  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.
