On the anomalous diffusion behavior in disordered media

A particle diffusing in a disordered medium with a Gaussian distribution of potential well depths is studied. The transition rates are related to the random potentials in the well-known Arrhenius form. A log-normal distribution of transition rates is naturally established. Finally, we show that the behaviour of diffusing particles in the one-dimensional case can be expressed as ln 〈X2〉1/2 ∼ (kT/σ)2(ln t)2, when kT/σ ∠ 1 (low temperature), where σ, T and k are the variance of the Gaussian distribution, the absolute temperature and the Boltzmann constant, respectively. When kT/σ ∠ 1 (high temperature) the normal type of diffusion behavior is exhibited. The anomalous types of diffusion behavior associated with higher dimensions are also investigated.