Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Anomalous long time tails due to trapping

A new mechanism is described for producing slow decays in the velocity correlation function of diffusive systems with directed trapping. If the directions for entering and leaving a trap are correlated and if the distribution of trapping times has a long tail then the velocity correlation function will have a corresponding long time tail. This new long time tail decays liket ^–(2 +), where is an exponent characterizing the tail of the distribution of trapping times. A simple random walk model which illustrates this mechanism is analyzed.     

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t^–(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket ^–d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ^d/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.     

Current noise and long time tails in biased disordered random walks

We calculate the mean velocity and the velocity correlation function for a random walk with a uniform bias on a disordered chain. We find a new long time tail in the velocity correlation function due to the combined effects of the bias and of the disorder in the site variables. This long time tail persists to a low-frequency cutoff inversely proportional to the square of the bias. By associating the velocity correlation function with the spectrum of current fluctuations, we calculate the excess low-frequency current noise associated with this long time tail. The spectrum of current fluctuations goes as(I ²/N)f ^–1/2, whereI is the DC current,N is the number of charge carriers, andf is the frequency. The possible connection to 1/f noise is discussed. The calculation is done by a perturbation expansion in the strength of the disorder, but is shown to be exact to all orders for weak enough bias.Supported by a fellowship of the German Academic Exchange Service (DAAD).Supported by the National Science Foundation through Grant No. DMR-8108328 and through the Cornell Materials Science Center.     

Kinetic theory of long time tails in velocity correlation functions in a moderately dense electron gas

The long time tails of the correlation functions that determine the self-diffusion coefficient and the kinetic parts of the shear viscosity and heat conductivity in a one-component plasma are calculated using a systematic kinetic theory. The results are in agreement with those obtained from the phenomenological mode coupling theory. The formal kinetic theory calculations of previous workers, who obtained incomplete long time tail results, are also discussed.     

Fractional nonlinear diffusion equation and first passage time

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passage time distribution, the mean first passage time, and the mean squared displacement corresponding to different time-dependent diffusion coefficient. In addition, we compare our results for the first passage time distribution and the mean first passage time with the one obtained by usual linear diffusion equation with time-dependent diffusion coefficient.     

On the dual symmetry between absorbing and amplifying random media

We re-examine the dual symmetry between absorbing and amplifying random media. By analysing the physically allowed choice of the sign of the square root to determine the complex wave vector in a medium, we draw a broad set of conclusions that enables us to resolve the apparent paradox of the dual symmetry and also to anticipate the large local electromagnetic field enhancements in amplifying random media.     

Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX _t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X _t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities _t =(d/dt) E (X _t ² –M ^–1 t and _t(w) = (d/dt)E_w(X _t ² – M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4 _st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since , with ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.     

Translational and rotational diffusion of an anisotropic particle in a molecular liquid: Long-time tails and Brownian limit

Formally exact equations are written down, describing the translational and rotational diffusion of an anisotropic tagged particle in a fluid of anisotropic particles. These equations are tractable in the long-time limit, and reduce to the solution of ordinary hydrodynamic equations supplemented by slip boundary conditions in the Brownian limit for a smooth tagged particle. No rotational viscosities or spin-diffusion constants appear in these results. The relation to other work is discussed.     

On the relation between conduction and diffusion in a random walk along self-similar clusters

The relation between diffusion and conduction in the random walk of a particle by means of Lévy hops is investigated. It is shown that on account of the unusual character of Lévy hops, the mobility of a particle is a nonlinear function of the electric field for arbitrarily weak fields. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 7, 518–520 (10 April 1998)     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论，数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题，对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度，结果显示，闪烁布朗马达定向流极大值出现在超扩散条件下；摇摆布朗马达定向流最大值出现在弹道扩散条件下.     

基于连续时间无规行走模型研究反常扩散

基于连续时间无规行走(CTRW)理论,数值研究了布朗粒子的欠扩散、正常扩散和超扩散三种扩散行为.解决了CTRW模型的跳跃步长和等待时间分布函数的可实现化问题,对Metropolis抽样方法进行了改进以适用于周期势.探讨了布朗马达依靠闪烁棘轮和摇摆棘轮整流反常扩散所获得的定向速度,结果显示,闪烁布朗马达定向流极大值出现在超扩散条件下;摇摆布朗马达定向流最大值出现在弹道扩散条件下. 关键词： 无规行走 反常扩散 Metropolis抽样 棘轮势     

Solutions of fractional nonlinear diffusion equation and first passage time: Influence of initial condition and diffusion coefficient

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with diffusion coefficient separable in time and space, D(t,x)=D(t)|x|^−θ, subject to absorbing boundary condition and the conventional initial condition p(x,0)=δ(x−x₀). We obtain explicit analytical expressions for the probability distribution, the first passage time distribution, the mean first passage time and the mean squared displacement, and discuss their behavior corresponding to different time dependent diffusion coefficients.     

Collective diffusion in a lattice gas automaton

We report on non-mean-field and ring-kinetic-theory calculations of both the momentum autocorrelation function and the collective diffusion coefficient in a diffusive lattice gas automaton. For both quantities the ring approximation is calculated exactly. A saddle point method yields a leadingt ^–2 and a subleadingt ^–5/2 long-time tail in the momentum autocorrelation function. The ring kinetic corrections to the mean field value of the diffusion coefficient are in good agreement with computer simulations.     

On the relationship between continuous time random walks and the non-equilibrium Ornstein-Zernike equation

An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.     

A random field model for anomalous diffusion in heterogeneous porous media

Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.     

The long time and Brownian limits for tagged particle motion in liquids

Formally exact theories of tagged particle motion in liquids are developed, based upon kinetic theory for hard spheres and mode coupling for smooth potentials. It is shown that the resulting equations are tractable in the long time and Brownian limits. The coefficient of the long time tail of the velocity correlation function is seen to differ from its low-density form by only the replacement of the low-density viscosity and diffusion constant by the true viscosity and diffusion constant. In the Brownian limit, the slip Stokes-Einstein law is obtained, with the true viscosity. The relation to other work is discussed.Supported by NSF Grant No. CHE81-11422 and by a Dreyfus Teacher-Scholar grant to TK.     

Non-Markoffian diffusion in a one-dimensional disordered lattice

Recent treatments of diffusion in a one-dimensional disordered lattice by Machta using a renormalization-group approach, and by Alexander and Orbach using an effective medium approach, lead to a frequency-dependent (or non-Markoffian) diffusion coefficient. Their results are confirmed by a direct calculation of the diffusion coefficient.Research supported by NSF Grant No. CHE 77-16308.     

Correlation function induced by a generalized diffusion equation with the presence of a harmonic potential

An integro-differential diffusion equation with linear force, based on the continuous time random walk model, is considered. The equation generalizes the ordinary and fractional diffusion equations, which includes short, intermediate and long-time memory effects described by the waiting time probability density function. Analytical expression for the correlation function is obtained and analyzed, which can be used to describe, for instance, internal motions of proteins. The result shows that the generalized diffusion equation has a broad application and it may be used to describe different kinds of systems.     

Polynomially decaying transmission for the nonlinear Schrödinger equation in a random medium

This is the first study of one of the transmission problems associate to the non-linear Schrödinger equation with a random potential. We show that for almost every realization of the medium the rate of transmission vanishes when increasing the size of the medium; however, whereas it decays exponentially in the linear regime, it decays polynomially in the nonlinear one.This work is part of a Thèse de Troisième Cycle by P. Devillard.⁽⁶⁾