## Ph.D. Research

My Ph.D. research focused on mathematical models of active matter.

Like ordinary matter, active matter consists of a large number of individual particles. Unlike ordinary matter, however, the particles in active matter are self-propelled. In more technical terms, energy is continually injected into the system at the scale of individual particles.

Many examples of active matter come from biology, ranging from flocks of birds and schools of fish all the way down to colonies of bacteria and structural components of cells. It is also possible to synthesize active materials in the lab, for instance using chemical reactions or light to imbue synthetic microscopic particles with self-propulsion.

Active matter behaves in fascinating and non-intuitive ways. One of many examples is flocking behavior in animal groups. It is well known that flocks of birds or schools of fish behave collectively – as if having a single "mind" – in order to navigate their environments or to evade predators. How are organisms without a significant capacity for abstract thought able to engage in such coordinated behavior? The surprising answer from statistical physics is that not much is required: all you need are basic interaction laws between nearby individuals, such as a rule that you stay close to and move in the same direction as your nearest neighbors.

This is why physicists are interested in active matter: it gives insight into how complex emergent behaviors can result from simple dynamical ingredients. With this understanding, there is the possibility to reverse-engineer the process: by selecting the right microscopic components and interaction laws, one can synthesize active materials with new and unique macroscopic properties. Such materials are sometimes called "biomimetic" since their design and function is similar to that of naturally occurring biological systems.

Active matter presents theoretical physicists with new and exciting challenges. Since active systems are dissipative and driven far from equilibrium, the usual statistical physics methods – dating back to 19th century thermodynamics and kinetic theory – no longer apply. Instead, new and frequently more difficult theoretical techniques are needed.

My research seeks to develop such techniques. An important sub-goal of this broader aim is to see if we can find a "thermodynamics" of active matter. Can active matter be described by simple macroscopic variables like an effective temperature or pressure? Or are fundamentally new concepts needed?

Such questions are most easily addressed in the context of a minimal models, i.e. the simplest models we can find which are still nontrivial. In active matter these are often systems of idealized spheres or rods. Along these lines, much of my work has focused on a model of spheres which self-propel at a constant velocity, termed "active Brownian particles." One could think of active Brownian systems as the "ideal gas" of active matter.

Even though they are simple, active Brownian particles behave unexpectedly. For instance, they can form dense clusters even without attractive interparticle interactions. They also accumulate at walls and spontaneously generate macroscopic flows, which can be harnessed to power microscopic motors, e.g. microscopic gears. The last two behaviors actually are present even in a very dilute system of particles, ignoring interparticle interactions. This raises the possibility of an exact mathematical description for these behaviors. Indeed, all that is needed to model non-interacting active Brownian particles is a single partial differential equation:

This is a substantial simplification over the description in the full interacting case, which consists of an infinite number of coupled partial differential equations. Still, this equation is hard to solve; it is a nonstandard differential equation which doesn't appear in the classic texts on mathematical physics. Much of my research has focused on developing mathematical tools to treat equations of this type. Since these equations are applicable to a wide range of systems besides active matter (e.g. neutron scattering, gas dynamics, certain astrophysical problems, and others), the techniques I've developed are of more general interest as well.