Dynamical phase transition in the first-passage probability of a Brownian motion

TL;DR

This paper investigates a dynamical phase transition in the first-passage time distribution of a Brownian particle, revealing a critical parameter that changes the distribution's shape, supported by theory, experiments, and simulations.

Contribution

It identifies and characterizes a phase transition in first-passage time distributions of Brownian motion, combining theoretical analysis, experimental validation, and numerical simulations.

Findings

Existence of a critical ratio b_c = L/σ that determines the shape of the distribution.

For b > b_c, the distribution has a maximum and minimum; for b < b_c, it decreases monotonically.

Experimental and numerical evidence confirm the theoretical predictions of the phase transition.

Abstract

We study theoretically, experimentally and numerically the probability distribution $F(t_f|x_0,L)$ of the first passage times $t_f$ needed by a freely diffusing Brownian particle to reach a target at a distance $L$ from the initial position $x_0$, taken from a normalized distribution $(1/\sigma)\, g(x_0/\sigma)$ of finite width $\sigma$. We show the existence of a critical value $b_c$ of the parameter $b=L/\sigma$, which determines the shape of $F(t_f|x_0,L)$. For $b>b_c$ the distribution $F(t_f|x_0,L)$ has a maximum and a minimum whereas for $b<b_c$ it is a monotonically decreasing function of $t_f$. This dynamical phase transition is generated by the presence of two characteristic times $\sigma^2/D$ and $L^2/D$, where $D$ is the diffusion coefficient. The theoretical predictions are experimentally checked on a Brownian bead whose free diffusion is initialized by an optical trap which determines the initial distribution $g(x_0/\sigma)$. The presence of the phase transition in 2d has also been numerically estimated using a Langevin dynamics.