等待时间分布函数ψ(t)的渐近行为与连续时间无规行走问题的渐近解

本文采用近年来在解释凝聚物质中的低频涨落、耗散和弛豫现象时普遍适用的一个新的非纯指数形式的等待时间分布函数(以下称WTDF)ψ(t),讨论了连续时间无规行走(以下称CTRW)问题的渐近解。得到了许多有意义的和与实验一致的结果。它们是:平均位移、色散迁移率、平均平方位移、色散扩散系数、Nernst-Einstein关系、方差与标准方差、点阵统计学、初始位置占有几率、色散电导率、色散电输运和记忆函数。所有结果表明:由这个WTDF ψ(t)所描写的CTRW过程在短时区内表现为非Markov的,在长时区内则表现为Markov的,即所有结果都含有一个与介质的微观结构有关的且决定着色散程度的单参数——红外发散指数n(0≤n<1)。n越大,色散越大。当n=0时,色散消失,所有结果立即退化成Markov即经典形式,这与已有的实验事实一致。 关键词：

ASYMPTOTIC BEHAVIORS OF WAITING TIME DISTRIBUTION FUNCTION (WTDF) ψ(t) AND ASYMPTOTIC SOLUTIONS OF CONTINUOUS-TIME RANDOM WALK (CTRW) PROBLEMS

In this paper, a new waiting time distribution function (WTDF), ψ(t), is adopted to discuss asymptotic solutions of the continuous-time random walk (CTRW) problems. This WTDF is not purely exponential and is universally valid for explaining the low-frequency (say, ω<10GHz) fluctuation, dissipation and relaxation properties of condensed matter. Many theoretically meaningful results are obtained, and they are in agreement with experiments, These results include the mean displacement, the dispersive mobility, the meansquared displacement, the dispersive diffusion coefficient, Nernst-Einstein relation, the variance and the standard variance, the lattice statistics, the initial site occupation probability, the dispersive conductivity, the dispersive electrical transport and the memory function. All results show that the CTRW process described by the WTDF ψ(t) behaves as non-Markovian over the very broad time domain and as Markovian only in long time limit, this is to say all results contain a single parameter, n, the infrared divergence exponent, which depends on the microscopic structure of condensed matter and determines the degree of dispersion. The larger the value of n is, the stronger the dispersion becomes. When n= 0, the dispersion disappears and all results reduce immediately to the classical Markovian forms.This is in agreement with receat experimental facts.