Random attraction in TASEP with time-varying hopping rates

TL;DR

This paper analyzes a time-varying TASEP model using nonautonomous random dynamical systems, demonstrating conditions for synchronization and stability of particle configurations over time.

Contribution

It formulates TASEP with time-varying rates as a nonautonomous random dynamical system and establishes conditions for the existence of unique attractors and synchronization.

Findings

Conditions for attractor existence are proven and shown to be tight.

Almost sure synchronization of particle paths is demonstrated.

Perturbations are shown to be filtered out in the long run.

Abstract

The totally asymmetric simple exclusion principle (TASEP) is a fundamental model in nonequilibrium statistical mechanics. It describes the stochastic unidirectional movement of particles along a 1D chain of ordered sites. We consider the continuous-time version of TASEP with a finite number of sites and with time-varying hopping rates between the sites. We show how to formulate this model as a nonautonomous random dynamical system (NRDS) with a finite state-space. We provide conditions guaranteeing that random pullback and forward attractors of such an NRDS exist and consist of singletons. In the context of the nonautonomous TASEP these conditions imply almost sure synchronization of the individual random paths. This implies in particular that perturbations that change the state of the particles along the chain are "filtered out" in the long run. We demonstrate that the required conditions are tight by providing examples where these conditions do not hold and consequently the forward attractor does not exist or the pullback attractor is not a singleton. The results in this paper generalize our earlier results for autonomous TASEP in https://doi.org/10.1137/20M131446X and contain these as a special case.