Analytical and Numerical Methods in Collisionless Kinetic Theory

This project develops analytic and computational methods to answer mathematical questions in the kinetic theory of plasma dynamics. Plasmas are charged gases, often referred to as the fourth state of matter, that account for more than 99% of all material in the universe and are of significant practical interest since they serve as excellent conductors of electricity.

Plasma engines have been developed by space agencies around the world and were recently employed to power selected spacecraft. The use of plasmas within nuclear fusion reactors is also being explored as a source of clean energy. Other notable examples of plasmas occurring in natural phenomena are the solar wind, the Earth's ionosphere, galactic nebulae, and the tails of comets.

A full understanding of the dynamics of the solar wind is of practical importance, as it dictates the intensity of space weather, which is often responsible for expensive damage to satellites orbiting the Earth. The motion of a plasma can be modeled by a system of partial differential equations, and among the goals of this project are to demonstrate that these equations possess realistic solutions, to determine their qualitative behavior, to compute their sensitivity with respect to model parameters, such as masses, charges, and temperature, and to approximate them computationally to predict future behavior with precision. The project provides graduate research training opportunities.

A plasma is a fully ionized gas in which electromagnetic forces are often strong enough to dominate collisional effects. The motion of a high temperature, low density collisionless plasma is described by the Vlasov-Maxwell equations, a nonlinear system of hyperbolic partial differential equations. In this setting, collisions are neglected while the charge and current densities, which drive the Maxwell system, are determined in a self-consistent manner from velocity averages of the distribution of ions in the system, which satisfies the Vlasov equation.

A major question this project addresses is if there are shocks in a collisionless plasma, that is, if a singularity can develop from smooth initial data as time progresses. In some cases, such as in lower dimensional relativistic formulations, smooth global solutions are known to exist. Additional questions concern the large time limiting behavior of the particle positions and momenta, charge and current densities, electromagnetic fields, and particle distribution functions within the system.

More specifically, the project determines whether dispersive effects in the equations cause these quantities to decay over time, or if there is sufficient interaction to sustain their strength even in the time asymptotic limit. Finally, the sensitivity of the fields and densities with respect to model input parameters are computed using global sensitivity metrics, corresponding dispersion relations, and particle-in-cell simulations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.