Time scale of thermally activated diffusion in random systems: A new law of thermal activation

Exact time scales are obtained for a class of stochastic models of thermally activated diffusion in random systems with a one-dimensional quantum reaction coordinate. The results are valid at any temperature. At low temperatures various forms of the thermal activation law of Arrhenius emerge, the dominant time scale of the diffusion depending on the temperature T according to $T^p$ exp(A/$k_B$T), where A is the activation energy and p an exponent depending on the details of the models. These correspond to different bounded energy spectra for the randomly selected energy levels along the reaction coordinate. As an example we consider the Wigner semicircular distribution of eigenvalues of large random matrices. Another example considered is a Gaussian distribution of energy levels, and in this case a new thermal-activation law is obtained, having the form exp[$B^2$/($k_B$T${)}^2$], where B is proportional to the width of the distribution. This form has recently been found in Monte Carlo experiments on large Ising-spin-glass models by Young.