Exact steady states of minimal models of nonequilibrium statistical mechanics

Abstract

Systems out of equilibrium with their environment are ubiquitous in nature. Of particular relevance to biological applications are models in which each microscopic component spontaneously generates its own motion. Known collectively as active matter, such models are natural effective descriptions of many biological systems, from subcellular motors to flocks of birds. One would like to understand such phenomena using the tools of statistical mechanics, yet the inherent nonequilibrium setting means that the most powerful classical results of that field cannot be applied. This circumstance has fuelled interest in exactly solvable models of active matter. The aim in studying such models is twofold. Firstly, as exactly solvable model are often minimal, it makes them good candidates as generic coarse-grained descriptions of real-world processes. Secondly, even if the model in question does not correspond directly to some situation realizable in experiment, its exact solution may suggest some general principles, which could also apply to more complex phenomena.

A typical tool to investigate the properties of a large system is to study the behaviour of a probe particle placed in such an environment. In this context, cases of interest are both an active particle in a passive environment or an active particle in an active environment. One model that has attracted much attention in this regard is the asymmetric simple exclusion process (ASEP), which is a prototypical minimal model of driven diffusive transport. In this thesis, I consider two variations of the ASEP on a ring geometry. The first is a system of symmetrically diffusing particles with one totally asymmetric (driven) defect particle. The second is a system of partially asymmetric particles, with one defect that may overtake the other particles. I analyze the steady states of these systems using two exact methods: the matrix product ansatz, and, for the second model the Bethe ansatz. This allows me to derive the exact density profiles and mean currents for these models, and, for the second model, the diffusion constant. Moreover, I use the Yang-Baxter formalism to study the general class of two-species partially asymmetric processes with overtaking. This allows me to determine conditions under which such models can be solved using the Bethe ansatz.