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In recent work, Baez, Fong and the author introduced a framework fordescribing Markov processes equipped with a detailed balanced equilibrium asopen systems of a certain type. These `open Markov processes' serve as thebuilding blocks for more complicated processes. In this paper, we describe thepotential application of this framework in the modeling of biological systemsas open systems maintained away from equilibrium. We show that non-equilibriumsteady states emerge in open systems of this type, even when the rates of theunderlying process are such that a detailed balanced equilibrium is permitted.It is shown that these non-equilibrium steady states minimize a quadratic formwhich we call `dissipation.' In some circumstances, the dissipation isapproximately equal to the rate of change of relative entropy plus a correctionterm. On the other hand, Prigogine's principle of minimum entropy productiongenerally fails for non-equilibrium steady states. We use a simple model ofmembrane transport to illustrate these concepts.

Open Markov processes: A compositional perspective on non-equilibrium steady states in biology