This project will focus on the mathematical derivation of the Balescu-Guernsey-Lenard (BGL) equation of plasma transport theory. This equation is commonly regarded as the most fundamental kinetic collisional model for classic Coulomb plasmas. It was proposed to describe the infinite-particle limit for the evolution of a system of particles that interact with long-range (e.g., Coulomb) forces. However, there is no rigorous mathematical proof that the BGL equation is the correct limit. This situation has made it much harder to develop more general models for the physical situations, where the BGL equation is known not to be accurate (e.g., anomalous transport in fusion reactors). This project will aim to show that there is a deep connection between the BGL equation and the theory of fluctuations in collisionless plasmas about the Vlasov limit. Specifically, the goal is to show that the BGL equation is simply the non-linear Fokker-Planck equation associated with long-time particle dynamics in the Vlasov fluctuation field of a spatially homogeneous plasma. The centerpiece of the study will be a new stochastic process which describes, loosely speaking, long-time, self-consistent dynamics of particles traveling in a random force field generated by the fluctuations of the particles' density about the mean-field limit.
The Balescu-Lenard-Guernsey equation is pivotal in the theory of plasmas and gravitating systems; it has many applications, especially in the mathematical modeling of magnetically-confined fusion reactors. However, the existing body of mathematical knowledge about it is very small. The chief goal of the project is to construct a rigorous mathematical proof that this model provides a good approximation for describing the behavior of a large assembly of particles, such as those found in fusion plasmas. Such a proof will validate many theoretical calculations and numerical simulations based on the BLG model, as well as their applications in plasma physics. Having a better understanding of the mathematics involved will be indispensable for modeling more complicated physical phenomena that arise in very hot plasmas in a fusion reactor.