Non-Markovian process with variable memory functions

TL;DR

This paper models non-Markovian processes using Mittag-Leffler functions as memory kernels in the GKFE, demonstrating a transition from exponential to power-law behavior and providing generalized equations.

Contribution

It introduces a novel approach to incorporate Mittag-Leffler functions into the GKFE to describe non-Markovian memory effects with a crossover analysis.

Findings

Solutions are identical for exponential and first-order stretched exponential for small τ.

The crossover from short to long time behavior is mathematically proven.

A generalized integro-differential form of the GKFE is derived.

Abstract

We present a treatment of non-Markovian character of memory by incorporating different forms of Mittag-Leffler (ML) function, which generally arises in the solution of fractional master equation, as different memory functions in the Generalized Kolmogorov-Feller Equation (GKFE). The cross-over from the short time (stretched exponential) to long time (inverse power law) approximations of the ML function incorporated in the GKFE is proven. We have found that the GKFE solutions are the same for negative exponential and for upto frst order expansion of stretched exponential function for very small $\tau \rightarrow 0$. A generalized integro-differential equation form of the GKFE along with an asymptotic case is provided.