Freezing transitions of Brownian particles in confining 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 $k_{\mathrm{opt}}$ that minimizes the MFPT to the target. However, when $c$ lies below the critical value, the optimal stiffness $k_{\mathrm{opt}}$ 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 $k_{\mathrm{opt}}$ 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.

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References in corpus (33)

• Diffusion with Stochastic Resetting
• Stochastic Resetting and Applications
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Diffusion with Optimal Resetting
• First Passage Under Restart
• Geometry-controlled kinetics
• First order transition for the optimal search time of Lévy flights with resetting
• Optimal stochastic restart renders fluctuations in first passage times universal
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• First passage under stochastic resetting in an interval
• Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics
• Stochastic multiplicative processes with reset events
• Restart could optimize the probability of success in a Bernoulli trial
• Effects of refractory period on stochastic resetting
• First passage of a particle in a potential under stochastic resetting: a vanishing transition of optimal resetting rate
• Péclet number governs transition to acceleratory restart in drift-diffusion
• Landau theory of restart transitions
• First Passage Problems in Biology
• Optimizing Brownian escape rates by potential shaping
• Intermittent resetting potentials
• Random acceleration process under stochastic resetting
• Resetting with stochastic return through linear confining potential
• First passage in the presence of stochastic resetting and a potential barrier
• Comparison of two models of tethered motion
• Resetting transition is governed by an interplay between thermal and potential energy
• Dynamical phase transition in the first-passage probability of a Brownian motion
• How a finite potential barrier decreases the mean first passage time
• Freezing transition in the barrier crossing rate of a diffusing particle
• Optimal potentials for diffusive search strategies
• Dichotomous flow with thermal diffusion and stochastic resetting
• Speeding up the first-passage for subdiffusion by introducing a finite potential barrier
• First-passage time of run-and-tumble particles with non-instantaneous resetting

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