Properties of Statistical Equilibrium Equations: Positivity and Uniqueness

Abstract

While linear statistical equilibrium equations play an important role in the description of nonequilibrium processes in astrophysics, some of their basic mathematical properties, such as uniqueness and positivity, have not been fully explored. In this paper these properties are related to concepts of connectivity from the theory of continuous-time Markov chains. For the irreducible case (in which every state is connected to every other state, either directly or through intermediate states), the solution is shown to be positive and unique when one positive normalization condition is provided. It is then shown how a general linear statistical equilibrium problem can be reduced by dividing the system into inessential and essential states and then partitioning the latter into separate irreducible subproblems. It is shown that: (1) The inessential states all have zero populations. If a positive normalization condition is imposed separately on each irreducible subproblem, then (2) the essential states all have positive populations and (3) the overall solution is unique.