Asymptotic behaviors of the waiting-time distribution function psi (t) and asymptotic solutions of continuous-time random-walk problems.

Abstract

In this paper, a novel waiting-time distribution function (WTDF), \ensuremath{\psi}(t), which is not purely exponential and is universally valid for explaining recently the low-frequency (say, \ensuremath{\omega}<10 GHz) fluctuation, dissipation, and relaxation properties of condensed matter, is used to discuss asymptotic solutions of the continuous-time random-walk (CTRW) problems. Many results of theoretical and experimental interest are obtained. These results include the mean displacement, the dispersive mobility, the mean-squared displacement, the dispersive diffusion coefficient, the 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 \ensuremath{\psi}(t) appears to be non-Markovian over the very broad time domain and is Markovian only in the long-time limit; in other words, all results contain a single parameter n, the infrared divergence exponent, which depends on the microscopic structure of condensed matter and governs the degree of dispersion. The larger the value of n, the stronger the dispersion. When n=0, the dispersion disappears and all results reduce immediately to the classical Markovian forms. This is in agreement with some experimental facts.