Freezing transitions of Brownian particles in confining potentials

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 (opt) that minimizes the MFPT to the target. However, when c lies below the critical value, the optimal stiffness k (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 (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 similar or equal to 0.171 85, n similar or equal to 0.395 39). For harmonic or higher order potentials (n > 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.

Automated summary

This study investigates the mean first passage time (MFPT) of a one-dimensional Brownian particle with an external potential in a finite domain. It shows that there is an optimal stiffness that minimizes the MFPT to an absorbing target when the target distance exceeds a critical value. However, when the target distance is below the critical value, the optimal stiffness disappears. The study also reveals that the freezing transition in the optimal stiffness becomes discontinuous when the reflective wall is replaced by a second absorbing target. The phase diagram exhibits three dynamical phases and metastability, with a 'triple' point at specific coordinates. Finally, for harmonic or higher order potentials, the MFPT always increases with stiffness at small values.