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

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 $1/2$.

Figures (5)

• Figure 1: Schematic representation of the double well potential for $a<a_c$ (light orange curve) and $a>a_c$ (dark green curve). The thick sections of the curves indicate the support of the stationary position distribution in the respective cases, and the arrows indicate the direction of the net force on the RTP.
• Figure 2: Schematic representation of the fixed points of the force $f(x)$ for $a>a_c$. Here we have taken $a=1.42$ and $v_0=1$ so that $a_c\simeq 1.3747$. The shaded green regions indicate the supports of the stationary distribution. The dotted lines indicate the boundaries of the basins of attractions of the two wells. The arrows along the $x$-axis indicate the direction of the force on the particle in the respective regions.
• Figure 3: Non-ergodic phase: Plot of $P_\text{st}(x)$ for $a=1.42> a_c$ and different values of initial position $x_0,$ for $\gamma=1$. The solid lines correspond to the analytical prediction Eq. \ref{['eq:P2w']} and symbols to numerical simulations. The inset shows the plot of the hitting probability $\pi(x_0)$ versus $x_0$ for different values of $\gamma$. Here we have taken $v_0=1$ so that $a_c\simeq 1.3747$.
• Figure 4: Position distribution in the ergodic regime: Plot of $P_\text{st}(x)$vs$x$ for $a=1$ for different values of $\gamma.$ The symbols correspond to the data obtained from numerical simulations while the dashed black lines correspond to the analytical prediction Eq. \ref{['eq:pst_gen']}.
• Figure 5: Barrier-crossing time in the ergodic phase: (a) Plot of $\log(\tau)$ versus $a_c-a$ in the log-log scale for different values of flip-rate $\gamma$. The solid lines indicate the data obtained from numerically integrating Eq. \ref{['eq:norm']} while the dashed black lines indicate the near-critical ${(a_c -a)}^{-1/2}$ behaviour predicted in Eq. \ref{['eq:log_tau']}. The symbols indicate the characteristic dwelling time measured from simulations. (b) Plot of escape rate as function of $\gamma$ for different values of $a$, obtained from numerically integrating Eq. \ref{['eq:norm']} and using Eq. \ref{['eq:tau_def']}.