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An introduction to the use of projection-operator methods for the derivationof classical fluid transport equations for weakly coupled, magnetized plasmasis given. In the present work, linear response is addressed. In theSchr\"odinger representation, projection onto the hydrodynamic subspace leadsto the conventional Braginskii fluid equations, while the orthogonal projectionleads to an alternative derivation of the Braginskii correction equations forthe nonhydrodynamic part of the distribution function f. Although ultimatelymathematically equivalent to Braginskii's calculations, the projection-operatorapproach provides a usefully intuitive way of discussing the derivation oftransport equations and interpreting the significance of the various parts ofthe perturbed distribution function; it is also technically more concise. Aspecial case of the Weinhold metric is used to provide a covariantrepresentation of the formalism; this allows a succinct demonstration of theOnsager symmetries for classical transport. The Heisenberg representation isused to derive a generalized Langevin system whose mean recovers the Braginskiiequations but that also includes fluctuating forces. Transport coefficients aresimply related to the two-time correlation functions of those forces, andphysical pictures of the various transport processes are naturally couched interms of them. A number of appendices review the traditional Chapman-Enskogprocedure; record results about the linearized Landau collision operator;discuss the covariant representation of the hydrodynamic projection; provide anexample of the calculation of some transport effects; describe thedecomposition of the stress tensor for magnetized plasma; introduce the lineareigenmodes of the Braginskii equations; and, with the aid of several examples,mention some caveats for the use of projection operators.

journal of plasma physics

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