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The mathematics of active matter
A flock of birds, a shoal of fish, a swarm of robots, a colony of swimming bacteria ; these are examples of systems composed of interacting units that consume energy and collectively generate motion and mechanical forces on their environment. They show a rich variety of collective behaviour, much of which remains mysterious. In recent years we have come to call such systems active matter. Clearly, biology (living systems) provides numerous examples of these active matter systems.
We call them active matter because they share some of the properties of the constituents of what we call matter, i.e. solids, liquids and gases in that they are made of many interacting components. However they have fundamental differences in that many conservation laws that govern the interactions of normal (passive) matter are not obeyed by their active components.
(Equilibrium) statistical mechanics has formed the framework for how we understand the properties of matter. I will argue that ideas developed in statistical mechanics must be augmented by a number of new mathematical structures to describe these systems. Then I will describe some recent theoretical work developing this framework for characterising the behaviour of active matter systems. Finally I will apply it to describe two examples of active systems, active Brownian particles and active nematics.