Thermodynamics of a Brownian particle in a nonconfining potential

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4 Scopus citations


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 t1/2 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/t1/2. 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 evidence that this scaling behavior of δF is a general feature of Brownian dynamics in nonconfining 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.

Original languageEnglish
Article number014105
JournalPhysical Review E
Issue number1
StatePublished - 1 Jul 2021

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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