Self-similar motion for modeling anomalous diffusion and nonextensive statistical distributions

TL;DR

This paper introduces a new class of self-similar motion models that generate anomalous diffusion and distributions aligning with nonextensive statistical mechanics, expanding understanding of complex dynamical systems.

Contribution

It presents a novel universality class of one-dimensional models with generalized Feigenbaum constants, linking self-similar motion to anomalous diffusion and q-Gaussian distributions.

Findings

Mean-square displacement exhibits anomalous diffusion behavior.

Displacement distributions match q-Gaussian and bimodal distributions.

Self-similar motion models can describe nonextensive statistical phenomena.

Abstract

We introduce a new universality class of one-dimensional iteration model giving rise to self-similar motion, in which the Feigenbaum constants are generalized as self-similar rates and can be predetermined. The curves of the mean-square displacement versus time generated here show that the motion is a kind of anomalous diffusion with the diffusion coefficient depending on the self-similar rates. In addition, it is found that the distribution of displacement agrees to a reliable precision with the q-Gaussian type distribution in some cases and bimodal distribution in some other cases. The results obtained show that the self-similar motion may be used to describe the anomalous diffusion and nonextensive statistical distributions.