Thermodynamics of a Brownian particle in a non-confining potential

TL;DR

This paper analyzes the long-time behavior of a Brownian particle in a non-confining potential, deriving the probability distribution and energy scaling laws, and revealing how the free energy approaches that of free diffusion.

Contribution

It provides analytical solutions for the PDF and energy decay in non-confining potentials, highlighting the faster convergence of free energy to the free particle case.

Findings

PDF outside the well converges to Gaussian form at large times

Average energy decays as 1/t^{1/2}

Free energy approaches that of free diffusion as 1/t

Abstract

We consider the overdamped Brownian dynamics of a particle starting inside a square potential well which, upon exiting the well, experiences a flat potential where it is free to diffuse. We calculate the particle's probability distribution function (PDF) at coordinate $x$ and time $t$, $P(x,t)$, by solving the corresponding Smoluchowski equation. The solution is expressed by a multipole expansion, with each term decaying $t^{1/2}$ faster than the previous one. At asymptotically large times, the PDF outside the well converges to the Gaussian PDF of a free Brownian particle. The average energy, which is proportional to the probability of finding the particle inside the well, diminishes as $E\sim 1/t^{1/2}$. Interestingly, we find that the free energy of the particle, $F$, approaches the free energy of a freely diffusing particle, $F_0$, as $\delta F=F-F_0\sim 1/t$, i.e., at a rate faster than $E$. We provide analytical and computational evidences that this scaling behavior of $\delta F$ is a general feature of Brownian dynamics in non-confining potential fields. Furthermore, we argue that $\delta F$ represents a diminishing entropic component which is localized in the region of the potential, and which diffuses away with the spreading particle without being transferred to the heat bath.