A word-counting technique for the solution of stochastic equations

A solution method for finding a stationary state of stochastic equations describing one-dimensional nonequilibrium systems with random sequential update is proposed. It is proved that if such a system (with periodic boundary condition) allows for a solution with finite-range correlations, then this solution can be written as a product of cluster functions. The solution can then be found by setting the coefficients of the linearly independent word counts occurring for a ring to zero. These sets of linearly independent word counts are described generally. The extension of these results to an open linear chain is indicated. General solutions for 1- and 2-clusters are derived. Completely solved examples include (i) a monomer–dimer model, (ii) the general pair reaction–diffusion system, (iii) an extension of the asymmetric exclusion process to two different types of particles (on a ring and on a chain) and (iv) the general exclusion process with dependence on the states of the neighbour sites. Many more examples are discussed more briefly. In all cases with particle conservation, the results are corroborated by simulations.