Abstract

Models of physical systems as sets of particles interacting through generalised 'forces' are common to a wide variety of sciences. While individual-based (microscopic) models of such systems are generally conceptually simple, they are often intractable analytically and computationally as most systems to be modelled contain an infeasibly large number of particles. It is common in such circumstances to derive a macroscopic continuum model, which tracks the evolution of population-averaged probability densities over an individual particle's phase space. In this talk we develop a general method for deriving continuum models of second-order (kinetic) systems with short-ranged interaction forces using matched asymptotic expansion in the small interaction-length parameter. We then extend this method to include clustering systems, which allows us to apply it to the Cucker-Smale velocity-averaging model to develop an equivalent continuum model of collective behaviour. The resulting model is evaluated against full particle simulations and shown to reproduce key system measures such as cluster velocity and size distributions more accurately than a standard mean-field approach.