Freezing transitions of Brownian particles in confining potentials

TL;DR

This paper investigates how the mean first passage time of a Brownian particle in a confining potential undergoes a continuous or discontinuous transition in optimal potential stiffness, revealing a phase diagram with multiple dynamical phases.

Contribution

It introduces the concept of a freezing transition in the optimal potential stiffness for Brownian particles in confining potentials, with detailed phase diagrams and critical points.

Findings

Optimal potential stiffness minimizes MFPT when domain size exceeds a critical value.

Freezing transition in stiffness is continuous for one target and discontinuous for two targets.

MFPT always increases with potential stiffness for harmonic or higher order potentials.

Abstract

We study the mean first passage time (MFPT) to an absorbing target of a one-dimensional Brownian particle subject to an external potential $v(x)$ in a finite domain. We focus on the cases in which the external potential is confining, of the form $v(x)=k|x-x_0|^n/n$, and where the particle's initial position coincides with $x_0$. We first consider a particle between an absorbing target at $x=0$ and a reflective wall at $x=c$. At fixed $x_0$, we show that when the target distance $c$ exceeds a critical value, there exists a nonzero optimal stiffness $k_{\rm opt}$ that minimizes the MFPT to the target. However, when $c$ lies below the critical value, the optimal stiffness $k_{\rm opt}$ vanishes. Hence, for any value of $n$, the optimal potential stiffness undergoes a continuous "freezing" transition as the domain size is varied. On the other hand, when the reflective wall is replaced by a second absorbing target, the freezing transition in $k_{\rm opt}$ becomes discontinuous. The phase diagram in the $(x_0,n)$-plane then exhibits three dynamical phases and metastability, with a "triple" point at $(x_0/c\simeq 0.17185$, $n\simeq 0.39539)$. For harmonic or higher order potentials $(n\ge 2)$, the MFPT always increases with $k$ at small $k$, for any $x_0$ or domain size. These results are contrasted with problems of diffusion under optimal resetting in bounded domains.