Relaxation in open one-dimensional systems

TL;DR

This paper introduces a new master equation model for open one-dimensional systems of interacting random walkers, revealing complex non-exponential energy relaxation behaviors influenced by system size and density exchange.

Contribution

A novel master equation approach for open single-file systems with fluctuating particle number and external bias, demonstrating complex relaxation dynamics.

Findings

Energy relaxation is highly non-exponential when system size is comparable to perturbation range.

Relaxation can exhibit stretched exponential and logarithmic time dependence.

Density exchange weakens non-exponential relaxation features.

Abstract

A new master equation to mimic the dynamics of a collection of interacting random walkers in an open system is proposed and solved numerically.In this model, the random walkers interact through excluded volume interaction (single-file system); and the total number of walkers in the lattice can fluctuate because of exchange with a bath.In addition, the movement of the random walkers is biased by an external perturbation. Two models for the latter are considered: (1) an inverse potential (V $\propto$ 1/r), where r is the distance between the center of the perturbation and the random walker and (2) an inverse of sixth power potential ($V \propto 1/r^6 $). The calculated density of the walkers and the total energy show interesting dynamics. When the size of the system is comparable to the range of the perturbing field, the energy relaxation is found to be highly non-exponential. In this range, the system can show stretched exponential ($ e^{-{(t/\tau_s)}^{\beta}} $) and even logarithmic time dependence of energy relaxation over a limited range of time. Introduction of density exchange in the lattice markedly weakens this non-exponentiality of the relaxation function, irrespective of the nature of perturbation.