On the Fokker-Planck approximation to the Boltzmann collision operator

Abstract

The Boltzmann equation (BE) is a mesoscopic model that provides a description of how gases undergoing a binary collision process evolve in time, however there is no general analytical approach for finding its solutions and direct numerical treatment using quadrature methods is prohibitively expensive due to the dimensions of the problem. For this reason, models that are able to capture the behaviour of solutions to the BE, but which are simpler to treat numerically and analytically are highly desirable. The Fokker-Planck collision operator is one such collision model, which is suited well to numerical solutions using stochastic particle methods, and is the subject of this thesis. The stochastic numerical solutions of the Fokker-Planck model suffer heavily from noise when the speed of the flow is low. We develop two methods that are able to reduced the variance of the estimators of the particle method. The first is a common random number method, which produces a correlated equilibrium solution where thermodynamic fields are known. The second is a importance sampling method, where weights are attached to the particles. This means that particles close to equilibrium do not contribute to the noise of the estimators. We also develop a randomised quasi-Monte Carlo scheme for solving the diffusion equation, which has a faster rate of convergence than simple Monte Carlo methods. The relative simplicity of the functional form of the Fokker-Planck collision operator makes it possible to find analytic solutions in simple cases. We consider a spatially homogeneous, isotropic gas with elastic collisions in the presence of forcing and dissipation and derive self-consistent non-equilibrium steady-state solutions. Previous numerical evidence exists that suggest such forcing and dissipation mechanisms, widely separated, give rise to steady-states of the BE that are close to Maxwellian, with a direct energy cascade and an inverse particle cascade. Using our analytic solutions, we are able to investigate the dependence of such solutions on the forcing and dissipation scales, and find that in the inertial range, the interaction is non-local. We then show that the “extreme driving” mechanism, responsible for a family of non-universal power-law solutions for inelastic granular gases, where the flux of energy is towards lower scales, is also able to produce inverse energy cascades for the elastic system