Velocity and displacement correlation functions for fractional generalized Langevin equations

Abstract

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.MSC 2010 : Primary 82C31; Secondary 26A33, 33E12

On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence

“…For the case of the functions (62), the theorems can be proved analogously using the specific asymptotic estimate (17). These are about to be published in [31].…”

“…Formulas for differentiation and integration of integer and fractional orders of E γ α β ( ) and the properties of the integral operator with such kernel in L( ) can be found in Kilbas, Saigo and Saxena [16]. For a recent use of the Prabhakar M-L type function (2) in the friction memory kernel and in the exact solutions of the fractional generalized Langevin equation, see for example Sandev, Metzler and Tomovski [17]. Recently, a class of special functions of Mittag-Leffler type that are multi-index analogues of E α β ( ) has been introduced and studied by Yakubovich and Luchko [18], Luchko [19], Kiryakova [7,20] …”

Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model

“…where B(β) = B(0) = B(1) = 1 is a normalization function and E γ is the Mittag-Leffler function [6][7][8][9][10][11][12]. The Mittag-Leffler kernel is a combination of both the exponential-law and power-law.…”

“…Recently, Abdon Atangana and Dumitru Baleanu proposed two fractional-order operators involving the generalized Mittag-Leffler function. The generalized Mittag-Leffler function was introduced in the literature to improve the limitations posed by the power-law [6][7][8][9][10][11][12]. The two-parametric, three-parametric, four-parametric and multiple Mittag-Leffler functions were presented by Wiman, Prabhakar, Shukla and Srivastava in [13][14][15][16][17][18].…”

Analytical solutions for the fractional diffusion-advection equation describing super-diffusion

“…The dynamical systems of fractional order are non-conservative and involve non-local operators [2][3][4][5][6][7]. Several approaches have been used for investigating anomalous diffusion, Langevin equations [8,9], random walks [10,11], or fractional derivatives, based on fractional calculus (FC) several works connected to anomalous diffusion processes may be found in [12][13][14][15][16][17][18][19]. Scher and Montroll [20] presented a stochastic model for the photocurrent transport in amorphous materials.…”