Generalized Langevin equation and the Prabhakar derivative
type: Item
instance of: scholarly article
title: Generalized Langevin equation and the Prabhakar derivative
zbMATH Open document ID: 1395.82195
author: Trifce Sandev
published in: Mathematics
publication date: 2018-07-05T00:00:00Z
review text: Summary: 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.
Mathematics Subject Classification ID: 82C31
Mathematics Subject Classification ID: 35Q82
Mathematics Subject Classification ID: 82C27
zbMATH DE Number: 6898695
zbMATH Keywords: generalized Langevin equation
zbMATH Keywords: regularized Prabhakar derivative
zbMATH Keywords: tempered memory kernel
zbMATH Keywords: Mittag-Leffler functions
describes a project that uses: ML
MaRDI profile type: Publication
full work available at URL: math5040066
OpenAlex ID: W2769965861
cites work: The Langevin equation. With applications to stochastic problems in physics, chemistry and electrical engineering.
cites work: Nonequilibrium statistical mechanics
cites work: Fractional Langevin Equation
cites work: Q5190149
cites work: Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
cites work: Generalized Langevin equation with tempered memory kernel
cites work: Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel
cites work: Beyond monofractional kinetics
cites work: Q5625726
cites work: Hilfer-Prabhakar derivatives and some applications
cites work: Some properties of Prabhakar-type fractional calculus operators
cites work: Generalized mittag-leffler function and generalized fractional calculus operators
cites work: Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models
cites work: Models of dielectric relaxation based on completely monotone functions
cites work: Prabhakar-like fractional viscoelasticity
cites work: Mathematical modeling of fractional differential filtration dynamics based on models with Hilfer-Prabhakar derivative
cites work: Q4625330
cites work: Analytic solution of generalized space time fractional reaction diffusion equation
cites work: Time-fractional particle deposition in porous media
cites work: General fractional calculus, evolution equations, and renewal processes
cites work: General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
cites work: Applications of fractional calculus in physics
cites work: Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative
cites work: Space-time fractional reaction-diffusion equations associated with a generalized Riemann-Liouville fractional derivative
cites work: Computational solutions of distributed order reaction-diffusion systems associated with Riemann-Liouville derivatives
cites work: Analytical solution of generalized space-time fractional cable equation
cites work: Q5547508
cites work: The Prabhakar or three parameter Mittag-Leffler function: theory and application
cites work: Aging properties of an anomalously diffusing particle
cites work: Convergence of series in three parametric Mittag-Leffler functions
cites work: From Bessel to multi-index Mittag-Leffler functions. Enumerable families, series in them and convergence
DOI: 10.3390/MATH5040066
description: scientific article; zbMATH DE number 6898695