Microscopic structure of travelling wave solutions in a class of stochastic interactingparticle systems

Abstract

We obtain exact travelling wave solutions for three families of stochastic one-dimensionalnon-equilibrium lattice models with open boundaries. These solutions describe the diffusivemotion and microscopic structure of (i) shocks in the partially asymmetric exclusionprocess with open boundaries, (ii) a lattice Fisher wave in a reaction–diffusion system, and(iii) a domain wall in non-equilibrium Glauber–Kawasaki dynamics with magnetizationcurrent. For each of these systems we define a microscopic shock position and calculate theexact hopping rates of the travelling wave in terms of the transition rates of themicroscopic model. In the steady state a reversal of the bias of the travelling wave marks afirst-order non-equilibrium phase transition, analogous to the Zel’dovich theory of kineticsof first-order transitions. The stationary distributions of the exclusion process withn shocks can be describedin terms of n-dimensionalrepresentations of matrix product states.