Miranda Holmes-Cerfon: Modelling and simulating mesoscale particles using sticky diffusion processes

Particles in soft-matter systems (such as colloids), which have sizes of nanometres to micrometres, tend to have attractive interactions that are very short-ranged compared to their diameters. This makes it hard to describe their dynamics, because their energy landscape is so rugged, and it also makes them hard to simulate, because of the presence of stiff forces. We propose a new framework for looking at such particles, based on taking the “sticky limit” as the range of the interaction goes to zero. In this limit, the energy landscape is described by a set of manifolds of different dimensions, while the dynamics are given by a diffusion process with “sticky” boundary conditions on each of the manifolds. A simple example of a sticky diffusion process is a Sticky Brownian Motion, which is a Brownian motion on a half-line that is slowed down in such a way that it spends a nonzero amount of time at {0}. I will show how the theory of sticky diffusions gives a new way to compute transition rates between clusters of colloids, which agrees quantitatively with our experiments. I will then show our progress on developing numerical methods to simulate sticky diffusion processes. The challenge lies in modifying existing methods for solving stochastic differential equations so the solution can spend finite time on lower-dimensional manifolds, and without introducing stiff forces.