Excluded-volume effects in stochastic models of diffusion - Bruna, M (2012)
Stochastic models describing how interacting individuals give rise to collective behaviour have become a widely used tool across disciplines—ranging from biology to physics to social sciences. Continuum population-level models based on partial differential equations for the population density can be a very useful tool (when, for large systems, particle-based models become computationally intractable), but the challenge is to predict the correct macroscopic description of the key attributes at the particle level (such as interactions between individuals and evolution rules). In this thesis we consider the simple class of models consisting of diffusive particles with short-range interactions. It is relevant to many applications, such as colloidal systems and granular gases, and also for more complex systems such as diffusion through ion channels, biological cell populations and animal swarms. To derive the macroscopic model of such systems, previous studies have used ad hoc closure approximations, often generating errors. Instead, we provide a new systematic method based on matched asymptotic expansions to establish the link between the individual- and the population-level models. We begin by deriving the population-level model of a system of identical Brownian hard spheres. The result is a nonlinear diffusion equation for the one-particle density function with excluded-volume effects enhancing the overall collective diffusion rate. We then expand this core problem in several directions. First, for a system with two types of particles (two species) we obtain a nonlinear cross-diffusion model. This model captures both alternative notions of diffusion, the collective diffusion and the self-diffusion, and can be used to study diffusion through obstacles. Second, we study the diffusion of finite-size particles through confined domains such as a narrow channel or a Hele–Shaw cell. In this case the macroscopic model depends on a confinement parameter and interpolates between severe confinement (e.g., a single- file diffusion in the narrow channel case) and an unconfined situation. Finally, the analysis for diffusive soft spheres, particles with soft-core repulsive potentials, yields an interaction-dependent non-linear term in the diffusion equation.