Large deviations in statistics of the local time and occupation time for a run and tumble particle

TL;DR

This paper analyzes the large deviation properties of local and occupation times for a run and tumble particle in one dimension, revealing phase transitions and extending large deviation formalism to active particles.

Contribution

It introduces a large deviation framework for local and occupation times of RTPs, uncovering nonanalytic rate functions and phase transitions, and extends the Donsker-Varadhan formalism to active particles.

Findings

Local time distribution satisfies a large deviation principle.

Presence of drift induces nonanalytic rate functions indicating phase transitions.

Occupation time exhibits a change from unimodal to bimodal endpoint distribution.

Abstract

We investigate the statistics of the local time $\mathcal{T} = \int_0^T \delta(x(t)) dt$ that a run and tumble particle (RTP) $x(t)$ in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution $P(\mathcal{T})$ satisfies the large deviation principle $P(\mathcal{T}) \sim \, e^{-T \, I(\mathcal{T} / T)} $ in the large observation time limit $T \to \infty$. Remarkably, we find that in presence of drift the rate function $I(\rho)$ is nonanalytic: We interpret its singularity as dynamical phase transitions of first order. We then extend these results by studying the statistics of the amount of time $\mathcal{R}$ that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability $P(\mathcal{R} = T)$ that the particle does not exit the interval up to time $T$. We find that the conditional endpoint distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.