The Markov Chain tree theorem expresses the steady state distribution of an irreducible Markov matrix in terms of directed spanning trees of its associated graph. It was probably originally discovered in the context of certain models for biological systems. After a short mathematical introduction, explaining a little graph theory so that everyone's on the same page, I will attempt to (succinctly) prove the theorem. In its original incarnation, the theorem appears an unhelpful brute force approach so I will try to motivate some of its uses in physics. Specifically, I will discuss some work by Christian Maes et. al which gives insight into Landauer's Blowtorch theorem. I will also introduce related work by his group on how the Markov tree theorem can be used to calculate steady states given a separation of scales e.g. at low temperatures.
This is a roughly weekly series of didactical blackboard talks focussing on some theoretical aspect of Condensed Matter, Biological, and Statistical Physics..