Matrix product solution for a partially asymmetric 1D lattice gas with a free defect

TL;DR

This paper studies a one-dimensional driven lattice gas with a mobile defect, deriving its phase diagram via mean field theory and exact solutions, revealing shock formation and finite size effects not captured by mean field.

Contribution

It provides an exact matrix product solution for the steady state of a partially asymmetric lattice gas with a free defect, extending understanding of nonequilibrium phase behavior.

Findings

Three phases identified in the phase diagram.

Exact solutions for density profiles and currents in the special case.

Shock front width scales as the square root of system size.

Abstract

A one-dimensional, driven lattice gas with a freely moving, driven defect particle is studied. Although the dynamics of the defect are simply biased diffusion, it disrupts the local density of the gas, creating nontrivial nonequilibrium steady states. The phase diagram is derived using mean field theory and comprises three phases. In two phases, the defect causes small localized perturbations in the density profile. In the third, it creates a shock, with two regions at different bulk densities. When the hopping rates satisfy a particular condition (that the products of the rates of the gas and defect are equal), it is found that the steady state can be solved exactly using a two-dimensional matrix product ansatz. This is used to derive the phase diagram for that case exactly and obtain exact asymptotic and finite size expressions for the density profiles and currents in all phases. In particular, the front width in the shock phase on a system of size $L$ is found to scale as $L^{1/2}$, which is not predicted by mean field theory. The results are found to agree well with Monte Carlo simulations.