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

We obtain exact travelling wave solutions for three families of stochastic one-dimensional non-equilibrium lattice models with open boundaries. These solutions describe the diffusive motion and microscopic structure of (i) shocks in the partially asymmetric exclusion process 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 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