Mittag-Leffler functions in superstatistics

TL;DR

This paper explores how Mittag-Leffler functions influence superstatistics in biophysics, leading to new probability distributions with power-law tails, using fractional calculus techniques.

Contribution

It introduces a novel distribution for intensive parameters in superstatistics based on Mittag-Leffler functions, extending the framework with fractional calculus methods.

Findings

Distributions exhibit power-law behavior at high energies.

Mittag-Leffler functions modify the superstatistics framework.

New generalized probability distributions are proposed.

Abstract

Nowadays, there is a series of complexities in biophysics that require a suitable approach to determine the measurable quantity. In this way, the superstatistics has been an important tool to investigate dynamic aspects of particles, organisms and substances immersed in systems with non-homogeneous temperatures (or diffusivity). The superstatistics admits a general Boltzmann factor that depends on the distribution of intensive parameters $\beta$ (inverse-diffusivity). Each value of intensive parameter is associated with a local equilibrium in the system. In this work, we investigate the consequences of Mittag-Leffler function on the definition of f-distribution of a complex system. Thus, using the techniques belonging to the fractional calculus with non-singular kernels, we constructed a distribution to intensive parameters using the Mittag-Leffler function. This function implies distributions with power-law behaviour to high energy values in the context of Cohen-Beck superstatistics. This work aims to present the generalised probabilities distribution in statistical mechanics under a new perspective of the Mittag-Leffler function inspired in Atangana-Baleanu and Prabhakar forms.