Projection-operator methods for classical transport in magnetized plasmas. II. Nonlinear response and the Burnett equations

TL;DR

This paper extends projection-operator methods to derive nonlinear plasma transport equations, unifying previous theories and emphasizing the role of non-Gaussian statistics in Burnett coefficient calculations.

Contribution

It introduces a formalism for calculating Burnett transport coefficients in magnetized plasmas using two-time correlation functions, unifying various plasma kinetic theories.

Findings

Consistency with Chapman-Enskog results for parallel viscosity

Demonstration of Burnett effects via a simple stochastic model

Unified formalism connecting turbulence theory and plasma transport

Abstract

The time-independent projection-operator formalism of Brey et al. [Physica 109A, 425-444 (1981)] for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for the transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular nontrivial example, that previously derived formulas from the two-time formalism [J. J. Brey, J. Chem. Phys. 79, 4585-598 (1983)] are consistent with the Chapman-Enskog methodology employed later by Catto & Simakov (CS) [Phys. Plasmas 11, 90-102 (2004)] for the contributions to the parallel viscosity driven by temperature gradients. The work serves to unify various previous work on plasma kinetic theory with formalism usually applied to turbulence theory. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts I and II of this series provide a tutorial introduction to projection-operator methods that should be broadly useful in theoretical plasma physics. A Supplement (bundled at the end of Part II) provides some technical details of the reduction of the general Burnett equations of Brey et al. (1981) to a one-component neutral fluid in order to support the result quoted by Brey (1983).