Symmetric Exclusion Process under Stochastic Power-law Resetting

TL;DR

This paper investigates how non-Markovian power-law stochastic resetting affects the symmetric exclusion process, revealing diverse behaviors in density profiles and currents depending on the resetting exponent.

Contribution

It analytically and numerically characterizes the impact of power-law resetting on the symmetric exclusion process, including density profiles, current growth, and distribution properties, across different regimes.

Findings

Density profile becomes uniform for α<1 and stationary for α>1.

Diffusive current exhibits different growth exponents depending on α.

Total current reaches a stationary distribution for α>1, with non-Gaussian features.

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.