The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

Journal Article · Sat Aug 15 00:00:00 EDT 2015 · Annals of Physics

DOI:https://doi.org/10.1016/J.AOP.2015.04.019· OSTI ID:22451200

Guo, Ran; Du, Jiulin

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution. - Highlights: • The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion is found. • The anomalous diffusion satisfies a generalized fluctuation–dissipation relation. • At long time the time-dependent solution approaches to a power-law distribution in nonextensive statistics. • Numerically we have demonstrated the accuracy and validity of the time-dependent solution.

• OSTI ID: 22451200
• Journal Information: Annals of Physics, Vol. 359; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
BROWNIAN MOVEMENT
DIFFUSION
FOKKER-PLANCK EQUATION
LANGEVIN EQUATION
STATISTICS
TIME DEPENDENCE