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 $\alpha$. 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 $\alpha<1$, the density profile eventually becomes uniform while for $\alpha >1$, 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 $\alpha \le 1/2$, $\theta = \alpha$ for $1/2 < \alpha \le 1$ and $\theta=1$ for $\alpha > 1$. 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 $\alpha \le 1$, $\phi=3/2-\alpha$ for $1 < \alpha \le 3/2$, while for $\alpha > 3/2$ the average total current reaches a stationary value, which we compute exactly. The variance of the total current also shows an algebraic growth with an exponent $\Delta=1$ for $\alpha \le 1$, and $\Delta=2-\alpha$ for $1 < \alpha \le 2$, whereas it approaches a constant value for $\alpha>2$. The total current distribution remains non-stationary for $\alpha<1$, while, for $\alpha>1$, it reaches a non-trivial and strongly non-Gaussian stationary distribution, which we also compute using the renewal approach.