# Probability density of the fractional Langevin equation with reflecting walls

## TL;DR

This paper studies how anomalous diffusion described by the fractional Langevin equation behaves in confined spaces with reflecting walls, revealing unique distribution patterns influenced by force correlations.

## Contribution

It provides a detailed analysis of the probability density evolution for fractional Langevin processes with reflecting boundaries, highlighting differences from normal diffusion and fractional Brownian motion.

## Key findings

- Probability density converges to uniform distribution on finite intervals.
- Persistent correlations cause particle accumulation at the wall.
- Anti-persistent correlations lead to particle depletion near the wall.

## Abstract

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 anti-persistent (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 non-thermal fractional Brownian motion with reflecting walls, and we discuss broader implications.