RANDOM TRANSITION RATE MODEL OF STRETCHED EXPONENTIAL RELAXATION

Stretched exponential regression to equilibrium is a frequently observed phenomenon in relaxation of non-equilibrium states. The question of the origins of the stretched exponentiality is addressed in terms of the probabilistic model of relaxation, based on the self-similarly distributed random transition rates. Each rate corresponds to one channel of relaxation and channels are assumed to operate in a parallel way, i.e., individual relaxation events are independent. As a consequence the effective transition rate obtained as a normalized sum of individual rates is found to be distributed according to the (asymmetric) Levy stable distribution, which is known to be a necessary and sufficient condition for stretched exponential relaxation. This known result is restated now within the framework of a model, which has the simple phenomenology of the parallel channels, but which however operates with self-similar dynamics. Moreover, the derivations are entirely carried out in terms of characteristic functions of untransformed random variables. The model closely resembles the existing probabilistic models and the differencies are mainly found in the way to motivate the self-similarity of dynamics and in different emphasis on the starting assumptions. The main motivation has been to point out the inherent relatedness of all probabilistic models operating with only one self-similar stochastic process, and to suggest that relaxation can be handled with a single class of well defined functions.