# Boundary driven Brownian gas

## TL;DR

This paper models a boundary-driven Brownian gas on an interval, characterizing its stationary distribution as a Poisson process and analyzing the empirical flow as a difference of two Poisson processes, revealing detailed flow statistics.

## Contribution

It constructs a Markov process for the Brownian gas with boundary reservoirs and characterizes the stationary distribution and empirical flow in detail.

## Key findings

- Stationary distribution is a Poisson point process with linear interpolation of boundary potentials.
- Empirical flow in the stationary regime is given by the difference of two independent Poisson processes.
- The flow statistics are explicitly identified and bounded for large times.

## Abstract

We consider a gas of independent Brownian particles on a bounded interval in contact with two particle reservoirs at the endpoints. Due to the Brownian nature of the particles, infinitely many particles enter and leave the system in each time interval. Nonetheless, the dynamics can be constructed as a Markov process with continuous paths on a suitable space. If $\lambda_0$ and $\lambda_1$ are the chemical potentials of the boundary reservoirs, the stationary distribution (reversible if and only if $\lambda_0=\lambda_1$) is a Poisson point process with intensity given by the linear interpolation between $\lambda_0$ and $\lambda_1$. We then analyze the empirical flow that it is defined by counting, in a time interval $[0,t]$, the net number of particles crossing a given point $x$. In the stationary regime we identify its statistics and show that it is given, apart an $x$ dependent correction that is bounded for large $t$, by the difference of two independent Poisson processes with parameters $\lambda_0$ and $\lambda_1$.