Microscopic Foundations of Kinetic Plasma Theory: The Relativistic Vlasov–Maxwell Equations and Their Radiation-Reaction-Corrected Generalization

Abstract

It is argued that the relativistic Vlasov--Maxwell equations of the kinetic theory of plasma approximately describe a relativistic system of $N$ charged point particles interacting with the electromagnetic Maxwell fields in a Bopp--Land\'e--Thomas--Podolsky (BLTP) vacuum, provided the microscopic dynamics lasts long enough.The purpose of this work is not to supply an entirely rigorous vindication, but to lay down a conceptual road map for the microscopic foundations of the kinetic theory of special-relativistic plasma, and to emphasize that a rigorous derivation seems feasible. Rather than working with a BBGKY-type hierarchy of $n$-point marginal probability measures, the approach proposed in this paper works with the distributional PDE of the actual empirical 1-point measure, which involves the actual empirical 2-point measure in a convolution term.The approximation of the empirical 1-point measure by a continuum density, and of the empirical 2-point measure by a (tensor) product of this continuum density with itself, yields a finite-$N$ Vlasov-like set of kinetic equations which includes radiation-reaction and nontrivial finite-$N$ corrections to the Vlasov--Maxwell-BLTP model. The finite-$N$ corrections formally vanish in a mathematical scaling limit $N\to\infty$ in which charges $\propto 1/\surd{N}$. The radiation-reaction term vanishes in this limit, too. The subsequent formal limit sending Bopp's parameter $\varkappa\to\infty$ yields the Vlasov--Maxwell model.