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Open Markov Processes and Reaction Networks
 Pollard, Blake Stephen
 Advisor(s): Baez, John C
Abstract
We begin by defining the concept of `open' Markov processes, which are continuoustime Markov chains where probability can flow in and out through certain `boundary' states. We study open Markov processes which in the absence of such boundary flows admit equilibrium states satisfying detailed balance, meaning that the net flow of probability vanishes between all pairs of states. External couplings which fix the probabilities of boundary states can maintain such systems in nonequilibrium steady states in which nonzero probability currents flow. We show that these nonequilibrium steady states minimize a quadratic form which we call `dissipation.' This is closely related to Prigogine's principle of minimum entropy production. We bound the rate of change of the entropy of a driven nonequilibrium steady state relative to the underlying equilibrium state in terms of the flow of probability through the boundary of the process.
We then consider open Markov processes as morphisms in a symmetric monoidal category by splitting up their boundary states into certain sets of `inputs' and `outputs.' Composition corresponds to gluing the outputs of one such open Markov process onto the inputs of another so that the probability flowing out of the first process is equal to the probability flowing into the second. Tensoring in this category corresponds to placing two such systems side by side.
We construct a `blackbox' functor characterizing the behavior of an open Markov process in terms of the space of possible steady state probabilities and probability currents along the boundary. The fact that this is a functor means that the behavior of a composite open Markov process can be computed by composing the behaviors of the open Markov processes from which it is composed. We prove a similar blackboxing theorem for reaction networks whose dynamics are given by the nonlinear rate equation. Along the way we describe a more general category of open dynamical systems where composition corresponds to gluing together open dynamical systems.
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