Kinetic modeling is a fundamental tool in the analysis and simulation of large particle-based systems. In a kinetic description, such systems are characterized by a positive distribution function which gives the number of particles with a given momentum at a given point in space and time. It is a mean-field description that, on the one hand, improves the accuracy of popular fluid models that are not valid away from equilibrium while, on the other hand, can incorporate important features of finer scale models, such as molecular dynamics and quantum mechanics without necessarily resolving these finer scales.
Traditional applications for kinetic modeling include dilute gases, plasmas, radiative transport, and multiphase flow. New and emerging applications include semi-classical descriptions of quantum mechanics, traffic and network modeling, and biological processes of self organization. In all these applications, kinetic transport shares a remarkably similar mathematical structure, featuring convection in phase space (with velocity/momentum advection and field-driven acceleration) and collision terms with integral operators.
The large phase space associated with the kinetic description has, in the past, made simulations impractical in most settings. However, recent advances in computer resources and numerical algorithms are making kinetic models more tractable. At Oak Ridge National Laboratory, the goal of the mathematics program is to continue and accelerate this trend by designing algorithms to leverage the computational power of the world's fastest computers. Particular emphasis is currently on three areas: (i) moment models, which track the evolution of a handful of weighted averages in the momentum space; (ii) asymptotic preserving methods, which capture asymptotic limits of kinetic equations without having to resolve unnecessary microscopic dynamics; and (iii) hybrid models that combine low-resolution and high-resolution models of particle systems in a single, efficient framework