Long time position distribution of an active Brownian particle in two dimensions

TL;DR

This paper analyzes the late-time position distribution of a two-dimensional active Brownian particle, revealing Gaussian behavior near the peak and non-Gaussian tails, with a focus on large deviations and the influence of activity.

Contribution

It provides an analytical and numerical study of the large deviation rate function for the particle's position, highlighting the persistent influence of activity at late times.

Findings

The position distribution approaches Gaussian near the peak at late times.

The distribution exhibits non-Gaussian tails characterized by a large deviation rate function.

The stationary distribution under harmonic confinement transitions from ring-shaped to Gaussian as activity decreases.

Abstract

We study the late time dynamics of a single active Brownian particle in two dimensions with speed $v_0$ and rotation diffusion constant $D_R$. We show that at late times $t\gg D_R^{-1}$, while the position probability distribution $P(x,y,t)$ in the $x$-$y$ plane approaches a Gaussian form near its peak describing the typical diffusive fluctuations, it has non-Gaussian tails describing atypical rare fluctuations when $\sqrt{x^2+y^2}\sim v_0 t$. In this regime, the distribution admits a large deviation form, $P(x,y,t) \sim \exp\left[-t\, D_R\, \Phi\left(\sqrt{x^2+y^2}/(v_0 t)\right)\right]$, where we compute the rate function $\Phi(z)$ analytically and also numerically using an importance sampling method. We show that the rate function $\Phi(z)$, encoding the rare fluctuations, still carries the trace of activity even at late times. Another way of detecting activity at late times is to subject the active particle to an external harmonic potential. In this case we show that the stationary distribution $P_\text{stat}(x,y)$ depends explicitly on the activity parameter $D_R^{-1}$ and undergoes a crossover, as $D_R$ increases, from a ring shape in the strongly active limit ($D_R\to 0$) to a Gaussian shape in the strongly passive limit $(D_R\to \infty)$.