Mittag-Leffler functions in superstatistics

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 β = 1 D (inverse-diffusivity). Each value of β 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 (β)-distribution of a complex system. Thus, using the techniques belonging to the fractional calculus with non-singular kernels, we constructed a distribution to β 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.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

“…Thereby, the overall probability defines the superstatistics of Gaussian statistics corresponding to different diffusivity packets. The above overall probability (Equation (4)) was analyzed in different scenarios in diffusion theory: (i) p(D) as a χ 2 gamma distribution in the context of random diffusivity, to compare with the diffusive diffusivity model [44], or to describe the movement of many individual small organisms that move with different diffusivities [54]; (ii) p(N) as a χ 2 -inverse gamma distribution that depends on particle size to investigate the aggregation and fragmentation process in the context of Laplace diffusion [50]; (iii) p(D) as a stretched exponential to construct a stretched Gaussian diffusion [17,55]; (iv) p(D) Lévy distribution to construct a Lévy process caused by large fluctuations in environment [52]; (v) p(β ∝ 1/T) as a Mittag-Leffler function to construct generalized Maxwell-Boltzmann distributions [56] and other generalized distributions [57,58], even as truncated-Mittag-Leffler function [59], which was applied in the analysis of the time series of oil price.…”

Random diffusivity scenarios behind anomalous non-Gaussian diffusion

“…Its impact can be addressed through the superstatistical approach proposed by C. Beck and E. G. D. Cohen to extend statistical mechanics to complex heterogeneous environments. Superstatistics has been applied to run-and-tumble particles [13], animal movement [14][15][16], metapopulation extinction dynamics [17], time series analyses [18][19][20][21], and many other cases [22][23][24][25][26][27][28]. The superstatistics of fBm has been recently developed, providing theoretical support for various experimental ob-servations, such as, protein diffusion in bacteria [29,30], micro-particles in a bi-dimensional system with disordered distribution of pillars [31], and tracer diffusion in mucin hydrogels [32] (see Refs.…”