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Title: Transport in the spatially tempered, fractional Fokker-Planck equation

Abstract

A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, lambda, and on the order of the fractional derivative in space, alpha. An expansion of the TFD operator for large lambda is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit lambda -> infinity, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (lambda > 0) truncations, and alpha not equal 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In themore » limit lambda -> infinity, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for alpha not equal 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any lambda > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with lambda, as J similar to lambda(-zeta), for alpha >= 1.75. However, for alpha <= 1.5, the steady-state current decays exponentially with lambda, as J similar to e(-xi lambda). In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the current on lambda and the physical parameters of the system.« less

Authors:
 [1];  [2]
  1. University of California, Los Angeles
  2. ORNL
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1048179
DOE Contract Number:  
DE-AC05-00OR22725
Resource Type:
Journal Article
Journal Name:
Journal of Physics A: Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 45; Journal Issue: 25; Journal ID: ISSN 1751-8113
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMMETRY; DECAY; DIFFUSION; DISTRIBUTION; FOKKER-PLANCK EQUATION; HARMONIC POTENTIAL; HARMONICS; PERIODICITY; POTENTIALS; PROBABILITY DENSITY FUNCTIONS; SYMMETRY; TEMPERING; TRANSPORT

Citation Formats

Kullberg, A, and Del-Castillo-Negrete, Diego B. Transport in the spatially tempered, fractional Fokker-Planck equation. United States: N. p., 2012. Web. doi:10.1088/1751-8113/45/25/255101.
Kullberg, A, & Del-Castillo-Negrete, Diego B. Transport in the spatially tempered, fractional Fokker-Planck equation. United States. https://doi.org/10.1088/1751-8113/45/25/255101
Kullberg, A, and Del-Castillo-Negrete, Diego B. 2012. "Transport in the spatially tempered, fractional Fokker-Planck equation". United States. https://doi.org/10.1088/1751-8113/45/25/255101.
@article{osti_1048179,
title = {Transport in the spatially tempered, fractional Fokker-Planck equation},
author = {Kullberg, A and Del-Castillo-Negrete, Diego B},
abstractNote = {A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker-Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, lambda, and on the order of the fractional derivative in space, alpha. An expansion of the TFD operator for large lambda is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit lambda -> infinity, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (lambda > 0) truncations, and alpha not equal 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald-Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit lambda -> infinity, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for alpha not equal 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any lambda > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with lambda, as J similar to lambda(-zeta), for alpha >= 1.75. However, for alpha <= 1.5, the steady-state current decays exponentially with lambda, as J similar to e(-xi lambda). In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the current on lambda and the physical parameters of the system.},
doi = {10.1088/1751-8113/45/25/255101},
url = {https://www.osti.gov/biblio/1048179}, journal = {Journal of Physics A: Mathematical and Theoretical},
issn = {1751-8113},
number = 25,
volume = 45,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 2012},
month = {Sun Jan 01 00:00:00 EST 2012}
}