Universal relaxation function in nonextensive systems

TL;DR

This paper derives a universal relaxation function for nonextensive systems using a cluster model based on Tsallis entropy, linking fractal and nonextensive parameters to relaxation behavior and aligning with the Jonscher universality principle.

Contribution

It introduces a physically motivated model connecting nonextensive entropy, fractal properties, and relaxation functions, providing a new interpretation of Weron's stochastic theory parameters.

Findings

Relaxation follows a power law related to fractal parameter α.

Intermediate times exhibit stretched exponential relaxation.

Asymptotic behaviors match Weron's dielectric function and Jonscher's universality.

Abstract

We have derived the dipolar relaxation function for a cluster model whose volume distribution was obtained from the generalized maximum Tsallis nonextensive entropy principle. The power law exponents of the relaxation function are simply related to a global fractal parameter $\alpha$ and for large time to the entropy nonextensivity parameter $q$. For intermediate times the relaxation follows a stretched exponential behavior. The asymptotic power law behaviors both in the time and the frequency domains coincide with those of the Weron generalized dielectric function derived from an extension of the Levy central limit theorem. They are in full agreement with the Jonscher universality principle. Moreover our model gives a physical interpretation of the mathematical parameters of the Weron stochastic theory and opens new paths to understand the ubiquity of self-similarity and power laws in the relaxation of large classes of materials in terms of their fractal and nonextensive properties.