TY - JOUR

T1 - Thermodynamics of a Brownian particle in a nonconfining potential

AU - Farago, Oded

N1 - Funding Information:
I thank Eli Barkai and Erez Aghion for critical comments on the manuscript. The support of the Israel Science Foundation (ISF) Grant No. 991/17 is acknowledged.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/7/1

Y1 - 2021/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85109301769&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.104.014105

DO - 10.1103/PhysRevE.104.014105

M3 - Article

C2 - 34412327

AN - SCOPUS:85109301769

SN - 2470-0045

VL - 104

JO - Physical Review E

JF - Physical Review E

IS - 1

M1 - 014105

ER -