Dynamic Density Functional Theory: A Tool For Mathematical Biology?
In recent years, a number of dynamic density functional theories (DDFTs) have been developed to model stochastic particle dynamics via deterministic PDEs. These DDFTs aim to overcome the high-dimensionality of systems with large numbers of particles by reducing to the dynamics of the one-body density, described by a PDE in only three spatial dimensions, independently of the number of particles. The standard derivations are via stochastic equations of motion, but there are fundamental differences in the underlying assumptions in each DDFT. I will begin by giving an overview of some DDFTs, highlighting the assumptions and range of applicability. Particular attention will be given to the inclusion of inertia and hydrodynamic interactions, both of which strongly influence non-equilibrium properties of the system. I will then demonstrate the very good agreement with the underlying stochastic dynamics for a wide range of systems. I will also discuss an accurate and efficient numerical code, based on pseudospectral techniques, which is applicable both to the integro-PDEs of DDFT and to many other systems. Finally I will discuss some existing and possible applications in mathematical biology (e.g. cancer tumour growth, drug delivery, the behaviour of yeast in brewing) and will welcome further suggestions.
Joint work with Serafim Kalliadasis, Greg Pavliotis, and Andreas Nold.