Current fluctuations in stochastically resetting particle systems

Abstract

We consider a system of non-interacting particles on a line with initial positions distributed uniformly with density ρ on the negative half-line. We consider two different models: (i) each particle performs independent Brownian motion with stochastic resetting to its initial position with rate r and (ii) each particle performs run and tumble motion, and with rate r its position gets reset to its initial value and simultaneously its velocity gets randomised. We study the effects of resetting on the distribution P (Q, t) of the integrated particle current Q up to time t through the origin (from left to right). We study both the annealed and the quenched current distributions and in both cases, we find that resetting induces a stationary limiting distribution of the current at long times. However, we show that the approach to the stationary state of the current distribution in the annealed and the quenched cases are drastically different for both models. In the annealed case, the whole distribution Pan(Q, t) approaches its stationary limit uniformly for all Q. In contrast, the quenched distribution Pqu(Q, t) attains its stationary form for Q < Qcrit(t), while it remains time-dependent for Q > Qcrit(t). We show that Qcrit(t) increases linearly with t for large t. On the scale where Q ∼ Qcrit(t), we show that Pqu(Q, t) has an unusual large deviation form with a rate function that has a third-order phase transition at the critical point. We have computed the associated rate functions analytically for both models. Using an importance sampling method that allows to probe probabilities as tiny as 10 −14000 , we were able to compute numerically this non-analytic rate function for the resetting Brownian dynamics and found excellent agreement with our analytical prediction.

Effusion of stochastic processes on a line

“…For noninteracting particles, a direct probabilistic approach was used recently in [10] that allowed to compute the current distribution for more general single particle dynamics, going beyond the simple diffusion. This includes, for instance, run-and-tumble particles (RTPs) [10] and more recently, stochastically resetting particles [11].…”

“…However, the distribution of N + (t) for the quenched case is highly nontrivial [9,10]. For example, the large deviation function characterizing the tail of the quenched distribution of N + (t) undergoes a third order phase transition for stochastically resetting dynamics (both Brownian and RTP) [11]. No signature of this phase transition is found in the annealed large deviation of N + (t) [11].…”

“…For example, the large deviation function characterizing the tail of the quenched distribution of N + (t) undergoes a third order phase transition for stochastically resetting dynamics (both Brownian and RTP) [11]. No signature of this phase transition is found in the annealed large deviation of N + (t) [11]. For a certain class of non-Poissonian initial conditions with a uniform density, the effect of initial conditions persists in the variance of the current even at late times [27][28][29].…”