GENERALIZED HUBER KINETICS FOR NONLINEAR RATE-PROCESSES IN DISORDERED-SYSTEMS - NONLINEAR ANALOGS OF STRETCHED EXPONENTIAL

This paper deals with one-variable nonlinear rate processes occurring in disordered systems. A general stochastic approach is introduced for these processes based on the following assumptions. The total rate co efficient is made up of the additive contributions of a large number o f individual reaction channels. These contributions are random functio ns of time and their stochastic properties are characterized by a func tional random point process. Exact analytical expressions for the time dependence of the average concentration are derived by using a charac teristic functional technique. These expressions are valid for systems with both dynamic and static disorder and are nonlinear analogs of th e general kinetic law derived by Huber [Phys. Rev. B 31, 6070 (1985); Phys. Rev. E 53, 6544 (1996)] for Linear rate processes in systems wit h static disorder. Fbr independent rate processes with static disorder and a self-similar distribution of reaction channels we derive a nonl inear analog of the stretched exponential. A closed analytic expressio n of the nonlinear stretched exponential is given in terms of Fox's H functions. As expected, when the reaction order of the process is one, the nonlinear kinetic law reduces to a stretched exponential with a s caling exponent characterizing the self-similar distribution of the in dividual reaction channels. For nonlinear processes the tail of the av eraged kinetic curve is self-similar and obeys a scaling law with a ne gative power law. Surprisingly, the scaling exponent of the tail depen ds only on the reaction order of the process and is independent of the scaling exponent that characterizes the self-similar distribution of the individual channels. We examine the possibilities of experimental evaluation of the statistical distribution of the total rate coefficie nt: The moments of different orders of the rate coefficient can be eva luated from the time derivatives of the survival function.