Thermodynamics of a Brownian particle in a non-confining potential

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 ∼ 1/t 1/2 . Interestingly, we find that the free energy of the particle, F , approaches the free energy of a freely diffusing particle, F0, as δF = F − F0 ∼ 1/t, i.e., at a rate faster than E. We provide analytical and computational evidences that this scaling behavior of δF is a general feature of Brownian dynamics in non-confining potential fields. Furthermore, we argue that δF 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.