Boundary conditions in stochastic master equations describing second-order non-equilibrium phase transition behaviour in steady states

Abstract

The role of boundary conditions in single-variable stochastic master equation systems displaying second-order-type non-equilibrium phase transition behaviour is analysed. The formal solution of the master equation is obtained in terms of the eigenvalues of the reaction matrix. The asymptotic behaviour of this solution is investigated and it is shown that the steady-state probability distribution depends upon two parameters which are decided by the detailed knowledge of the initial conditions and the eigenspectrum of the reaction matrix. Considering the lack of information regarding the initial conditions and the wide range of possible initial conditions leading to the same macroscopic state a generalised entropy maximisation principle is introduced to obtain the unique steady-state probability distribution. Explicit application to a simple semiconductor generation-recombination model is discussed.