Statistical mechanics of boundary driven systems

Statistical mechanics is concerned with finding the macroscopic behaviour of a physical system given its microscopic characteristics. At equilibrium there is a general framework given in terms of the various statistical ensembles that describes how to calculate the macroscopic quantity that is desired. Out of equilibrium there is no such framework, leading to the treatment of microscopic models on an individual basis and the investigation of arbitrarily defined models. However, there exists a recent theory of boundary driven steady states and an associated nonequilibrium counterpart to detailed balance
due to Evans.

In this thesis I first review this theory of boundary driven steady states and the associated nonequilibrium counterpart to detailed balance due to Evans, before applying the theory to some toy models of driven athermal systems. These initial attempts do not reproduce the qualitative behaviour of granular systems such as jamming but are a valuable and novel starting point for a
more thorough investigation of this technique.

I then move on to the general theory of boundary driven systems and formulate a nonequilibrium free energy principle. The physical content of this is illustrated through a simple diffusion model. I then provide a reformulation of the principle which is more suitable for calculation and demonstrate its validity in a more complex model.

Finally I investigate a particular example of a boundary driven system, a toy model of a complex fluid called the rotor model. I first use simulation to investigate the model and its phase behaviour, before using an analytical
approach to do the same. This approach takes the form of a nonequilibrium real space renormalisation group calculation, and qualitatively reproduces some of the features seen in the simulations.