Microscopic structure of travelling wave solutions in a class of stochastic interacting particle systems

TL;DR

This paper derives exact microscopic traveling wave solutions for various stochastic lattice models, revealing phase transitions and shock dynamics in non-equilibrium systems.

Contribution

It introduces a method to obtain exact traveling wave solutions and shock positions in three classes of stochastic models, linking microscopic dynamics to phase transitions.

Findings

Exact shock positions and hopping rates are calculated.

Reversal of wave bias indicates a first-order phase transition.

Stationary distributions are described by matrix product states.

Abstract

We obtain exact travelling wave solutions for three families of stochastic one-dimensional nonequilibrium lattice models with open boundaries. These solutions describe the diffusive motion and microscopic structure of (i) of shocks in the partially asymmetric exclusion process with open boundaries, (ii) of a lattice Fisher wave in a reaction-diffusion system, and (iii) of a domain wall in non-equilibrium Glauber-Kawasaki dynamics with magnetization current. For each of these systems we define a microscopic shock position and calculate the exact hopping rates of the travelling wave in terms of the transition rates of the microscopic model. In the steady state a reversal of the bias of the travelling wave marks a first-order non-equilibrium phase transition, analogous to the Zel'dovich theory of kinetics of first-order transitions. The stationary distributions of the exclusion process with $n$ shocks can be described in terms of $n$-dimensional representations of matrix product states.