The stochastic nature of complexity evolution in the fractional systems

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.