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.