Ergodicity Breaking in Active Run-and-Tumble Particles in a Double-Well Potential

TL;DR

This paper studies how active run-and-tumble particles in a double-well potential exhibit ergodicity breaking above a critical barrier height, with exact calculations of hitting probabilities and a divergence in crossing times near the transition.

Contribution

It provides an analytical demonstration of ergodicity breaking in active particles and derives explicit formulas for stationary distributions and crossing times, contrasting with passive Brownian particles.

Findings

Ergodicity breaks down when barrier exceeds critical height.

Stationary distribution depends on initial conditions above the threshold.

Crossing times diverge near the critical barrier following a Vogel-Fulcher-Tammann-like law.

Abstract

We investigate the dynamics of a run-and-tumble particle in a double-well potential and demonstrate that, in stark contrast to Brownian particles, active dynamics can lead to strong ergodicity breaking. When the barrier height exceeds a critical threshold, the long-time position distribution depends crucially on the initial condition: if the particle starts within the basin of attraction of one well, it remains trapped there, while if it begins between the two basins, it can reach either well with a finite probability, which we compute exactly via hitting probabilities. Below the critical barrier height, ergodicity is restored and the system converges to a unique stationary distribution, which we derive analytically. Using this result, we also estimate the characteristic barrier crossing time and show that it violates Kramer's-Arrhenius law, and displays a divergence near the critical height following a Vogel-Fulcher-Tammann-like form with an anomalous exponent $1/2$.