Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Abstract

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) as an infinite series in three parameter Mittag-Leffler functions. According to the models, the mean squared displacement of the process is analytically obtained and numerically verified, in particular, the trend of its transition is shown; furthermore the stochastic representation of the process is presented and the positiveness of the PDF of the position of the particles is strictly proved. Finally, the fractional moments of the model are calculated, and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.

Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting

“…The Prabhakar fractional derivative has revealed a class of interesting behaviors in the context of the viscoelasticity theory [52] and in anomalous advection-dispersion transport [53]. On the other hand, the fractional Prabhakar derivative has been an efficient tool in physical models to approach the transition among anomalous diffusions [54,55].…”

“…Our generalization consists of the combination of the fractional diffusion [28,54] and stochastic resetting with memory [13]. To do this we consider the CTRW theory with an additional term g(x, t) proposed by Henry and Wearne in [71].…”

“…in which ℘ ς,ν α,µ [s + κ] = (s+κ) ας−µ (ν+(s+κ) α ) ς . The Equation (18) was reported and analyzed in [54]. Using Equation (18) in Equation 17, we obtain…”

“…Equation 22has many situations that are relevant. In [54] the authors derived the tempered Prabhakar diffusion equation making use of CTRW theory. For simplicity, in this section, we will consider that the system is governed by an equation that is free of external forces.…”

Control of the transient subdiffusion exponent at short and long times

“…then the FP equation is expressed in terms of the Prabhakar integral operator [29,30]. Here the Prabhakar-type kernel is expressed in terms of the three-parameter Mittag-Leffler function, having the following Taylor series…”