RANDOM TRANSITION RATE MODEL OF STRETCHED EXPONENTIAL RELAXATION

Stretched exponential regression to equilibrium is a frequently observ ed 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 dis tributed 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 t he effective transition rate obtained as a normalized sum of individua l rates is found to be distributed according to the (asymmetric) Levy stable distribution, which is known to be a necessary and sufficient c ondition for stretched exponential relaxation. This known result is re stated now within the framework of a model, which has the simple pheno menology of the parallel channels, but which however operates with sel f-similar dynamics. Moreover, the derivations are entirely carried out in terms of characteristic functions of untransformed random variable s. The model closely resembles the existing probabilistic models and t he differencies are mainly found in the way to motivate the self-simil arity of dynamics and in different emphasis on the starting assumption s. The main motivation has been to point out the inherent relatedness of all probabilistic models operating with only one self-similar stoch astic process, and to suggest that relaxation can be handled with a si ngle class of well defined functions.