Projection-operator methods for classical transport in magnetized plasmas. I. Linear response, the Braginskii equations, and fluctuating hydrodynamics

Abstract

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetized plasmas is given. In the present work, linear response is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional Braginskii fluid equations, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the nonhydrodynamic part of the distribution function f. Although ultimately mathematically equivalent to Braginskii's calculations, the projection-operator approach provides a usefully intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman-Enskog procedure; record results about the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetized plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

Version 1: 67 pages, 4 figures. Version 2: 68 pages, 4 figures. Minor edits to grammar and punctuation, and some minor rewriting for improved clarity. Augmented the discussion of the mu-space projection operator for better correspondence with the gamma-space one used in Part II. Version 3: 71 pages, 4 figures. Corrected the proof of Onsager symmetry to properly deal with cases of mixed parity

References in corpus (2)

• An introduction to the physics of the Coulomb logarithm with emphasis on quantum-mechanical effects
• Projection-operator methods for classical transport in magnetized plasmas. II. Nonlinear response and the Burnett equations

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