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We establish a simple algebraic criterion for the existence of an <jats:italic>equilibrium<\/jats:italic>, that is to say, an invariant probability that satisfies the <jats:italic>detailed balance<\/jats:italic> condition familiar from the thermodynamics of reaction networks. We also find that when such a probability exists, it can be described by a free energy function that combines an internal energy term and an entropy term. 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