Effective Eigenvalues for the Josephson Junction / 5.6：

Linear Response Using the Effective Eigenvalues / 5.7：

Spectrum of the Josephson Radiation / 5.8：

Translational Brownian Motion in a Double-Well Potential / Chapter 6：

Relaxation Time of the Position Correlation Function / 6.1：

Comparison of Characteristic Times and Evaluation of the Position Correlation Function / 6.3：

Three-Dimentional Rotational Brownian Motion in an External Potential: Application to the Theory of Dielectric and Magnetic Relaxation / Chapter 7：

Rotational Diffusion in an External Potential: The Langevin Equation Approach / 7.1：

Gilbert's Equation Augmented by a Random Field Term / 7.3：

Langevin equation approach / 7.3.1：

Fokker-Planck equation approach / 7.3.2：

Brownian Rotation in the Uniaxial Potential / 7.4：

Longitudinal relaxation / 7.4.1：

Susceptibility and relaxation times / 7.4.2：

Integral form and asymptotic expansions / 7.4.3：

Transverse response / 7.4.4：

Complex susceptibilities / 7.4.5：

Brownian Rotation in a Uniform DC External Field / 7.5：

Longitudinal response / 7.5.1：

Comparison with experimental data / 7.5.3：

Anisotropic Noninertial Rotational Diffusion of an Asymmetric Top in an External Potential / 7.6：

Solution of the Euler-Langevin equation for an asymmetric top in the noninertial limit / 7.6.1：

Linear response of an assembly of asymmetric tops / 7.6.2：

Response in superimposed ac and strong dc bias fields: perturbation solution / 7.6.3：

Rotational Brownian Motion in Axially Symmetric Potentials: Matrix Continued Fraction Solutions / Chapter 8：

Application to the Single Axis Rotator / 8.1：

Relaxation times / 8.2.1：

Rotation in Three Dimensions: Longitudinal Response / 8.3：

Uniaxial particle in an external field / 8.3.1：

Characteristic times and magnetic susceptibility / 8.3.2：

Magnetic stochastic resonance / 8.3.3：

Transverse Response of Uniaxial Particles / 8.4：

Matrix continued fraction solution / 8.4.1：

Transverse complex magnetic susceptibility / 8.4.2：

Nonlinear Transient Responses in Dielectric and Kerr Effect Relaxation / 8.5：

Nonlinear Dielectric Relaxation of Polar Molecules in a Strong AC Electric Field: Steady State Response / 8.6：

Dielectric Relaxation and Rotational Brownian Motion in Nematic Liquid Crystals / 8.7：

Rotational Brownian Motion in Non-Axially Symmetric Potentials / Chapter 9：

Uniaxial Superparamagnetic Particles in an Oblique Field / 9.1：

Recurrence equations / 9.2.1：

Smallest nonvanishing eigenvalue, the relaxation time, and the complex susceptibility / 9.2.2：

Cubic Anisotropy / 9.3：

Complex susceptibility and relaxation times / 9.3.1：

Inertial Langevin Equations: Application to Orientational Relaxation in Liquids / Chapter 10：

Step-On Solution for Noninertial Rotation about a Fixed Axis / 10.1：

Inertial Rotation about a Fixed Axis / 10.3：

Inertial effects and nonlinear response / 10.3.1：

Inertial Rotational Brownian Motion of a Thin Rod in Space / 10.3.2：

Derivation of recurrence equations / 10.4.1：

Evaluation of C[subscript 1] / 10.4.2：

Evaluation of C[subscript 2] / 10.4.3：

Evaluation of C[subscript l] for an arbitrary l / 10.4.4：

Rotational Brownian Motion of a Symmetrical Top / 10.5：

Evaluation of C[subscript 1] and C[subscript 2] / 10.5.1：

Itinerant Oscillator Model of Rotational Motion in Liquids / 10.6：

Generalisation of the Onsager model - Relation to the cage model / 10.6.1：

Dipole correlation function / 10.6.3：

Exact solution for the complex susceptibility using matrix continued fractions / 10.6.4：

Results and comparison with experimental data / 10.6.5：

Application of the Cage to Ferrofluids / 10.7：

Statistical Averages of the Hermite Polynomials of the Angular Velocity Components for Linear Molecules / Appendix A：

Averages of the Angular Velocities Components / Appendix B：

Evaluation of cos[theta]([varepsilon]) in the Low Damping Limit / Appendix C：

Sack's Continued Fraction Solution for the Sphere / Appendix D：

Discrete and Continuous Time Random Walks / Chapter 11：

A Fractional Diffusion Equation for the Continuous Time Random Walk Model / 11.2：

Solution of fractional diffusion equations in configuration space / 11.2.1：

Anomalous diffusion of a planar rotator in a mean field potential / 11.2.2：

Divergence of Global Characteristic Times in Anomalous Diffusion / 11.3：

First passage time for normal diffusion / 11.3.1：

First passage time distribution for anomalous diffusion / 11.3.2：

Inertial Effects in Anomalous Relaxation / 11.4：

Slow transport process governed by trapping / 11.4.1：

Calculation of the complex susceptibility / 11.4.2：

Comment on the use of the telegraph equation as an approximate description of the configuration space distribution function including inertial effects / 11.4.3：

Barkai and Silbey's Form of the Fractional Klein-Kramers Equation / 11.5：

Complex susceptibility / 11.5.1：

Fractional kinetic equation for the needle model / 11.5.2：

Anomalous Diffusion in a Periodic Potential / 11.6：

Calculation of the spectra / 11.6.1：

Fractional Langevin Equation / 11.7：

Fractal Dimension, Anomalous Exponents and Random Walks / Appendix：

Index

Preface to the Second Edition

Preface to the First Edition

Contents

Historical Background and Introductory Concepts / Chapter 1：

Brownian Motion / 1.1：

Einstein's Explanation of the Brownian Movement / 1.2：

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