EQUATION LANGEVIN. Physics, Chemistry and Electrical Engineering. World Scientific. With Applications to Stochastic Problems in. William T.
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1 SHANGHAI HONG WorlrfScientific Series krtonttimfjorary Chemical Physics-Vol. 27 THE LANGEVIN EQUATION With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering Third Edition William T. Coffey Trinity College, Ireland Yuri P. Kalmykov Universite de Perpignan Via Domitia, France World Scientific NEW JERSEY LONDON SINGAPORE BEIJING KONG TAIPEI CHENNAI
2 CONTENTS Preface to the Third Edition Contents vii xv Chapter 1 Historical Background and Introductory Concepts Brownian motion Einstein's explanation of Brownian movement The Langevin equation Calculation of Avogadro's number Einstein's Method Essential concepts in Statistical Mechanics Ensemble of systems Phase space Representative point Ergodic hypothesis Calculation of averages Liouville equation Reduction of the Liouville equation Langevin equation for a system with one degree of freedom Intuitive derivation of the Klein-Kramers equation Conditions under which a Maxwellian distribution in the may be deemed to be attained Very-high-damping (VHD) regime Very-low-damping (VLD) regime Probability theory Random variables and probability distributions The Gaussian distribution Moment-generating functions Central limit theorem Random processes Wiener-Khinchin theorem Application to the Langevin equation Wiener process Variance of the Wiener process Wiener integrals The Fokker-Planck equation Drift and diffusion coefficients 69 xv
3 xvi The Langevin Equation Solution of the one-dimensional Fokker-Planck equation The Smoluchowski equation Escape of particles over potential barriers: Kramers' theory Escape rate in the IHD limit Kramers' calculation of the escape rate in the VLD limit Range of validity of the IHD and VLD formulas Extension of Kramers' theory to many dimensions in the IHD limit Langer's treatment of the IHD limit Kramers' formula as a special case of Langer's formula Kramers'turnover problem Applications of the theory of Brownian movement in a potential Rotational Brownian motion: application to dielectric relaxation Breakdown of the Debye theory at high frequencies Superparamagnetism: magnetic after-effect Brown's treatment of Neel relaxation Asymptotic expressions for the Neel relaxation time Magnetization reversal time in a uniaxial superparamagnet: application of Kramers' method Escape rate formulas for superparamagnets Ferrofluids Depletion effect in a biased bistable potential./ Stochastic resonance Anomalous diffusion Empirical formulas for the complex dielectric permittivity Theoretical justification for anomalous relaxation behavior Anomalous dielectric relaxation of an assembly of dipolar molecules 160 References 163 Chapter 2 Langevin Equations and Methods of Solution Criticisms ofthe Langevin equation Doob's interpretation of the Langevin equation Nonlinear Langevin equation with a multiplicative noise term: Ito and Stratonovich rules Derivation of differential-recurrence relations from the one-dimensional Langevin equation Nonlinear Langevin equation in several dimensions Average of me multiplicative noise term in the Langevin equation Multiplicative noise term for rotation in space Multiplicative noise terms in the non-inertial limit Explicit average of the noise-induced terms for a planar rotator 188
4 Contents xvii 2.7. Methods of solution of differential-recurrence relations arising from the nonlinear Langevin equation Matrix method Initial conditions Matrix continued-fraction solution of recurrence equations Linear response theory Integral relaxation time Linear response theory for systems with dynamics governed by singlevariable Fokker-Planck equations Smallest non-vanishing eigenvalue: continued-fraction approach Evaluation of A\ from a scalar three-term recurrence relation Evaluation of Ai from a matrix three-term recurrence relation Effective relaxation time Evaluation of the dynamic susceptibility using zmx, ref, and Xx Nonlinear transient response of a Brownian particle 222 References 225 Chapter 3 Brownian Motion of a Free Particle and a Harmonic Oscillator Introduction Ornstein-Uhlenbeck theory ofbrownian motion Stationary solution of the Langevin equation: the Wiener-Khinchin theorem Application to phase diffusion in MRI Brownian motion of a harmonic oscillator Rotational Brownian motion of a fixed-axis rotator Torsional oscillator model: example of the use of the Wiener integral 246 References 250 Chapter 4 Rotational Brownian Motion About a Fixed Axis in A'-Fold Cosine Potentials Introduction Langevin equation for rotation about a fixed axis Longitudinal and transverse effective relaxation times Polarizabilities and relaxation times of a fixed-axis rotator with two equivalent sites Matrix solution for the longitudinal response Continued-fraction solution for the longitudinal response Comparison of the longitudinal relaxation time with the Kramers theory Continued-fraction solution for the transverse response Effect ofa d.c. bias field on the orientational relaxation of a fixed-axis rotator with two equivalent sites Longitudinal response 277
5 xviii The Langevin Equation Transverse response Relaxation times 281 References 283 Chapter 5 Brownian Motion in a Tilted Periodic Potential: Application to the Josephson Tunnelling Junction Introduction Langevin equations Josephson junction: dynamic model Reduction of the averaged Langevin equation for the junction to a set of differential-recurrence relations Current-voltage characteristics Linear response to an applied alternating current Effective eigenvalues for the Josephson junction Linear impedance Spectrum of the Josephson radiation Nonlinear effects in d.c. and a.c. current-voltage characteristics Concluding remarks 317 References 318 Chapter 6 Translational Brownian Motion in a Double-Well Potential Introduction Characteristic times of the position correlation function Converging continued fractions for the correlation functions Two-mode approximation Stochastic resonance Concluding remarks 340 References 340 Chapter 7 Non-inertial Rotational Diffusion in Axially Symmetric External Potentials: Applications to Orientational Relaxation of Molecules in Fluids and Liquid Crystals Introduction Rotational diffusion in a potential: Langevin equation approach Averaging the non-inertial Euler-Langevin equation Differential-recurrence equations for spherical harmonics Examples of differential-recurrence equations Brownian rotation in a uniaxial potential Longitudinal relaxation Susceptibility and relaxation times: continued-fraction solution Integral form and asymptotic expansions of the exact solution Transverse response: continued-fraction solution Complex susceptibilities 368
6 Contents xix 7.4. Brownian rotation in a uniform d.c. external field Longitudinal response: continued-fraction solution Transverse response: continued-fraction solution Comparison with experimental data Nonlinear transient responses in dielectric and Kerr-effect relaxation Nonlinear transient dielectric and Kerr-effect relaxation times Nonlinear step-on transient response: matrix continued-fraction solution Nonlinear dielectric relaxation of polar molecules in a strong a.c. electric field: steady-state response Concluding remarks 397 References 398 Chapter 8 Anisotropic Non-inertial Rotational Diffusion in an External Potential: Application to Linear and Nonlinear Dielectric Relaxation and the Dynamic Kerr Effect Introduction Anisotropic non-inertial rotational diffusion of an asymmetric top in an external potential Euler-Langevin equation in the non-inertial limit Differential-recurrence equation for Wigner's D functions Application to dielectric relaxation Linear dielectric response of an assembly of asymmetric tops Nonlinear response in superimposed a.c. and strong d.c. bias fields: perturbation solution Rotational Brownian motion and dielectric relaxation in nematic liquid crystals Kerr-effect relaxation Basic relations Dynamic Kerr effect: matrix formulation Nonlinear dielectric and Kerr-effect transients in strong fields Concluding remarks 453 References 454 Chapter 9 Brownian Motion of Classical Spins: Application to Magnetization Relaxation in Superparamagnets Introduction Brown's model: Langevin equation approach Derivation of the differential-recurrence equation for the statistical moments from the stochastic Gilbert equation Fokker-Planck equation approach Differential-recurrence equations for a uniaxial superparamagnet in an external magnetic field 469
7 XX The Langevin Equation Comparison of the Gilbert, Landau-Lifshitz, and Kubo kinetic models of the Brownian rotation of classical spins Calculation of the observables Magnetization relaxation in uniaxial superparamagnets Longitudinal relaxation Characteristic times and magnetic susceptibility Magnetic stochastic resonance Dynamic magnetic hysteresis Transverse response: ferromagnetic resonance Reversal time of the magnetization in superparamagnets with nonaxially symmetric potentials: escape-rate theory approach Basic equations for the escape rate Discrete orientation model Reversal time for cubic anisotropy Uniaxial superparamagnet in a uniform d.c. magnetic field Biaxial superparamagnet in a uniform d.c. magnetic field applied along the easy axis Mixed anisotropy: breakdown of the paraboloid approximation Reversal time ofa superantiferromagnetic nanoparticle Switching-field curves and surfaces Magnetization relaxation in superparamagnets with non-axially symmetric anisotropy: matrix continued-fraction approach Uniaxial superparamagnet in an oblique field Cubic anisotropy Nonlinear a.c. stationary response of superparamagnets Concluding remarks 581 References» 585 Chapter 10 Inertial Effects in Rotational and Translational Brownian Motion for a Single Degree offreedom Introduction Inertial effects in nonlinear dielectric response Linear response for non-inertial rotation about a fixed axis Differential-recurrence equations for nonlinear transients Matrix continued-fraction solution Analytic equations for the integral relaxation time Inertial effects Brownian motion of a fixed-axis rotator in a double-well potential Matrix continued-fraction solution Turnover formula for the escape rate Analytic formulas for the longitudinal correlation time 625
8 Contents xxi Brownian motion of a fixed-axis rotator in an asymmetric double-well potential Basic equations...^ Matrix continued-fraction solution Turnover formula for and VHD and VLD asymptotes for rb. Complex susceptibility Brownian motion in a tilted periodic potential Matrix continued-fraction solution Turnover equation for the decay rate Translational Brownian motion in a double-well potential j Matrix continued-fraction solution Longest and integral relaxation times. Dynamic susceptibility Concluding remarks 663 References 664 Chapter 11 Inertial Effects in Rotational Diffusion in Space: Application to Orientational Relaxation in Molecular Liquids and Ferrofluids Introduction Inertial rotational Brownian motion of a thin rod in space Recurrence equations for the correlation functions First-rank correlation function Second-rank correlation function Orientational correlation function of an arbitrary rank / Rotational Brownian motion of a symmetrical top Recurrence equations for the correlation functions First-and second-rank correlation functions Inertial rotational Brownian motion of a rigid dipolar rotator in a uniaxial biased potential Basic relations Asymptotic formulas for the longest relaxation time Itinerant oscillator model ofrotational motion in liquids Generalization of the Onsager model: relation to the cage model Dipole correlation function Matrix continued-fraction solution for the complex susceptibility Comparison with experimental data Application of the cage model to ferrofluids 718 Appendix A: Statistical averages of the Hermite polynomials of the components of the angular velocity for linear molecules 729 Appendix B: Averages of the components of the angular velocities 730 Appendix C: Sack's continued-fraction solution for the sphere 733 References 734
9 xxii The Langevin Equation Chapter 12 Anomalous Diffusion and Relaxation Discrete-and continuous-time random walks Fractional diffusion equation for the continuous-time random walk model Solution of fractional diffusion equations Anomalous diffusion of a fixed-axis rotator in a potential Fractional diffusion ofa particle in a double-well potential Characteristic times of anomalous diffusion Mean first-passage time for normal diffusion First-passage-time distribution for anomalous diffusion Spectral definition of the characteristic times Inertial effects in anomalous relaxation Slow transport process governedby trapping Calculation of the complex susceptibility Comment on the use of the telegraph equation Barkai and Silbey's fractional kinetic equation Complex susceptibility Fractional kinetic equation for the needle model Anomalous diffusion in a periodic potential Fractional Langevin equation Application to phase diffusion in MRI Concluding remarks 811 Appendix: Fractal dimension, anomalous exponents, and random walks 814 References 816 Index 821
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