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Microscopic foundations of kinetic plasma theory: the relativistic Vlasov-Maxwell equations and their radiation-reaction-corrected generalization. (English) Zbl 1448.82038

Summary: 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 \(N\rightarrow \infty\) 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.

MSC:

82D10 Statistical mechanics of plasmas
82C40 Kinetic theory of gases in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
35Q83 Vlasov equations
35Q61 Maxwell equations
35Q20 Boltzmann equations
81V70 Many-body theory; quantum Hall effect
83A05 Special relativity
83C22 Einstein-Maxwell equations
83C50 Electromagnetic fields in general relativity and gravitational theory

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