We investigate anomalous diffusion processes governed by the fractional Langevin equation and confined to a finite or semi-infinite interval by reflecting potential barriers. As the random and damping forces in the fractional Langevin equation fulfill the appropriate fluctuation-dissipation relation, the probability density on a finite interval converges for long times towards the expected uniform distribution prescribed by thermal equilibrium. In contrast, on a semi-infinite interval with a reflecting wall at the origin, the probability density shows pronounced deviations from the Gaussian behavior observed for normal diffusion. If the correlations of the random force are persistent (positive), particles accumulate at the reflecting wall while antipersistent (negative) correlations lead to a depletion of particles near the wall. We compare and contrast these results with the strong accumulation and depletion effects recently observed for nonthermal fractional Brownian motion with reflecting walls, and we discuss broader implications.



Research Center/Lab(s)

Center for High Performance Computing Research


This work was supported in part by the NSF under Grant Nos. DMR-1506152 and DMR-1828489 (T.V.) and by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. ME 1535/7-1 (R.M.).

Keywords and Phrases

Brownian movement; Probability distributions, Anomalous diffusion; Fluctuation-dissipation relation; Fractional brownian motion; Fractional langevin equations; Probability densities; Semi-infinite intervals; Thermal equilibriums; Uniform distribution, Differential equations, article; diffusion; motion; probability

International Standard Serial Number (ISSN)

2470-0045; 2470-0053

Document Type

Article - Journal

Document Version

Final Version

File Type





© 2019 American Physical Society (APS), All rights reserved.

Publication Date

01 Oct 2019

Included in

Physics Commons