Active matter systems consist of large collections of self-propelled particles and arise in a wide range of biological and physical settings, including bird flocks, fish schools, and bacterial suspensions. In contrast to passive systems, whose dynamics are typically driven by gradients of an energy, active systems are intrinsically out of equilibrium and can display rich and complex forms of collective behaviour.

In these two lectures, I will introduce stochastic particle models for active matter and discuss mathematical techniques for studying interacting particle systems and their collective behaviour. The first hour will provide an introduction to the main ideas and tools, while the second will focus on their application to a specific class of active matter systems. The second hour will be largely self-contained, so that participants already familiar with the area may attend it independently.

In the second hour, I will focus on a lattice gas consisting of a mixture of active and passive particles with excluded-volume interactions. Starting from the microscopic stochastic dynamics, I will derive the exact hydrodynamic limit for the particle densities, given by a system of nonlinear partial differential equations. I will then analyse the resulting macroscopic model and discuss the different types of solutions it exhibits, including phase-separated states and travelling solutions.

TBP