Journal Article

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

Kiessling, M. K.-H., Elskens, Y.

2020

Overview

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–Landé–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 → ∞ in which charges ∝ 1 / √ N . The radiation-reaction term vanishes in this limit, too. The subsequent formal limit sending Bopp’s parameter ϰ → ∞ yields the Vlasov–Maxwell model.

• Publisher: Springer US,Springer,Springer Nature B.V,Springer Verlag
• Subject: Analysis / Charged particles / Convolution / Density / Electromagnetism / Foundations / Kinetic equations / Kinetic theory / Mathematical and Computational Physics / Mathematical models / Mathematical Physics / Maxwell's equations / Physical Chemistry / Physics / Physics and Astronomy / Plasma Physics / Quantum Physics / Radiation / Relativistic effects / Relativistic plasmas / Relativistic theory / Specific gravity / Statistical Physics and Dynamical Systems / Tensors / Theoretical