Ergodicity of observable and ergodic hypothesis in Markovian kinetics

Abstract

The time average of any observable, which evolves following an irreducible Markov process with countable states each of which is a persistent state, equals the ensemble average of the same quantity if and only if the corresponding master equation does not exhibit the accumulation of its eigenvalues around the infinitesimal neighborhood of the point with the value zero and if the steady distribution as the eigenfunction with the eigenvalue zero is uniform. The nonuniform steady distribution invalidates the ergodicity of observable. The principle of a priori equal weight is identical to the ergodicity of observable.