Symmetric exclusion process under stochastic power-law resetting

Abstract

We study the behaviour of a symmetric exclusion process in the presence of non-Markovian stochastic resetting, where the configuration of the system is reset to a step-like profile at power-law waiting times with an exponent α. We find that the power-law resetting leads to a rich behaviour for the currents, as well as density profile. We show that, for any finite system, for α < 1, the density profile eventually becomes uniform while for α > 1, an eventual non-trivial stationary profile is reached. We also find that, in the limit of thermodynamic system size, at late times, the average diffusive current grows $\sim\! t^\theta$ with $\theta = 1/2$ for $\alpha \leqslant 1/2$, $\theta = \alpha$ for $1/2 \lt \alpha \leqslant 1$ and θ = 1 for α > 1. We also analytically characterize the distribution of the diffusive current in the short-time regime using a trajectory-based perturbative approach. Using numerical simulations, we show that in the long-time regime, the diffusive current distribution follows a scaling form with an $\alpha-$dependent scaling function. We also characterise the behaviour of the total current using renewal approach. We find that the average total current also grows algebraically $\sim\! t^{\phi}$ where $\phi = 1/2$ for $\alpha \leqslant 1$, $\phi = 3/2-\alpha$ for $1 \lt \alpha \leqslant 3/2$, while for $\alpha \gt 3/2$ the average total current reaches a stationary value, which we compute exactly. The standard deviation of the total current also shows an algebraic growth with an exponent $\Delta = \frac{1}{2}$ for $\alpha \leqslant 1$, and $\Delta = 1-\frac{\alpha}{2}$ for $1 \lt \alpha \leqslant 2$, whereas it approaches a constant value for α > 2. The total current distribution remains non-stationary for α < 1, while, for α > 1, it reaches a non-trivial and strongly non-Gaussian stationary distribution, which we also compute using the renewal approach.