The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (World Scientific Series in Contemporary Chemical Physics Vol. 14) - Second Edition

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. 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 "This volume is the second edition of the 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."--Jacket

this Volume Is The Second Edition Of The 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.