A note on confined diffusion

TL;DR

This paper investigates the confined diffusion of Brownian particles, deriving exact series solutions for specific geometries and analyzing the short-time behavior of mean square displacement.

Contribution

It provides explicit solutions for the series describing confined diffusion in circular and spherical domains, enabling precise analysis of short-time dynamics.

Findings

Exact series solutions for 2D circular and 3D spherical confinement

Explicit short-time mean square displacement expressions

Series can sometimes be summed exactly

Abstract

The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion operator. Unfortunately, the latter depend on space dimensionality and on the particular shape of the domain, and an analytical expression is in most circumstances not available. In this article, it is shown that the series may in some circumstances sum up exactly. Explicit calculations are performed for 2D diffusion restricted to a circular domain and 3D diffusion inside a sphere. In both cases, the short-time behaviour of the mean square displacement is obtained.