Dynamics of continuous-time random walk, fractal time dispersion, and fractional exponential time relaxation.

Abstract

This paper attempts to establish the dynamics of a microscopic model for a continuous-time random walk. The waiting-time distribution Q(t) is derived from the time-dependent perturbation theory of quantum mechanics for the walker's motion coupled with the media. The walker's motion includes the hopping of a localized particle and a spin (or dipole) flip. The medium is modeled as a harmonic heat bath. The walker moves among a set of degenerate localized states. The scaling behavior of the effective spectrum at low frequency with index \ensuremath{\beta} is modeled by using stochastic theory. It is found that Q(t)=exp(-${\mathrm{at}}^{(2\mathrm{\ensuremath{-}}\ensuremath{\beta})}$) for 0\ensuremath{\le}\ensuremath{\beta}2 and Q(t)\ensuremath{\sim}${t}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ for \ensuremath{\beta}=2. The applications of our theory include dispersion diffusion, the transient drift of hopping control light excitation in a-Si:H, and thermoremanent magnetization relaxation in spin glasses.