Thermodynamics of a Brownian particle in a non-confining potential

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.