29 September 2017:

This week's progress:

1) My paper with Nina Otter, Operads and phylogenetic trees, was finally accepted for publication by*Theory and Applications of Categories!* Steve Lack was the editor here. The referee had taken over a year to read our paper, and then given us a huge list of suggestions, so this is great news.

2) Blake Pollard's thesis is now on the arXiv:

**Open Markov Processes and Reaction Networks*

> **Abstract.** We define the concept of an 'open' Markov process, a continuous-time Markov chain equipped with specified boundary states through which probability can flow in and out of the system. External couplings which fix the probabilities of boundary states induce non-equilibrium steady states characterized by non-zero probability currents flowing through the system. We show that these non-equilibrium 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 non-equilibrium 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. We construct a 'black-box' 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 black-boxing theorem for reaction networks whose dynamics are given by the non-linear rate equation. Along the way we describe a more general category of open dynamical systems where composition corresponds to gluing together open dynamical systems.

3) Daniel Cicala submitted the final version of his paper Categorifying the zx-calculus to the conference proceedings Quantum Physics and Logic. This was already accepted for publication, so we can count this as yet another touchdown.

This week's progress:

1) My paper with Nina Otter, Operads and phylogenetic trees, was finally accepted for publication by

2) Blake Pollard's thesis is now on the arXiv:

*

> **Abstract.** We define the concept of an 'open' Markov process, a continuous-time Markov chain equipped with specified boundary states through which probability can flow in and out of the system. External couplings which fix the probabilities of boundary states induce non-equilibrium steady states characterized by non-zero probability currents flowing through the system. We show that these non-equilibrium 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 non-equilibrium 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. We construct a 'black-box' 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 black-boxing theorem for reaction networks whose dynamics are given by the non-linear rate equation. Along the way we describe a more general category of open dynamical systems where composition corresponds to gluing together open dynamical systems.

3) Daniel Cicala submitted the final version of his paper Categorifying the zx-calculus to the conference proceedings Quantum Physics and Logic. This was already accepted for publication, so we can count this as yet another touchdown.