Archimedes' principle for Brownian liquid

TL;DR

This paper investigates the stationary distribution of Brownian particles with gravity in confined spaces, illustrating Archimedes' principle through mathematical analysis of reflecting Brownian motion.

Contribution

It introduces a novel analysis of reflecting Brownian motion with drift in arbitrary domains, connecting stochastic processes to classical physical principles.

Findings

Stationary distribution characterized for Brownian particles with gravity

Illustration of Archimedes' principle in a stochastic setting

Application of sphere packing results in two dimensions

Abstract

We consider a family of hard core objects moving as independent Brownian motions confined to a vessel by reflection. These are subject to gravitational forces modeled by drifts. The stationary distribution for the process has many interesting implications, including an illustration of the Archimedes' principle. The analysis rests on constructing reflecting Brownian motion with drift in a general open connected domain and studying its stationary distribution. In dimension two we utilize known results about sphere packing.