Minimal Models of Nonequilibrium Phase Transitions

Phase transitions are processes in which the properties of a physical system change qualitatively, like when a pot of boiling water turns into steam or a bar of iron becomes magnetic. In both those examples, the only energy input comes from the environment surrounding the system and eventually the system and environment reach an equilibrium. Physicists have understood phase transitions of systems in equilibrium for a long time through the frameworks of classical thermodynamics and Landau theory. However, in nature we often encounter systems which are not in equilibrium with their environments. For example, in biological systems, organisms convert the energy from their food into motion and thus can start and stop moving without inputs from their surroundings. These systems can also exhibit phase transitions, like when a flock of birds starts flying as a single unit, but unlike the equilibrium examples, there is currently no general framework to describe them. In this project we study minimal mathematical models of nonequilibrium systems. Such models are defined by simple rules and can be sometimes be solved exactly, unlike more realistic and complex models. The purpose of studying these models is twofold. Firstly, they can sometimes be used as approximations of real physical systems. In that case, it is possible to qualitatively describe the behaviour of a complex system without the need for extensive numerical computations. Secondly, by stripping a model of excessive detail, we can often reveal more simple underlying mechanisms which are responsible for some complicated phenomenon. We can then describe many real-world systems by one minimal model. The goal of this project is to study such minimal models to develop our understanding of the mechanisms causing nonequilibrium phase transitions.