Wait-and-switch relaxation model: Relationship between nonexponential relaxation patterns and random local properties of a complex system

Abstract

The wait-and-switch stochastic model of relaxation is presented. Using the "random-variable" formalism of limit theorems of probability theory we explain the universality of the short- and long-time fractional-power laws in relaxation responses of complex systems. We show that the time evolution of the nonequilibrium state of a macroscopic system depends on two stochastic mechanisms: one, which determines the local statistical properties of the relaxing entities, and the other one, which determines the number (random or deterministic) of the microscopic and mesoscopic relaxation contributions. Within the proposed framework we derive the Havriliak-Negami and Kohlrausch-Williams-Watts functions. We also discuss the influence of the random-walk characteristics of migrating defects on the homogeneous and heterogeneous relaxation scenarios and show the origins of the stretched-exponential integral kernel in the integral representation of the ensemble-averaged relaxation function.