## 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 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 language | English |
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Article number | 014105 |

Journal | Physical Review E |

Volume | 104 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2021 |

## ASJC Scopus subject areas

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