Generalized Langevin Equation and the Prabhakar Derivative

Trifce Sandev

doi:10.3390/math5040066

Abstract

We consider a generalized Langevin equation with regularized Prabhakar derivative operator. We analyze the mean square displacement, time-dependent diffusion coefficient and velocity autocorrelation function. We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative. Various diffusive behaviors are observed. We show the importance of the three parameter Mittag-Leffler function in the description of anomalous diffusion in complex media. We also give analytical results related to the generalized Langevin equation for a harmonic oscillator with generalized friction. The normalized displacement correlation function shows different behaviors, such as monotonic and non-monotonic decay without zero-crossings, oscillation-like behavior without zero-crossings, critical behavior, and oscillation-like behavior with zero-crossings. These various behaviors appear due to the friction of the complex environment represented by the Mittag-Leffler and tempered Mittag-Leffler memory kernels. Depending on the values of the friction parameters in the system, either diffusion or oscillations dominate.

Highlights

The Langevin equation for a Brownian particle with mass m = 1 is represented by the following equaion [1,2,3]ẍ (t) + γ ẋ (t) = ξ (t), ẋ (t) = v(t), (1)where x (t) is the particle displacement, v(t) is the particle velocity, γ is the friction coefficient, and ξ (t) is a stationary random force with zero mean and correlation hξ (t)ξ (t0 )i = 2γk B Tδ(t0 − t), where 2γk B T is the so-called spectral density
We further introduce the so-called tempered regularized Prabhakar derivative and analyze the corresponding generalized Langevin equation with friction term represented through the tempered derivative
We show that the generalized Langevin equation with a friction represented in terms of the regularized Prabhakar derivative generates decelerating subdiffusion

Summary

Introduction

Is the Caputo fractional derivative of order 0 < μ < 1 [5], and γμ is the generalized friction coefficient This equation is a special case of the generalized Langevin equation (GLE) with the power-law memory kernel γ(t) = γμ t−μ /Γ(1 − μ) (see Section 3). In this paper we introduce the GLE with a friction term represented through the regularized Prabhakar derivative (see Section 2 for details), i.e., δ,μ ẍ (t) + γμ,ρ,δ C Dρ,−ν,t x (t) = ξ (t),. Where 0 < μ, δ < 1, 0 < μ/δ < 1, 0 < μ/δ − ρ < 1, ν = τ −μ , τ is a time parameter, and γμ,ρ,δ is the generalized friction coefficient This equation is a generalization of the fractional Langevin. We further introduce a GLE with a tempered regularized Prabhakar friction term and analyze the normalized displacement correlation function in case of harmonic potential.

Prabhakar Derivatives

Free Particle

Tempered Friction

Harmonically Bounded Particle in Presence of Prabhakar Friction Term

Summary