The Stochastic Nature of Complexity Evolution in the Fractional Systems

TL;DR

This paper presents a probabilistic model explaining nonexponential relaxation in complex systems, linking stochastic processes with fractional calculus, and clarifying parameters in empirical response functions across various relaxation phenomena.

Contribution

It introduces a stochastic formalism based on limit theorems to explain complex relaxation behaviors and connects fractional derivatives with empirical response functions.

Findings

Nonexponential relaxation results from asymptotic self-similarity.

The model justifies Jonscher's energy criterion.

Links fractional derivatives to empirical response parameters.

Abstract

The stochastic scenario of relaxation in the complex systems is presented. It is based on a general probabilistic formalism of limit theorems. The nonexponential relaxation is shown to result from the asymptotic self-similar properties in the temporal behavior of such systems. This model provides a rigorous justification of the energy criterion introduced by Jonscher. The meaning of the parameters into the empirical response functions is clarified. This treatment sheds a fresh light on the nature of not only the dielectric relaxation but also mechanical, luminescent and radiochemical ones. In the case of the Cole-Cole response there exists a direct link between the notation of the fractional derivative (appearing in the fractional macroscopic equation often proposed) and the model. But the macroscopic response equations, relating to the Cole-Davidson and Havriliak-Negami relaxations, have a more general integro-differential form in comparison with the ordinary fractional one.