Microscopic foundations of kinetic plasma theory: The relativistic Vlasov--Maxwell equations and their radiation-reaction-corrected generalization

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–Lande–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 $$N\rightarrow \infty $$ in which charges $$\propto 1/\surd {N}$$ . The radiation-reaction term vanishes in this limit, too. The subsequent formal limit sending Bopp’s parameter $$\varkappa \rightarrow \infty $$ yields the Vlasov–Maxwell model.