Projection-operator methods for classical transport in magnetized plasmas. Part 1. 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, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly 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 linearized 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 some properties of 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 magnetised 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.

1 Introduction

The review article by Braginskii (published in Russian in 1963 and in English translation in 1965) on classical transport in weakly coupled, magnetised, multispecies plasmas has served as an invaluable reference for multiple generations of plasma physicists. For a two-component plasma with small electron-to-ion mass ratio, $\unicode[STIX]{x1D707}\doteq m_{e}/m_{i}\ll 1$ ( $\doteq$  denotes a definition), Braginskii described a path to an approximate derivation of the so-called correction equations for the non-hydrodynamic parts of the distribution function, from which the classical transport coefficients are ultimately derived. The methodology, first published by Braginskii (1957), ¹ can be traced back to the pioneering work of Chapman (1916) and Enskog (1917) on the kinetic theory of rarefied gases; for many details, see Chapman & Cowling (1952). The traditional Chapman–Enskog procedure is reviewed for the simple case of a one-component plasma (OCP) in appendix A.

Although the relevant mathematics was described clearly by Braginskii (1957), experience shows that many students do not take the time to work through those calculations and consequently do not always grasp the beautiful underlying structure of the transport problem. The techniques described in this article provide an alternative, heuristically appealing, and technically efficient approach that for neutral fluids is known to unify a number of threads of non-equilibrium statistical mechanics. That unification carries over to the more complicated plasma. Appreciation of the methods enables one to avoid reinventing the wheel and provides one with a concise yet workable formalism on which non-trivial generalizations can be built. An example of such a generalization is the calculation of second-order (Burnett) transport coefficients, addressed in Part 2 of this series of articles (Krommes 2018b ).

In the same year that the English translation of Braginskii’s review appeared, Mori (1965) published a seminal paper in which the transport problem was reformulated with the aid of projection-operator methods. The general approach had been anticipated by Zwanzig (1961a , b ), and the methodology is now known as the Mori–Zwanzig formalism. The purpose of the present article is to describe the application of the Mori–Zwanzig formalism to the problem of classical transport in weakly coupled, magnetised plasmas. This leads one to an alternative derivation of the Braginskii correction equations.

A great strength of Braginskii’s review article is its focus on the physical interpretation of the transport coefficients, knowledge of which is essential for researchers on magnetised plasmas. But although Braginskii’s interpretations of the mathematics are entirely correct, here too the present projection methods are helpful in providing additional intuition. They enable an extension of Braginskii’s equations for the macroscopic, mean hydrodynamic variables (density, flow, and temperature) to generalized Langevin equations that include fluctuating forces. Such equations have previously been derived from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy (Bixon & Zwanzig 1969; Hinton 1970), but the projection-operator methods are arguably more transparent and efficient. Fluctuating forces appear implicitly in Braginskii’s heuristic explanations of the various transport processes, and their effects are contained in his systematic mathematics. However, explicit definition of those forces brings additional clarity to the transport calculations.

A projection operator  $\text{P}$ is linear and idempotent (i.e.  $\text{P}^{2}=\text{P}$ ). It extends to linear algebra and functional analysis the notion of graphical projection onto an axis. An example of a projection operator is the ensemble average ²   $\langle \ldots \rangle$ . Thus, all of statistical closure theory (Krommes 2002, 2015) can be said to involve projection operators. However, this is stretching the point. The specifically Mori–Zwanzig-style projection-operator techniques have a particular flavour and do not naturally generalize to all possible methods of statistical closure. They have been little used in plasma physics. Krommes (1975) and Krommes & Oberman (1976) employed them to discuss the phenomenon of long-time tails in magnetised, thermal-equilibrium plasmas, earlier identified in the molecular-dynamics neutral-fluid computer experiments of Alder & Wainwright (1970). ³ They are mentioned briefly by Diamond, Itoh & Itoh (2010, § 6.2). However, they have not been used systematically in the context of plasma transport theory. In particular, there is no published account of their application to the derivation of the Braginskii transport equations. ⁴

One possible reason that the Mori–Zwanzig formalism has not seen much use in plasma physics is that the most general non-equilibrium statistical–mechanical treatments of the transport problem produce formulas for the transport coefficients in terms of averages over the $N$ -particle equilibrium ensemble and dynamics evolved with a modified $N$ -particle Liouville propagator, where $N$  is the total number of particles in the system. Such formulas go back to Green (1954) and Kubo (1957). (Subtleties with the interpretation of the Green–Kubo formulas are discussed in § G.3.) While elegant, they are difficult to work out in the general case. However, in the special but very important regime of weak coupling, those formulas simplify dramatically. Moreover, with the further approximation of linear response an alternative approach becomes possible wherein a kinetic equation for the one-particle distribution function is first derived, then processed to produce formulas for the transport coefficients. It is that processing that is clarified by the projection-operator techniques described in the present paper. I shall show that a suitably modified Mori–Zwanzig formalism can be easily applied to the linearized plasma kinetic equation, leads efficiently to standard results, clarifies the structure of the hydrodynamic system and fosters heuristic understanding of the first-order transport processes.

By $n$ th-order processes, I mean contributions to dissipative fluxes of $n$ th order in the gradients. Thus, ‘first order’ is to be distinguished from ‘linear response’. The latter describes a formally infinitesimal perturbation of thermal equilibrium. In either linear response theory or a more general nonlinear theory, dissipative fluxes exist that are driven by arbitrary powers of the gradients. The proportionality factors are the ‘ $n$ th-order transport coefficients’ (more precisely, the order-unity transport coefficients that enter in fluxes involving $n$ th-order gradients). The first-order coefficients are called the Braginskii transport coefficients by plasma physicists; they are known in non-equilibrium statistical mechanics as the Navier–Stokes coefficients. For example, if small perturbations are denoted by  $\unicode[STIX]{x0394}$ , the first-order heat flux in an unmagnetised plasma has the form $\unicode[STIX]{x0394}\boldsymbol{q}=-n\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}\unicode[STIX]{x0394}T$ ( $n$  is the density and $T$  is the temperature); $\unicode[STIX]{x1D705}$  is the first-order thermal conductivity. This relation is both linear ( $\unicode[STIX]{x0394}\boldsymbol{q}\propto \unicode[STIX]{x0394}T$ ) and first order ( $\unicode[STIX]{x0394}\boldsymbol{q}\propto \unicode[STIX]{x1D735}$ ). The second-order coefficients are called the Burnett transport coefficients. Some Burnett coefficients exist already in the linear approximation. For example, one contribution to the linear Burnett heat flux has the form $\unicode[STIX]{x0394}\boldsymbol{q}=n\unicode[STIX]{x1D705}^{\prime }\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x0394}\boldsymbol{u}$ , where $\boldsymbol{u}$  is the fluid velocity. For nonlinear response, additional Burnett terms arise. For example, a contribution to the nonlinear Burnett heat flux has the form $\unicode[STIX]{x0394}\boldsymbol{q}=n\unicode[STIX]{x1D705}^{\prime \prime }(\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u})\unicode[STIX]{x1D735}\unicode[STIX]{x0394}T$ .

This paper is Part 1 of a two-part series of articles whose goal is to show how to formulate classical, weakly coupled plasma transport theory by using projection-operator methods. Part 1 describes a self-contained rederivation of the Braginskii correction equations for the special case of linear perturbations of an absolute Maxwellian equilibrium. This is less than what Braginskii accomplished (his equations are nonlinear), but it serves to familiarize one with the basic methodology in the simplest possible context. (Following Braginskii, I restrict my attention to effects of first order in the gradients. Although the present methodology could also be used to study linear Burnett effects, I shall defer such calculation to the more general discussion given in Part 2, where nonlinear Burnett effects are also considered.) I shall also derive a set of generalized Langevin equations that extend Braginskii’s equations to include fluctuating forces. That is a core topic that provides important insight into the structure of the transport theory.

In Part 2, the problem is reformulated in order to embrace nonlinearity and arbitrarily strong coupling. The methodology used there is a generalization of the formulation of Brey, Zwanzig & Dorfman (1981) to include a background magnetic field and multiple species. The nonlinear Braginskii equations are recovered with ease. The techniques also allow one to obtain the next-higher-order Burnett equations, including both of the linear and nonlinear effects as described above. Those were originally obtained for plasmas described by the Landau collision operator by Mikhaǐlovskiǐ (1967), Mikhaǐlovskiǐ & Tsypin (1971, 1984) and Catto & Simakov (2004), none of whom remarked on connections to previous calculations of Burnett coefficients for neutral gases. In Part 2 it is shown that the Navier–Stokes and Burnett transport coefficients calculated by Catto and Simakov are special cases of the complete set of coefficients for which Brey et al. (1981) gave general expressions in terms of integrals of two-time correlation functions. ⁵ Thus, the present series of articles serves to unify a variety of previous research threads.

I now return to an overview of the present Part 1. I illustrate the basic approach in § 2 by deriving linearized transport equations for the unmagnetised one-component plasma, restricting attention to dissipative fluxes that are of first order in the gradients. Then in § 3 I extend the calculations to embrace multispecies and magnetised plasmas, and I derive the linearized version of Braginskii’s correction equations.

The formalism employed in §§ 2 and 3 uses what is known as the Schrödinger representation, which means that one takes time-independent velocity moments of the time-dependent distribution function  $f$ (which is taken to evolve according to the Landau kinetic equation). Alternatively, one can use the Heisenberg representation, in which equations are written for time-dependent, random hydrodynamic observables whose statistical behaviour evolves from statistics (velocity dependence) that reside in the distribution function at some initial time. The averaged equations are the same as before. However, the raw equations for the random variables can be cast into the form of generalized Langevin equations, which include fluctuating forces and imply a theory of hydrodynamic correlation functions. This is done in § 4. The body of the paper concludes with a brief discussion in § 5.

Several appendices are included. In appendix A I review the traditional Chapman–Enskog calculation for the one-component plasma, in appendix B I record various properties of the linearized Landau collision operator, in appendix C I describe a technically efficient covariant representation of the transport equations, in appendix D I provide an example of the evaluation of the general formulas by considering the electron heat flow in the limit of small collisionality, in appendix E I review Braginskii’s tensorial decomposition of the stress tensor, in appendix F I consider some important special cases of the linear eigenmodes of the Braginskii equations and in appendix G I focus on some important caveats regarding the use of projection operators. Key notation is summarized in appendix 2:I, ⁶ which merges the definitions from both Part 1 and Part 2.

This paper is intended to be useful to both graduate students just beginning their study of classical plasma transport as well as seasoned researchers interested in advanced techniques. Although it rederives some of Braginskii’s principal results and quotes others of them for completeness, it is neither a complete replacement for Braginskii’s article nor a comprehensive review of classical plasma transport theory. For initial study of classical transport, a reasonable plan of attack would be to begin by skimming the first half of Braginskii (1965) (through the end of §5, p. 262). Additional perspective should then be provided from the present paper by §§ 1–3 and 5 as well as appendices A, B, D, E and F, with § 4 and appendices C and G containing more advanced material. To fully appreciate the material in appendix G, the reader may find it useful to first study the concise modern introduction to non-equilibrium statistical mechanics given by Zwanzig (2001), particularly chap. 8 on projection operators.

The material in Part 1 is intended to be introductory, and the restrictions to linear response and first-order transport coefficients are greatly simplifying. In fact, some deep conceptual issues lurk in what may appear at first glance to be straightforward analysis. If concerns arise, consult § 5 for brief discussion and see Part 2 for a more thorough treatment.

2 Linearized hydrodynamics for the one-component plasma

In a one-component, weakly coupled ⁷ plasma (for example, a discrete ion plasma with smooth electron neutralizing background) in which the collision frequency  $\unicode[STIX]{x1D708}$ is ordered large, it is well known that Chapman–Enskog theory singles out the number density, momentum density, and kinetic-energy density (or temperature, or entropy density) as preferred hydrodynamic variables. That is, those quantities are conserved by the Landau collision operator. Mathematically, with $\overline{n}$  being the spatially uniform mean density, the velocity-dependent functions  $\overline{n}$ , $\overline{n}m\boldsymbol{v}$ , and $(1/2)\overline{n}mv^{2}$ are the left null eigenfunctions of the collision operator (in terms of a natural scalar product to be defined shortly). No dissipative transport is associated with the null eigenfunctions. Rather, transport is carried by the non-hydrodynamic corrections to the local Maxwellian distribution function. The traditional Chapman–Enskog approach is reviewed in appendix A.

2.1 Basic idea of the hydrodynamic projection

These ideas can be codified by dividing a complete set of functions on the single-particle velocity space into two orthogonal subspaces: the hydrodynamic subspace  ${\mathcal{H}}$ , spanned by the null eigenfunctions of the collision operator, and the non-hydrodynamic, orthogonal, or vertical subspace  $O$ . ⁸

The fluid evolution equations ‘live in the hydrodynamic subspace’. The non-hydrodynamic part of the distribution function determines the values of the transport coefficients. Transport coefficients enter the fluid equations because the dynamics of the hydrodynamic and non-hydrodynamic subspaces are coupled by the evolution equation for  $f$ . That coupling is a special case of the statistical closure problem (Krommes 2015) for passive equations with random coefficients. (Such equations are said to possess a stochastic nonlinearity.) Here the random variable is velocity and the stochastic nonlinearity is the $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f$ term in the kinetic equation.

The decomposition into hydrodynamic and non-hydrodynamic parts can be expressed by the introduction of appropriate projection operators. If $\text{P}$  projects into  ${\mathcal{H}}$ and $\text{Q}\doteq 1-\text{P}$ (where $1$  is the identity operator) therefore projects into  $O$ , and if the perturbed (denoted by  $\unicode[STIX]{x0394}$ ) distribution function is written as $\unicode[STIX]{x0394}f=\unicode[STIX]{x0394}f_{\text{h}}+\unicode[STIX]{x0394}f_{\bot }$ , then $\unicode[STIX]{x0394}f_{\text{h}}=\text{P}\unicode[STIX]{x0394}f$ and $\unicode[STIX]{x0394}f_{\bot }=\text{Q}\unicode[STIX]{x0394}f$ . The basic idea is illustrated in figure 1; the precise realization of  $\text{P}$ is given in the next section.

Figure 1. Illustration of the $\text{P}$ and  $\text{Q}$ projections. The kinetic-energy axis of the hydrodynamic subspace is omitted for clarity.

The formulation described in the present paper focuses on the decomposition of the one-particle distribution function  $f_{s}(\boldsymbol{x},\boldsymbol{v},t)$ , which lives in the so-called $\unicode[STIX]{x1D707}$  space (the six-dimensional (6-D) phase space for one generic particle of species  $s$ ). In contrast, the instantaneous state of the entire collection of $N$  particles can be described by a single point in the so-called $\unicode[STIX]{x1D6E4}$  space (of $6N$  dimensions). A well-known consequence of the reduction from $\unicode[STIX]{x1D6E4}$  space to $\unicode[STIX]{x1D707}$  space is that the kinetic evolution equation for  $f$ is nonlinear (it involves the product $\boldsymbol{E}[f]\,f$ , where the electric field  $\boldsymbol{E}$ is a linear functional of  $f$ , as well as a nonlinear collision operator), which leads to various complications. $\unicode[STIX]{x1D6E4}$ -space dynamics, on the other hand, is formally linear, being described by the Liouville equation. That linearity combined with the linearity of projection operators was usefully exploited by Mori, who was able to write exact equations for two-time correlation functions with apparent ease; ultimately, the transport coefficients are expressed in terms of various integrals over those correlation functions. However, the simplicity is only formal. Mori’s projection operators involve the full $N$ -particle dynamics, and his equations contain nonlinearities to all orders; in the general case, it is difficult to extract from them quantitative expressions for the transport coefficients. As I shall show in Part 2, the formulas simplify in the limit of weak coupling, which fortunately is appropriate for research on magnetic fusion and various other applications. However, that still leaves the dichotomy between the linearity of projection operators and the nonlinearity of $\unicode[STIX]{x1D707}$ -space dynamics to be dealt with. It might seem that a projection-operator formalism is restricted to the derivation of linearized transport equations. That is not correct, although the generalization is non-trivial. I shall address nonlinear transport in Part 2. In the present paper, I restrict my attention to linear response, where the basic features of the projection-operator methodology can be explained in the simplest possible context.

2.2 Transport equations for the unmagnetised, one-component, weakly coupled plasma

In this section I shall introduce the formalism by using projection methods to derive the linearized fluid equations for the one-component plasma (OCP) in the limit of weak coupling and in the electrostatic approximation with background magnetic field $\boldsymbol{B}=\mathbf{0}$ , restricting my attention to fluxes that are of first order in the gradients. (An even simpler example, the Brownian test particle, is discussed in § G.4.) It is well known and easy to show that the fully nonlinear moment equations for the density  $n$ , flow velocity  $\boldsymbol{u}$ , and temperature  $T$ of the one-component, weakly coupled plasma are

(2.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}n(\boldsymbol{x},t)+\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,(n\boldsymbol{u})=0, & \displaystyle\end{eqnarray}$$

(2.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle nm\frac{\text{d}\boldsymbol{u}(\boldsymbol{x},t)}{\text{d}t}=nq\boldsymbol{E}-\unicode[STIX]{x1D735}p-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x1D745}, & \displaystyle\end{eqnarray}$$

(2.1c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}n\frac{\text{d}T(\boldsymbol{x},t)}{\text{d}t}=-p\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{u}-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{q}-(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{ : }\unicode[STIX]{x1D745}. & \displaystyle\end{eqnarray}$$

Here ⁹

(2.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle n(\boldsymbol{x},t)\doteq \int \!\text{d}\boldsymbol{v}\,\overline{n}f(\boldsymbol{x},\boldsymbol{v},t), & \displaystyle\end{eqnarray}$$

(2.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle n\boldsymbol{u}(\boldsymbol{x},t)\doteq \int \!\text{d}\boldsymbol{v}\,\overline{n}\boldsymbol{v}f(\boldsymbol{x},\boldsymbol{v},t), & \displaystyle\end{eqnarray}$$

(2.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}nT(\boldsymbol{x},t)\doteq \int \!\text{d}\boldsymbol{v}\,\frac{1}{2}\overline{n}m\,w^{2}f(\boldsymbol{x},\boldsymbol{v},t), & \displaystyle\end{eqnarray}$$

where $\boldsymbol{w}\doteq \boldsymbol{v}-\boldsymbol{u}$ is the so-called peculiar velocity and $p\doteq nT$ is the pressure of an ideal gas. The total advective time derivative is

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}t}\doteq \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}, & \displaystyle\end{eqnarray}$$

and the stress tensor  $\unicode[STIX]{x1D745}$ and the heat-flow vector  $\boldsymbol{q}$ are defined by

(2.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D745}\doteq \int \!\text{d}\boldsymbol{v}\,(\overline{n}m\boldsymbol{w})\boldsymbol{w}f-p\unicode[STIX]{x1D644}, & \displaystyle\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{q}\doteq \int \!\text{d}\boldsymbol{v}\,\left(\frac{1}{2}\overline{n}mw^{2}\right)\boldsymbol{w}f. & \displaystyle\end{eqnarray}$$

These equations are closed in the collisional limit by the results

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D745}\doteq -nm\unicode[STIX]{x1D707}\left((\unicode[STIX]{x1D735}\boldsymbol{u})+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-{\textstyle \frac{2}{3}}(\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{u})\unicode[STIX]{x1D644}\right), & \displaystyle\end{eqnarray}$$

where $\text{T}$  denotes transpose and $\unicode[STIX]{x1D707}$  is the kinematic viscosity ¹⁰ ; and

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{q}\doteq -n\unicode[STIX]{x1D705}\unicode[STIX]{x1D735}T, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$  is the first-order thermal conductivity. The linearizations of these equations around an absolute thermal equilibrium (which possesses no gradients or flows) of density $n_{0}=\overline{n}$ and temperature  $T_{0}$ are

(2.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{\unicode[STIX]{x0394}n}{n_{0}}\right)+\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle n_{0}m\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\boldsymbol{u}}{\unicode[STIX]{x2202}t}=n_{0}q\unicode[STIX]{x0394}\boldsymbol{E}-\unicode[STIX]{x1D735}\unicode[STIX]{x0394}p-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\unicode[STIX]{x1D745}, & \displaystyle\end{eqnarray}$$

(2.7c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}n_{0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}T}{\unicode[STIX]{x2202}t}=-p_{0}\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{q}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}p\doteq T_{0}\unicode[STIX]{x0394}n+n_{0}\unicode[STIX]{x0394}T$ and

(2.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D745}\doteq -n_{0}m\unicode[STIX]{x1D707}_{0}((\unicode[STIX]{x1D735}\unicode[STIX]{x0394}\boldsymbol{u})+(\unicode[STIX]{x1D735}\unicode[STIX]{x0394}\boldsymbol{u})^{\text{T}}-{\textstyle \frac{2}{3}}(\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u})\unicode[STIX]{x1D644}), & \displaystyle\end{eqnarray}$$

(2.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\boldsymbol{q}\doteq -n_{0}\unicode[STIX]{x1D705}_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x0394}T. & \displaystyle\end{eqnarray}$$

Specific formulas for the transport coefficients  $\unicode[STIX]{x1D707}_{0}$ and  $\unicode[STIX]{x1D705}_{0}$ are available (appendix A). The immediate goal is to rederive these results from the projection-operator formalism.

The starting point is the nonlinear kinetic equation

(2.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{D}f}{\text{D}t}\doteq \frac{\unicode[STIX]{x2202}f}{\unicode[STIX]{x2202}t}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f+\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f=-\text{C}[f], & \displaystyle\end{eqnarray}$$

where $\boldsymbol{\unicode[STIX]{x2202}}\doteq (q/m)\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\boldsymbol{v}$ and $\text{C}[f]$ is the collision operator, ¹¹ written here as a (spatially local) nonlinear functional (denoted by the square brackets) of  $f$ . For present purposes, one may think of  $C$ as the nonlinear Landau operator. However, although use of the Landau kinetic equation is commonplace, it requires a non-trivial justification. It appears to be adequate for the treatment of first-order and linear response in a weakly coupled plasma with a statistically homogeneous background; more generally, however, it is unclear that a spatially local collision operator even exists. Further explication of this point is relegated to Part 2, as it lies well outside of the scope of this introductory discussion.

The electric field  $\boldsymbol{E}=-\unicode[STIX]{x1D735}\unicode[STIX]{x1D719}$ (I consider only the electrostatic approximation) is obtained from Poisson’s equation

(2.10) $$\begin{eqnarray}\displaystyle & \displaystyle -\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}(\boldsymbol{x},t)=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}=4\unicode[STIX]{x03C0}\mathop{\sum }_{\overline{s}}(\overline{n}q)_{\overline{s}}\int \!\text{d}\overline{\boldsymbol{v}}\,f_{\overline{s}}(\boldsymbol{x},\overline{\boldsymbol{v}},t). & \displaystyle\end{eqnarray}$$

For the OCP, one allows perturbations to only one species, so a species label is dropped in this section; the other species merely serve to provide an overall charge-neutral background. A stationary and stable solution of the kinetic-equation–Poisson system is the absolute Maxwellian,

(2.11) $$\begin{eqnarray}\displaystyle & \displaystyle f_{\text{M}}(\boldsymbol{v})\doteq (2\unicode[STIX]{x03C0}v_{\text{t}}^{2})^{-3/2}\text{e}^{-v^{2}/2v_{\text{t}}^{2}}, & \displaystyle\end{eqnarray}$$

where ¹²   $v_{\text{t}}\doteq (T/m)^{1/2}$ . Infinitesimal perturbations to that equilibrium obey

(2.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}f+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x0394}f+\unicode[STIX]{x0394}\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}f_{\text{M}}=-\widehat{\text{C}}\unicode[STIX]{x0394}f, & & \displaystyle\end{eqnarray}$$

where $\widehat{\text{C}}$  is the linearized Landau operator (discussed in appendix B). This is the basic dynamical equation used in this article. (I shall add a constant background magnetic field in § 3.) As a linear dynamical evolution equation, it is analogous to the Schrödinger equation of quantum mechanics. There are, of course, important differences: the Schrödinger equation is time reversible whereas the present equation is time irreversible due to the presence of  $\widehat{\text{C}}$ , and the quantum-mechanical wave function is a probability amplitude (whose squared modulus is a probability) whereas $f$  is directly an actual probability (density).

2.2.1 Definition of the projection operator

In this and the next several sections I shall use the Schrödinger representation, in which averages of time-independent functions of velocity are taken with the time evolved  $f(t)$ , which plays the role of a time-dependent state vector. It is useful to treat those averages as projections, and it is convenient to realize those projections by using a time-independent scalar product. I shall use a Dirac bra–ket notation with a hidden weight function  $f_{\text{M}}$ . Thus, for any functions $A(\boldsymbol{v})$ and  $B(\boldsymbol{v})$ ,

(2.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A|\doteq A(\boldsymbol{v}), & \displaystyle\end{eqnarray}$$

(2.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle |A\rangle \doteq A(\boldsymbol{v})f_{\text{M}}(\boldsymbol{v}), & \displaystyle\end{eqnarray}$$

(2.13c ) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A|\text{L}|B\rangle \doteq \int \!\text{d}\boldsymbol{v}\,\text{d}\overline{\boldsymbol{v}}\,A(\boldsymbol{v})L(\boldsymbol{v},\overline{\boldsymbol{v}})B(\overline{\boldsymbol{v}})f_{\text{M}}(\overline{\boldsymbol{v}}), & \displaystyle\end{eqnarray}$$

where L is a linear operator with two-point kernel ¹³   $L(\boldsymbol{v},\boldsymbol{v}^{\prime })$ . When multispecies plasmas are discussed later, the scalar product will be extended to include species summations. Note that no complex conjugate appears in this definition, unlike the analogous situation in quantum mechanics. This scalar product is ‘natural’ because (appendix B) $\overline{n}\widehat{\text{C}}$  is self-adjoint with respect to it:

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A|\overline{n}\widehat{\text{C}}|B\rangle =\langle B|\overline{n}\widehat{\text{C}}|A\rangle . & \displaystyle\end{eqnarray}$$

A special case of (2.13c ) is obtained by specializing  $\text{L}$ to the identity operator $\text{I}\rightarrow \unicode[STIX]{x1D6FF}(\boldsymbol{v}-\overline{\boldsymbol{v}})$ :

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A|B\rangle =\int \!\text{d}\boldsymbol{v}\,A(\boldsymbol{v})B(\boldsymbol{v})f_{\text{M}}(\boldsymbol{v}), & \displaystyle\end{eqnarray}$$

just the equilibrium velocity average of the product $AB$ : $\langle A|B\rangle =\langle AB\rangle _{\text{M}}$ . I shall often drop the $\text{M}$  subscript.

Note that the scalar product used in this paper is defined on just the velocity space. That this is useful is a consequence of the assumptions that the collision operator is spatially local and that no potential-energy contributions are included in the definition of the hydrodynamic moments (see footnote 9 on page 9 and the discussion below). In Part 2, I shall remove those restrictions and employ a more general scalar product that generalizes the present one to also involve spatial integration.

Even for spatially local  $\widehat{\text{C}}$ , other operators as well as the bras and kets will in general depend on space. The spatial dependence will often be removed by Fourier transformation, so constructions such as $\langle A|B\rangle$ will be labelled by an (often implicit)  $\boldsymbol{x}$ or  $\boldsymbol{k}$ . ¹⁴

If one writes $\unicode[STIX]{x0394}f\doteq \unicode[STIX]{x0394}\unicode[STIX]{x1D712}\,f_{\text{M}}$ for unknown (dimensionless)  $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}$ , (2.12) becomes

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +|\boldsymbol{\unicode[STIX]{x2202}}\ln f_{\text{M}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{E}=-\widehat{\text{C}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle , & \displaystyle\end{eqnarray}$$

where $|\boldsymbol{\unicode[STIX]{x2202}}\ln f_{\text{M}}\rangle =-(q/T)|\boldsymbol{v}\rangle$ . In collisional transport theory, the conservation properties of the collision operator are crucial. For the Landau operator, those are ¹⁵

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \overline{n}\breve{\boldsymbol{A}}|\widehat{\text{C}}=0, & \displaystyle\end{eqnarray}$$

where $\overline{n}$  is the spatially constant density of the equilibrium,

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \breve{\boldsymbol{A}}\doteq (1\;\;\boldsymbol{P}^{\text{T}}\;\;K)^{\text{T}}, & \displaystyle\end{eqnarray}$$

$\boldsymbol{P}\doteq m\boldsymbol{v}$ , $K\doteq (1/2)mv^{2}$ , and the breve accent is used to distinguish non-orthogonal functions from more convenient orthogonal ones to be introduced shortly. ¹⁶ Thus, $\langle \breve{\boldsymbol{A}}|$ are the five null left eigenfunctions of  $\overline{n}\widehat{\text{C}}$ . (In fact, there is no need to distinguish left and right eigenfunctions because for the Landau operator $\overline{n}\widehat{\text{C}}$  is self-adjoint with respect to the chosen scalar product; see § B.1.) Those eigenfunctions span a preferred five-dimensional hydrodynamic subspace  ${\mathcal{H}}$ (see figure 1).

Clearly the definition of  ${\mathcal{H}}$ is tied to the form of the collision operator that is assumed. A consequence of assuming the Landau operator (appropriate for weak coupling) is that no potential-energy contributions appear in the eigenfunctions. Thus, the present  ${\mathcal{H}}$ is not the true hydrodynamic subspace of the many-particle system. This approximation is relaxed in Part 2.

From the definitions of  $n$ , $\boldsymbol{u}$ and  $T$ , it follows that

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \langle 1|\,f/f_{\text{M}}\rangle =n/\overline{n}, & \displaystyle\end{eqnarray}$$

so

(2.20) $$\begin{eqnarray}\displaystyle & \displaystyle \langle 1|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\unicode[STIX]{x0394}n/\overline{n}. & \displaystyle\end{eqnarray}$$

Similarly,

(2.21) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{v}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\unicode[STIX]{x0394}\boldsymbol{u} & \displaystyle\end{eqnarray}$$

(the absolute equilibrium has no flow). Finally,

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \langle K|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\frac{3}{2}\unicode[STIX]{x0394}T+\frac{3}{2}T\left(\frac{\unicode[STIX]{x0394}n}{\overline{n}}\right). & \displaystyle\end{eqnarray}$$

Thus, the perturbed (kinetic) temperature can be extracted by

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{3}{2}}\unicode[STIX]{x0394}T=\langle K^{\prime }|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle , & \displaystyle\end{eqnarray}$$

where

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle K^{\prime }\doteq K-\langle K\rangle ={\textstyle \frac{1}{2}}mv^{2}-{\textstyle \frac{3}{2}}T. & \displaystyle\end{eqnarray}$$

This motivates the introduction of

(2.25) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{A}\doteq (1\;\;{\boldsymbol{P}^{\prime }}^{\text{T}}\;\;K^{\prime })^{\text{T}}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{P}^{\prime }=\boldsymbol{P}-\langle \boldsymbol{P}\rangle =\boldsymbol{P}$ ; with this definition, the components of  $\boldsymbol{A}$ are orthogonal. Then the deviations of the statistically averaged hydrodynamic variables from their equilibrium values are given by

(2.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\boldsymbol{a}\doteq \langle \boldsymbol{A}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\left(\frac{\unicode[STIX]{x0394}n}{\overline{n}}\;\;\unicode[STIX]{x0394}\boldsymbol{p}^{\text{T}}\;\;\frac{3}{2}\unicode[STIX]{x0394}T\right)^{\text{T}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}\boldsymbol{p}\doteq m\unicode[STIX]{x0394}\boldsymbol{u}$ .

These results lead one to define the hydrodynamic projection operator as

(2.27) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}\doteq |\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\,\unicode[STIX]{x1D648}^{-1}\boldsymbol{\cdot }\langle \boldsymbol{A}|, & \displaystyle\end{eqnarray}$$

where the normalization matrix ¹⁷

(2.28) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D648}\doteq \langle \boldsymbol{A}\boldsymbol{A}^{\text{T}}\rangle =\left(\begin{array}{@{}ccc@{}}1 & \mathbf{0}^{\text{T}} & 0\\ \boldsymbol{ 0} & N_{\boldsymbol{p}}\unicode[STIX]{x1D644} & \mathbf{0}\\[3.0pt] 0 & \mathbf{0}^{\text{T}} & N_{T}\end{array}\right) & \displaystyle\end{eqnarray}$$

ensures that $\text{P}^{2}=\text{P}$ ; one easily calculates that

(2.29a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle N_{\boldsymbol{p}}\doteq (mv_{\text{t}})^{2},\quad N_{T}\doteq {\textstyle \frac{3}{2}}T^{2}. & \displaystyle\end{eqnarray}$$

The projection of the state vector is thus

(2.30) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}\boldsymbol{\cdot }\langle \boldsymbol{A}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}, & \displaystyle\end{eqnarray}$$

and the hydrodynamic variables can be extracted by taking the scalar product of  $\boldsymbol{A}$ with that projection:

(2.31) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\boldsymbol{a}(t)=\langle \boldsymbol{A}|\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle . & \displaystyle\end{eqnarray}$$

The projection formalism can be couched in a manifestly covariant fashion that turns out to be very convenient. Namely, if the components of  $\boldsymbol{A}$ are labelled as $A^{\unicode[STIX]{x1D707}}$ , where $\unicode[STIX]{x1D707}=1,\ldots ,5$ (or $\unicode[STIX]{x1D707}\in \{n,\;\boldsymbol{p},\;T\}$ ), and if $(\unicode[STIX]{x1D648}^{-1})_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$ is treated as a metric tensor $g_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$ that lowers indices according to $A_{\unicode[STIX]{x1D707}}=g_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}A^{\unicode[STIX]{x1D708}}$ (where Einstein’s convention for summation over repeated indices is adopted), then $\text{P}$  can be written as

(2.32) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}=|A_{\unicode[STIX]{x1D707}}\rangle \langle A^{\unicode[STIX]{x1D707}}|. & \displaystyle\end{eqnarray}$$

One has

(2.33) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A^{\unicode[STIX]{x1D707}}|A_{\unicode[STIX]{x1D707}^{\prime }}\rangle =\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D707}^{\prime }}^{\unicode[STIX]{x1D707}}, & \displaystyle\end{eqnarray}$$

the hydrodynamic projection of the state vector is

(2.34) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle =|A_{\unicode[STIX]{x1D707}}\rangle \unicode[STIX]{x0394}a^{\unicode[STIX]{x1D707}}(t) & \displaystyle\end{eqnarray}$$

(cf. (2.30)), and the hydrodynamic variables themselves are

(2.35) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}a^{\unicode[STIX]{x1D707}}(t)=\langle A^{\unicode[STIX]{x1D707}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle =\langle A^{\unicode[STIX]{x1D707}}|\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle & \displaystyle\end{eqnarray}$$

(cf. (2.26) and (2.31)). This representation, which identifies the  $\unicode[STIX]{x0394}a^{\unicode[STIX]{x1D707}}$ values as the contravariant components of a hydrodynamic vector, is further discussed in appendix C.

2.2.2 Projecting the kinetic equation; the statistical closure problem

Because $\text{P}$  is a time-independent linear operator, one can extract the linearized fluid moment equations by applying  $\text{P}$ to (2.16), or equivalently by taking the time derivative of (2.31). Define

(2.36a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}\doteq \text{L}+\text{L}_{\boldsymbol{E}}\quad \text{and}\quad \text{L}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}, & \displaystyle\end{eqnarray}$$

where

(2.37a-c ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\text{L}_{\boldsymbol{v}}\doteq \boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735},\quad \text{i}\text{L}_{\text{C}}\doteq \widehat{\text{C}},\quad \text{i}\text{L}_{\boldsymbol{E}}\doteq -(q/T)|\boldsymbol{v}\rangle \boldsymbol{\cdot }\langle \pmb{\mathbb{E}}|, & \displaystyle\end{eqnarray}$$

and $\pmb{\mathbb{E}}$  is the linear operator that solves Poisson’s equation for  $\unicode[STIX]{x0394}\boldsymbol{E}$ in terms of  $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}$ according to $\unicode[STIX]{x0394}\boldsymbol{E}=\langle \pmb{\mathbb{E}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ ; in Fourier space, the kernel of  $\pmb{\mathbb{E}}$ is

(2.38a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \pmb{\mathbb{E}}_{\boldsymbol{k}}(\boldsymbol{v},\boldsymbol{v}^{\prime })=\unicode[STIX]{x1D750}_{\boldsymbol{k}}\overline{n}q,\quad \unicode[STIX]{x1D750}_{\boldsymbol{k}}\doteq -4\unicode[STIX]{x03C0}\,\text{i}\boldsymbol{k}/k^{2} & \displaystyle\end{eqnarray}$$

( $\unicode[STIX]{x1D750}_{\boldsymbol{k}}$  is the Fourier transform of the electric field of a unit point charge). Then the hydrodynamic projection of the linearized kinetic equation is

(2.39) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\text{P}\text{i}{\mathcal{L}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =0. & \displaystyle\end{eqnarray}$$

The ultimate goal is to obtain a closed system for $\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ ; that is called ‘solving the closure problem’. That would be accomplished if one could move the  $\text{P}$ past the  ${\mathcal{L}}$ (i.e. if $\text{P}$ and  ${\mathcal{L}}$ commute – $[\text{P},\,{\mathcal{L}}]\doteq \text{P}{\mathcal{L}}-{\mathcal{L}}\text{P}=0$ ). However, this is not entirely possible. From (2.37), one sees that the $\unicode[STIX]{x0394}\boldsymbol{E}$  term is proportional to $|\boldsymbol{v}\rangle$ and thus, given the definition (2.27) of  $\text{P}$ , lies entirely in the hydrodynamic subspace [which is why $\text{L}_{\boldsymbol{E}}$  was broken out separately in (2.36)]. A consequence is that $\text{P}$ and  $\text{L}_{\boldsymbol{E}}$ commute, so there is no closure issue stemming from  $\text{L}_{\boldsymbol{E}}$ . However, the $\text{P}\text{i}\text{L}$ term is problematical because $\text{P}$ and  $\text{L}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}$ do not commute. Although for the OCP it is true that $[\text{P},\,\widehat{\text{C}}]=0$ as a consequence of the conservation laws, that is not true in general, ¹⁸ and it is never the case that $[\text{P},\,\boldsymbol{v}]=0$ . In order to see this, note that while $\text{P}\boldsymbol{v}\propto |\boldsymbol{A}\rangle$ lies in the hydrodynamic subspace by construction, one has

(2.40) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}\text{P}\propto |\boldsymbol{v}\boldsymbol{A}\rangle =\left(\begin{array}{@{}c@{}}|\boldsymbol{v}\rangle \\ |\boldsymbol{v}(m\boldsymbol{v})\rangle \\ |\boldsymbol{v}(\displaystyle \textstyle \frac{1}{2}mv^{2}-\textstyle \frac{3}{2}T)\rangle \end{array}\right). & \displaystyle\end{eqnarray}$$

While the density component of this vector lies in the hydrodynamic subspace, as do the diagonal elements of the momentum component [ $\boldsymbol{v}\boldsymbol{v}=(1/3)v^{2}\unicode[STIX]{x1D644}+(\boldsymbol{v}\boldsymbol{v}-(1/3)v^{2}\unicode[STIX]{x1D644})$ ] and the $(3/2)|\boldsymbol{v}\!\rangle T$ term, the off-diagonal terms and the kinetic-energy flux $|(1/2)mv^{2}\boldsymbol{v}\!\rangle$ lie in the orthogonal subspace. This is the familiar problem of moment closure, formalized here in terms of projection operators. It is one instance of the famous statistical closure problem much discussed in analytical turbulence theory (Krommes 2002, 2015, and references therein), here involving averages over a random velocity variable.

To deal with this difficulty, I follow Mori and insert the identity $\text{P}+\text{Q}=1$ so that

(2.41) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}{\mathcal{L}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\text{P}{\mathcal{L}}(\text{P}+\text{Q})|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\text{P}{\mathcal{L}}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\text{P}{\mathcal{L}}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle . & \displaystyle\end{eqnarray}$$

(In writing the last form, I used $\text{P}^{2}=\text{P}$ and $\text{Q}^{2}=\text{Q}$ . I also slightly abused the notation ¹⁹ to write $\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle \equiv |\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ .) Let us define the frequency matrix  $\unicode[STIX]{x1D734}$ as

(2.42) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D734}\doteq \langle \boldsymbol{A}|{\mathcal{L}}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}; & \displaystyle\end{eqnarray}$$

this is a normalized version of the hydrodynamic matrix element of  $\text{P}{\mathcal{L}}\text{P}$ . The nomenclature is consistent with the fact that under Fourier transformation $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\rightarrow \text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ , which corresponds to the streaming frequency $\unicode[STIX]{x1D6FA}_{\boldsymbol{v}}\doteq \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ . In general, the ‘frequency’ is complex because ${\mathcal{L}}$  involves the dissipative term  $\text{L}_{\text{C}}\doteq -\text{i}\widehat{\text{C}}$ . However, for the OCP $\widehat{\text{C}}$  does not contribute to the frequency matrix since $\text{P}\widehat{\text{C}}=0$ as a consequence of the conservation laws. In any event, if the $\text{P}{\mathcal{L}}\text{Q}$ term in (2.41) were to vanish, then one would have a matrix equation for $|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ (or for the $\unicode[STIX]{x0394}\boldsymbol{a}$ values) and the closure problem would be solved. Unfortunately, this is not the case because, as shown above, $[\text{P},\,\boldsymbol{v}]\neq \mathbf{0}$ . (This is one way of stating that the kinetic equation is stochastically nonlinear in the velocity variable.) Thus, the dynamics in the hydrodynamic subspace,

(2.43) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\,\unicode[STIX]{x1D648}^{-1}\boldsymbol{\cdot }\text{i}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\langle \boldsymbol{A}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\text{P}\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =0 & \displaystyle\end{eqnarray}$$

or, upon applying $\langle \boldsymbol{A}|$ to (2.43),

(2.44) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}\boldsymbol{a}+\text{i}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}+\langle \boldsymbol{A}|\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =0, & \displaystyle\end{eqnarray}$$

is coupled to that in the orthogonal subspace. Note that only  $\text{L}$ ( ${\mathcal{L}}$ sans  $\text{L}_{\boldsymbol{E}}$ ) enters the last term of (2.44); $\text{Q}\text{L}_{\boldsymbol{E}}=\text{L}_{\boldsymbol{E}}\text{Q}=0$ since $\text{L}_{\boldsymbol{E}}~\propto |\boldsymbol{v}\rangle \langle 1|$  has components only in  ${\mathcal{H}}$ .

In order to obtain a closed system, it is necessary to eliminate  $|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ in favour of some function of  $\unicode[STIX]{x0394}\boldsymbol{a}$ . Upon applying  $\text{Q}$ to (2.16), one obtains

(2.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\,\text{Q}\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\,\text{Q}\text{i}\text{L}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =0. & \displaystyle\end{eqnarray}$$

Since $\text{Q}\text{i}\text{L}\text{Q}$ is a linear operator, equation (2.45) can be solved by means of a Green’s function:

(2.46) $$\begin{eqnarray}\displaystyle & \displaystyle |\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle (t)=\text{G}_{\text{Q}}(t;0)|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle -\int _{0}^{t}\!\text{d}\overline{t}\,\text{G}_{\text{Q}}(t;\overline{t})\text{Q}\text{i}\text{L}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(\overline{t})\rangle , & \displaystyle\end{eqnarray}$$

where

(2.47) $$\begin{eqnarray}\displaystyle & \displaystyle \text{G}_{\text{Q}}(t;t^{\prime })\doteq H(t-t^{\prime })\exp [-\text{Q}\text{i}\text{L}\text{Q}(t-t^{\prime })] & \displaystyle\end{eqnarray}$$

(since  $\text{Q}$ and  $\text{L}$ are time independent). Here $H(\unicode[STIX]{x1D70F})$  is the unit step function. ²⁰ Only the time arguments are written explicitly in the previous two equations, although in reality the kernels of two-point operators such as  $\text{G}_{\text{Q}}$ also depend on two space and two velocity variables, so convolutions over those variables are implied. Because the background is spatially homogeneous and the equations are linear, one can Fourier transform in space. (This is not necessary, but it makes some of the notation less cumbersome.) It is conventional to ignore the initial condition $|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle$ . Physically, this means that the system is prepared to lie entirely in the hydrodynamic subspace. Then one obtains

(2.48) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)+\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{k}}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)+\int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t-\overline{\unicode[STIX]{x1D70F}})=0, & \displaystyle\end{eqnarray}$$

where ²¹

(2.49) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})\doteq \langle \boldsymbol{A}|\text{L}_{\boldsymbol{k}}\text{Q}\text{G}_{\text{Q},\boldsymbol{k}}(\unicode[STIX]{x1D70F})\text{Q}\text{L}_{\boldsymbol{k}}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}. & \displaystyle\end{eqnarray}$$

Although (2.48) is now closed in terms of  $\unicode[STIX]{x0394}\boldsymbol{a}$ , that system does not yet have the form of conventional linearized fluid equations because it is non-local in time. To obtain the conventional form, one makes a Markovian approximation. In the time domain, that means that one assumes that $\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)$  varies much more slowly than the characteristic decay time of  $\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ . Then (ignoring spatial or wavenumber dependence for the moment)

(2.50) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}(\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}(t-\overline{\unicode[STIX]{x1D70F}})\approx \left(\int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}(\overline{\unicode[STIX]{x1D70F}})\right)\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}(t)\approx \left(\int _{0}^{\infty }\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}(\overline{\unicode[STIX]{x1D70F}})\right)\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}(t).\quad & \displaystyle\end{eqnarray}$$

The temporal non-locality can also be represented in frequency space. The one-sided temporal Fourier transform of (2.48) (denoted by a hat) is

(2.51) $$\begin{eqnarray}\displaystyle & \displaystyle [-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D644}+\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{k}}+\widehat{\unicode[STIX]{x1D72E}}_{\boldsymbol{k}}(\unicode[STIX]{x1D714})]\boldsymbol{\cdot }\unicode[STIX]{x0394}\widehat{\boldsymbol{a}}_{\boldsymbol{k}}(\unicode[STIX]{x1D714})=\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(0). & \displaystyle\end{eqnarray}$$

The temporal Markovian approximation in (2.50) becomes $\widehat{\unicode[STIX]{x1D72E}}(\unicode[STIX]{x1D714})\approx \widehat{\unicode[STIX]{x1D72E}}(0)$ .

Because $\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ depends on  $\boldsymbol{k}$ , the inverse spatial Fourier transform of (2.48) becomes a convolution in space and thus demonstrates several kinds of spatial non-locality – one from the  $\text{L}_{\boldsymbol{k}}$ operators, and one from  $\text{G}_{\text{Q},\boldsymbol{k}}$ . The contribution from the  $\text{L}_{\boldsymbol{k}}$ operators is easy to understand. For the OCP, it is straightforward to see that only $\text{L}_{\boldsymbol{v},\boldsymbol{k}}\doteq \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ contributes to (2.49) because the components of $\boldsymbol{A}$ are the null eigenvectors of  $\widehat{\text{C}}$ . That means that $\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})\propto k^{2}$ . This is the Fourier representation of a diffusion process, corresponding to the spatial operator $-\unicode[STIX]{x1D6FB}^{2}$ , and is the expected result for a conventional transport theory. ( $\unicode[STIX]{x1D6FB}^{2}$  is non-local because it couples neighbouring points in space.)

Remaining to discuss is the $\boldsymbol{k}$  dependence of  $\text{G}_{\text{Q},\boldsymbol{k}}$ . Completely analogous to the temporal Markovian approximation described above, conventional collisional transport theory will emerge if one can make the spatial Markovian approximation $\text{G}_{\text{Q},\boldsymbol{k}}\approx \text{G}_{\text{Q},\mathbf{0}}$ .

To see whether these Markovian approximations are possible, let us look more closely at  $\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ . For scale lengths that are long compared to the mean free path, one can approximate $\exp [-\text{Q}(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}+\widehat{\text{C}})\text{Q}\unicode[STIX]{x1D70F}]\rightarrow \exp [-\text{Q}(\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}+\widehat{\text{C}})\text{Q}\unicode[STIX]{x1D70F}]\approx \exp (-\text{Q}\widehat{\text{C}}\text{Q}\unicode[STIX]{x1D70F})$ . (This $k\rightarrow 0$ limit is the spatial Markovian approximation.) For the OCP, one has $\text{P}\widehat{\text{C}}=\widehat{\text{C}}\text{P}=0$ , so $\exp (-\text{Q}\widehat{\text{C}}\text{Q}\unicode[STIX]{x1D70F})\rightarrow \exp (-\widehat{\text{C}}\unicode[STIX]{x1D70F})$ . If this operator were to act on an element of  ${\mathcal{H}}$ , which contains the null eigenfunctions of  $\widehat{\text{C}}$ , it would not decay and one would be in grave difficulty. However, in  $\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})$ this exponential is sandwiched between two $\text{Q}$  operators, giving the construction $\text{Q}\exp (-\widehat{\text{C}}\unicode[STIX]{x1D70F})\text{Q}$ . Those  $\text{Q}$ operators ‘protect’  $\widehat{\text{C}}$ from the null space. Thus, the null eigenvalues of  $\widehat{\text{C}}$ do not contribute and the effect of  $\widehat{\text{C}}$ is roughly to provide a damping $\text{e}^{-\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}}$ , where $\unicode[STIX]{x1D706}$  is of the order of the collision frequency  $\unicode[STIX]{x1D708}$ . (For discussion and examples of this assertion, which for the Landau operator is non-trivial, see § B.4.) Thus, since $\unicode[STIX]{x1D72E}(\unicode[STIX]{x1D70F})$  has a finite decay time  ${\sim}\unicode[STIX]{x1D708}^{-1}$ , the temporal Markovian approximation is justifiable and we have been led to a conventional set of time-local (Markovian) fluid equations:

(2.52) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)+(\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{k}}+\boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)=0, & \displaystyle\end{eqnarray}$$

where (for generality and later application to multispecies plasmas I reinstate the original construction $\exp (-\text{Q}\widehat{\text{C}}\text{Q}\unicode[STIX]{x1D70F})$ )

(2.53) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\boldsymbol{k}}\doteq \int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\unicode[STIX]{x1D70F})\approx \langle \boldsymbol{A}|\text{L}_{\boldsymbol{k}}\text{Q}(\text{Q}\widehat{\text{C}}\text{Q})^{-1}\text{Q}\text{L}_{\boldsymbol{k}}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}, & \displaystyle\end{eqnarray}$$

or covariantly

(2.54) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708},\boldsymbol{k}}^{\unicode[STIX]{x1D707}}=\langle A^{\unicode[STIX]{x1D707}}|\text{L}_{\boldsymbol{k}}\text{Q}(\text{Q}\widehat{\text{C}}\text{Q})^{-1}\text{Q}\text{L}_{\boldsymbol{k}}|A_{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

The result just obtained represents the lowest-order term in the Taylor expansion of $\unicode[STIX]{x0394}\boldsymbol{a}(\overline{\unicode[STIX]{x1D70F}})$ in (2.50). In principle, one could carry that expansion to higher order; at second order, one would ultimately obtain linear Burnett effects. However, I shall defer such a calculation to Part 2.

To reemphasize and reiterate the role of the  $\text{Q}$ operators, I return to the formal expression for  $\boldsymbol{\unicode[STIX]{x1D702}}$ , which involves the inverse of the operator $\text{Q}\widehat{\text{C}}\text{Q}$ applied to a quantity in the orthogonal subspace  $O$ . It is important to understand the sense in which this inverse exists. A quantity of the form $|\unicode[STIX]{x1D713}\rangle \doteq (\text{Q}\widehat{\text{C}}\text{Q})^{-1}\text{Q}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ can be obtained as the solution to the equation $(\text{Q}\widehat{\text{C}}\text{Q})|\unicode[STIX]{x1D713}\rangle =\text{Q}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ . For this equation to have a solution, the source term on the right-hand side must be orthogonal to the left null eigenfunctions of the operator on the left. That is true; both sides are annihilated by  $\text{P}$ . Thus, by Fredholm’s alternative theorem, the equation has a solution, but it is not unique because an arbitrary superposition $\sum _{i}\unicode[STIX]{x1D6FC}_{i}|i\rangle$ of the null eigenfunctions  $|i\rangle$ can be added to $|\unicode[STIX]{x1D713}\rangle$ without changing the equation. However, according to (2.53) or (2.54), one requires $\text{Q}|\unicode[STIX]{x1D713}\rangle$ , so the $\unicode[STIX]{x1D6FC}_{i}$  terms do not contribute. Thus, those terms may be set to 0, leaving one with a unique solution.

To see that the formalism recovers the conventional linearized fluid equations for the OCP, one needs to work out the various matrix elements. For variety, I shall here use the spatial form of the operators, but the discussion could equally well be conducted in $\boldsymbol{k}$  space. Let us begin with the frequency matrix:

(2.55) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D734}=\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}}+\text{i}\unicode[STIX]{x1D734}_{\text{C}}+\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{E}}, & \displaystyle\end{eqnarray}$$

where

(2.56a ) $$\begin{eqnarray}\displaystyle \displaystyle \text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}} & \doteq & \displaystyle \langle \boldsymbol{A}|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1},\end{eqnarray}$$

(2.56b ) $$\begin{eqnarray}\displaystyle \displaystyle \text{i}\unicode[STIX]{x1D734}_{\text{C}} & \doteq & \displaystyle \langle \boldsymbol{A}|\widehat{\text{C}}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1},\end{eqnarray}$$

(2.56c ) $$\begin{eqnarray}\displaystyle \displaystyle \text{i}\unicode[STIX]{x1D734}_{\boldsymbol{E}} & \doteq & \displaystyle \langle \boldsymbol{A}|\text{i}\text{L}_{\boldsymbol{E}}|\boldsymbol{A}^{\text{T}}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}.\end{eqnarray}$$

In these expressions, the role of the  $\unicode[STIX]{x1D648}^{-1}$ is to change contravariant indices to covariant ones. Thus, for example,

(2.57) $$\begin{eqnarray}\displaystyle & \displaystyle (\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}})_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}=\langle A^{\unicode[STIX]{x1D707}}|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

(Note that this mixed tensor has the dimensions of frequency.) For the OCP, $\unicode[STIX]{x1D734}_{\text{C}}$  vanishes because $\langle \boldsymbol{A}|$ is a left null eigenfunction of  $\widehat{\text{C}}$ . The $\unicode[STIX]{x1D734}_{E}$ term is readily shown to give rise to the electric-force term in the momentum equation. Finally, with $i$ , $j$ , and  $k$ denoting Cartesian components of the momentum vector, one finds

(2.58) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{A}|v_{k}|\boldsymbol{A}^{\text{T}}\rangle =\left(\begin{array}{@{}ccc@{}}0 & T\unicode[STIX]{x1D6FF}_{jk} & 0\\ T\unicode[STIX]{x1D6FF}_{ik} & \unicode[STIX]{x1D7EC} & T^{2}\unicode[STIX]{x1D6FF}_{ik}\\ 0 & T^{2}\unicode[STIX]{x1D6FF}_{jk} & 0\end{array}\right). & \displaystyle\end{eqnarray}$$

Upon multiplying on the right by $\unicode[STIX]{x1D648}^{-1}$ , one finds

(2.59) $$\begin{eqnarray}\displaystyle & \displaystyle (\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}})_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}=\left(\begin{array}{@{}ccc@{}}0 & m^{-1}\unicode[STIX]{x1D735}_{j} & 0\\ T\unicode[STIX]{x1D735}_{i} & \unicode[STIX]{x1D7EC} & \frac{2}{3}\unicode[STIX]{x1D735}_{i}\\[4.0pt] 0 & Tm^{-1}\unicode[STIX]{x1D735}_{j} & 0\end{array}\right). & \displaystyle\end{eqnarray}$$

Note that the spatial Fourier transform of this matrix is proportional to  $\boldsymbol{k}$ .

To illustrate the content of  $\unicode[STIX]{x1D734}$ , consider the density component of (2.52). It is easy to see that there is no $\unicode[STIX]{x1D702}$ contribution because $\langle 1|(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}+\widehat{\text{C}})\text{Q}=\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\langle \boldsymbol{v}|\,\text{Q}+\langle 1|\widehat{\text{C}}\text{Q}=0$ . The last equality follows since $|\boldsymbol{v}\!\rangle$  is in the hydrodynamic subspace, which is orthogonal to  $\text{Q}$ , and since $\widehat{\text{C}}$  conserves number. But $\unicode[STIX]{x1D6FA}_{\boldsymbol{p}}^{n}$ couples density to momentum, and one obtains

(2.60) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{\unicode[STIX]{x0394}n}{\overline{n}}\right)+\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

which reproduces the linearized continuity equation (2.7a ). As another illustration, consider the velocity component of (2.52). One readily finds that

(2.61) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\boldsymbol{p}}{\unicode[STIX]{x2202}t}=q\unicode[STIX]{x0394}\boldsymbol{E}-\overline{n}^{-1}\unicode[STIX]{x1D735}\unicode[STIX]{x0394}p+\cdots =0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x0394}p=T\unicode[STIX]{x0394}n+\overline{n}\unicode[STIX]{x0394}T$ and the centred dots denote the contribution from  $\boldsymbol{\unicode[STIX]{x1D702}}$ , which will be discussed in the next paragraph. The explicitly written terms reproduce the correct non-dissipative (Euler) part of the linearized momentum equation (2.7b ). Similarly, the temperature projection leads to

(2.62) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}T}{\unicode[STIX]{x2202}t}=-T\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}+\cdots \,, & \displaystyle\end{eqnarray}$$

the explicitly written terms being the Euler part of the linearized temperature equation (2.7c ).

Now consider the dissipative contributions from  $\boldsymbol{\unicode[STIX]{x1D702}}$ . I shall discuss the stress tensor in detail, leaving the heat-flow vector as an exercise for the reader. One has (now dropping the prime on  $\boldsymbol{v}^{\prime }$ since there is no flow in the equilibrium, and specifically considering the OCP)

(2.63) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\unicode[STIX]{x1D708}}^{\boldsymbol{p}}=-\langle m\boldsymbol{v}|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\text{Q}\widehat{\text{C}}-1\text{Q}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

The density component  $\unicode[STIX]{x1D702}_{n}^{\boldsymbol{p}}$ vanishes because $A_{n}=1$ and $\text{Q}|\boldsymbol{v}\rangle =\mathbf{0}$ . The temperature contribution vanishes by a symmetry argument that uses the fact that $\widehat{\text{C}}$  is rotationally invariant. For the self-coupling  $\unicode[STIX]{x1D702}_{\boldsymbol{p}}^{\boldsymbol{p}}$ , note that

(2.64) $$\begin{eqnarray}\displaystyle & \displaystyle \text{Q}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\boldsymbol{p}}\rangle =(1-\text{P})\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\boldsymbol{p}}\rangle . & \displaystyle\end{eqnarray}$$

One has

(2.65a ) $$\begin{eqnarray}\displaystyle \displaystyle \text{P}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\boldsymbol{p}}\rangle & = & \displaystyle |A_{\unicode[STIX]{x1D707}}\rangle \langle A^{\unicode[STIX]{x1D707}}|\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\boldsymbol{p}}\rangle\end{eqnarray}$$

(2.65b ) $$\begin{eqnarray}\displaystyle \displaystyle & = & \displaystyle |A_{n}\rangle (\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}})_{\boldsymbol{p}}^{n}+|A_{T}\rangle (\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}})_{\boldsymbol{p}}^{\text{T}}\end{eqnarray}$$

(2.65c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle {\textstyle \frac{1}{3}}T^{-1}|v^{2}\rangle \unicode[STIX]{x1D735}.\end{eqnarray}$$

Thus,

(2.66) $$\begin{eqnarray}\displaystyle \text{Q}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}|A_{\boldsymbol{p}}\rangle =T^{-1}\left|\boldsymbol{v}\,\boldsymbol{v}-{\textstyle \frac{1}{3}}v^{2}\unicode[STIX]{x1D644}\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735},\end{eqnarray}$$

and then

(2.67) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{\boldsymbol{p}}^{\boldsymbol{p}}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{p}=\overline{n}^{-1}\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\unicode[STIX]{x1D745}, & \displaystyle\end{eqnarray}$$

where the linearized stress tensor is

(2.68) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\unicode[STIX]{x1D745}=-\overline{n}m\unicode[STIX]{x1D662}\boldsymbol{ : }\unicode[STIX]{x1D735}\unicode[STIX]{x0394}\boldsymbol{u} & \displaystyle\end{eqnarray}$$

and the fourth-rank tensor  $\unicode[STIX]{x1D662}$ is

(2.69) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D662}\doteq v_{\text{t}}^{-2}\left\langle \boldsymbol{v}\,\boldsymbol{v}-{\textstyle \frac{1}{3}}v^{2}\unicode[STIX]{x1D644}\right|\widehat{\text{C}}^{-1}\left|\boldsymbol{v}\,\boldsymbol{v}-{\textstyle \frac{1}{3}}v^{2}\unicode[STIX]{x1D644}\right>. & \displaystyle\end{eqnarray}$$

This is the same result that follows from the traditional approach described in appendix A. There the matrix element is written in terms of $\boldsymbol{w}\doteq \boldsymbol{v}-\boldsymbol{u}$ , where $\boldsymbol{u}$  is the lowest-order flow velocity. Here we are perturbing from an equilibrium with no flow, so $\boldsymbol{w}\rightarrow \boldsymbol{v}$ .

The quantity $|m\boldsymbol{v}\boldsymbol{v}-(1/3)mv^{2}\unicode[STIX]{x1D644}\rangle$ (see (2.66)) is called the subtracted momentum flux; the subtraction arises from the  $\text{P}$ in $\text{Q}=1-\text{P}$ . All dissipative transport coefficients are defined in terms of subtracted fluxes; the physical reason is that the $\text{P}$  terms represent the fluxes that already exist in local thermal equilibrium, and those must be subtracted from the total flux in order to obtain the gradient-driven corrections that are responsible for the net transport (which determines the relaxation of a small perturbation to thermal equilibrium). In classical Chapman–Enskog theory (appendix A), subtracted fluxes arise when the first-order solvability conditions (Euler equations) are used to simplify the correction equations by eliminating time derivatives in favour of spatial gradients. The use of projection operators executes that task substantially more efficiently.

The reduction of  $\unicode[STIX]{x1D662}$ to a form involving a single scalar viscosity coefficient  $\unicode[STIX]{x1D707}$ is given in appendix A. One thus recovers the proper expression (2.8a ) for the linearized stress tensor  $\unicode[STIX]{x0394}\unicode[STIX]{x1D745}$ of the unmagnetised OCP, where $\unicode[STIX]{x1D745}$  is defined by (2.5). Similar considerations lead to (2.8b ) for the linearized heat flow.

I remind the reader that the present calculation of the dissipative fluxes is correct only to first order in the gradients. Second-order fluxes are discussed in Part 2.

3 The Braginskii equations: Multispecies, magnetised, classical, collisional fluid equations

I now turn to the important problem of classical transport in multispecies, magnetised plasma (in the limit of weak coupling). This is somewhat more technically complicated than is the OCP because of interspecies collisional coupling and the loss of symmetry due to the magnetic field, but the basic idea of projection into hydrodynamic and orthogonal subspaces still applies. Again, in this Part 1 I consider only linear response and first-order dissipative effects. Nonlinear equations and Burnett corrections will be considered in Part 2.

3.1 Exact form of the moment equations, and summary of the results for a two-species plasma

Before I consider perturbations from thermal equilibrium, it is useful to have the exact form of the moment equations in mind. The starting point will be the Landau kinetic equation for the one-particle distribution, including the effect of an external magnetic field  $\boldsymbol{B}$ (assumed here to be constant in space):

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\,f_{s}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{s}+(\boldsymbol{E}+c^{-1}\boldsymbol{v}\times \boldsymbol{B})\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}_{s}\,f=-\text{C}_{s}[f], & \displaystyle\end{eqnarray}$$

where as usual $\boldsymbol{E}_{\boldsymbol{k}}=\sum _{\overline{s}}(nq)_{\overline{s}}\int \!\text{d}\overline{\boldsymbol{v}}\,\unicode[STIX]{x1D750}_{\boldsymbol{k}}f_{\overline{s},\boldsymbol{k}}(\overline{\boldsymbol{v}})$ , $\boldsymbol{\unicode[STIX]{x2202}}_{s}\doteq (q/m)_{s}\unicode[STIX]{x2202}_{\boldsymbol{v}}$ , and $\text{C}$  is the Landau collision operator defined in (B 1). That operator conserves number density (separately for each species), total (summed over species) momentum density and total kinetic-energy density:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{\overline{s}}\overline{n}_{\overline{s}}\!\int \!\text{d}\overline{\boldsymbol{v}}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D6FF}_{s\overline{s}}\\ m_{\overline{s}}\overline{\boldsymbol{v}}\\ \textstyle \frac{1}{2}m_{\overline{s}}\overline{v}^{2}\end{array}\right)\text{C}_{\overline{s}}[f]=0. & \displaystyle\end{eqnarray}$$

Upon taking the number-density, momentum-density and kinetic-energy-density moments of (3.1), one is led to

(3.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}n_{s}+\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,(n_{s}\boldsymbol{u}_{s})=0, & \displaystyle\end{eqnarray}$$

(3.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle (nm)_{s}\frac{\text{d}_{s}\boldsymbol{u}_{s}}{\text{d}t}=(nq)_{s}(\boldsymbol{E}+c^{-1}\boldsymbol{u}_{s}\times \boldsymbol{B})-\unicode[STIX]{x1D735}p_{s}-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x1D745}_{s}+\boldsymbol{R}_{s}, & \displaystyle\end{eqnarray}$$

(3.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}n_{s}\frac{\text{d}_{s}T_{s}}{\text{d}t}=-p_{s}\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{u}_{s}-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{q}_{s}-\unicode[STIX]{x1D745}_{s}\boldsymbol{ : }\unicode[STIX]{x1D64E}_{s}+Q_{s}, & \displaystyle\end{eqnarray}$$

where

(3.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle p\doteq nT, & \displaystyle\end{eqnarray}$$

(3.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}_{s}}{\text{d}t}\doteq \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{s}\boldsymbol{\cdot }\unicode[STIX]{x1D735}, & \displaystyle\end{eqnarray}$$

(3.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D64E}\doteq {\textstyle \frac{1}{2}}[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}], & \displaystyle\end{eqnarray}$$

and the stress tensor  $\unicode[STIX]{x1D745}$ , the interspecies momentum transfer  $\boldsymbol{R}$ , the heat-flow vector  $\boldsymbol{q}$ and the heat generation  $Q$ are defined by

(3.5a ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D745}_{s} & \doteq & \displaystyle \int \!\text{d}\boldsymbol{v}\,(\overline{n}m\boldsymbol{w})_{s}\boldsymbol{w}_{s}\,f_{s}-p_{s}\unicode[STIX]{x1D644},\end{eqnarray}$$

(3.5b ) $$\begin{eqnarray}\displaystyle \displaystyle \boldsymbol{R}_{s} & \doteq & \displaystyle -\int \!\text{d}\boldsymbol{v}\,(\overline{n}m\boldsymbol{w})_{s}\text{C}_{s}[f],\end{eqnarray}$$

(3.5c ) $$\begin{eqnarray}\displaystyle \displaystyle \boldsymbol{q}_{s} & \doteq & \displaystyle \int \!\text{d}\boldsymbol{v}\left(\frac{1}{2}\overline{n}mw^{2}\right)_{s}\boldsymbol{w}_{s}\,f_{s},\end{eqnarray}$$

(3.5d ) $$\begin{eqnarray}\displaystyle \displaystyle Q_{s} & \doteq & \displaystyle -\int \!\text{d}\boldsymbol{v}\left(\frac{1}{2}\overline{n}mw^{2}\right)_{s}\text{C}_{s}[f],\end{eqnarray}$$

with $\boldsymbol{w}_{s}\doteq \boldsymbol{v}-\boldsymbol{u}_{s}$ . Braginskii uses the notation $\boldsymbol{R}\equiv \boldsymbol{R}_{e}$ . Conservation of total momentum ensures that $\sum _{i}\boldsymbol{R}_{i}=-\boldsymbol{R}$ . For a one-component plasma, $\boldsymbol{R}$  vanishes because the collision operator conserves momentum for like-species collisions; similarly, $Q$  vanishes for the OCP by energy conservation. In the general case, application of the conservation laws leads to

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \mathop{\sum }_{i}Q_{i}=-Q_{e}-\mathop{\sum }_{i}\boldsymbol{R}_{i}\boldsymbol{\cdot }(\boldsymbol{u}_{e}-\boldsymbol{u}_{i}), & \displaystyle\end{eqnarray}$$

which reduces to

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{e}=-Q_{i}+\boldsymbol{R}\boldsymbol{\cdot }\boldsymbol{u} & \displaystyle\end{eqnarray}$$

( $\boldsymbol{u}\doteq \boldsymbol{u}_{e}-\boldsymbol{u}_{i}$ ) for a single species of ions.

For a two-species, strongly magnetised electron–ion plasma (charges $q_{e}=-e$ and $q_{i}=Ze$ ) possessing overall charge neutrality and with mass ratio $\unicode[STIX]{x1D707}\ll 1$ and $(\unicode[STIX]{x1D708}/|\unicode[STIX]{x1D714}_{\text{c}}|)_{s}\ll 1$ , Braginskii quotes the following results, which constitute an approximate hydrodynamic closure (the numerical coefficients are valid for $Z=1$ ). Further discussion of the limits of validity of this closure is given at the end of this section.

1. (i) The electron and ion collision times are

   (3.8a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70F}_{e}=\frac{3m_{e}^{1/2}T_{e}^{3/2}}{4\sqrt{2\unicode[STIX]{x03C0}}\ln \unicode[STIX]{x1D6EC}\,Z^{2}e^{4}n_{i}},\quad \unicode[STIX]{x1D70F}_{i}=\frac{3m_{i}^{1/2}T_{i}^{3/2}}{4\sqrt{2\unicode[STIX]{x03C0}}\ln \unicode[STIX]{x1D6EC}\,Z^{4}e^{4}n_{i}}. & \displaystyle\end{eqnarray}$$

2. (ii) The gyrofrequencies are $\unicode[STIX]{x1D714}_{\text{c}s}\doteq (qB/mc)_{s}$ . (In my convention, unlike Braginskii’s, $\unicode[STIX]{x1D714}_{\text{c}e}$  is negative.) The gyroradii are $\unicode[STIX]{x1D70C}_{s}\doteq v_{\text{t}s}/|\unicode[STIX]{x1D714}_{\text{c}s}|$ , where $v_{\text{t}s}\doteq (T/m)_{s}^{1/2}$ .

3. (iii) The interspecies momentum transfer is denoted by $\boldsymbol{R}_{e}\equiv \boldsymbol{R}$ and $\boldsymbol{R}_{i}=-\boldsymbol{R}$ . It consists of two parts, $\boldsymbol{R}=\boldsymbol{R}_{\boldsymbol{u}}+\boldsymbol{R}_{T}$ :

   1. (1) The friction force  $\boldsymbol{R}_{\boldsymbol{u}}$ is, with $\boldsymbol{u}\doteq \boldsymbol{u}_{e}-\boldsymbol{u}_{i}$ ,

      (3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{R}_{\boldsymbol{u}}=-(mn)_{e}\unicode[STIX]{x1D70F}_{e}^{-1}(\unicode[STIX]{x1D6FC}\boldsymbol{u}_{\Vert }+\boldsymbol{u}_{\bot }), & \displaystyle\end{eqnarray}$$
      where $\unicode[STIX]{x1D6FC}\doteq 0.51$ .

   2. (2) The thermal force  $\boldsymbol{R}_{T}$ is

      (3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{R}_{T}=-\unicode[STIX]{x1D6FD}n_{e}\unicode[STIX]{x1D735}_{\Vert }T_{e}+\frac{3}{2}\frac{n_{e}}{\unicode[STIX]{x1D714}_{\text{c}e}\unicode[STIX]{x1D70F}_{e}}\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}T_{e}, & \displaystyle\end{eqnarray}$$
      where $\unicode[STIX]{x1D6FD}\doteq 0.71$ .

4. (iv) The electron heat flux is $\boldsymbol{q}_{e}=\boldsymbol{q}_{e,\boldsymbol{u}}+\boldsymbol{q}_{e,T}$ , where

   (3.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{q}_{e,\boldsymbol{u}}=\unicode[STIX]{x1D6FD}(nT)_{e}\boldsymbol{u}_{\Vert }-\frac{3}{2}\frac{(nT)_{e}}{\unicode[STIX]{x1D714}_{\text{c}e}\unicode[STIX]{x1D70F}_{e}}\widehat{\boldsymbol{b}}\times \boldsymbol{u}, & \displaystyle\end{eqnarray}$$

   (3.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{q}_{e,T}=-n_{e}\left(\unicode[STIX]{x1D705}_{\Vert e}\unicode[STIX]{x1D735}_{\!\Vert }T_{e}+\unicode[STIX]{x1D705}_{\bot e}\unicode[STIX]{x1D735}_{\!\bot }T_{e}+\frac{5}{2}\frac{v_{\text{t}e}^{2}}{\unicode[STIX]{x1D714}_{\text{c}e}}\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}T_{e}\right), & \displaystyle\end{eqnarray}$$
   where $\unicode[STIX]{x1D705}_{\Vert e}=3.16v_{\text{t}e}^{2}\unicode[STIX]{x1D70F}_{e}$ and $\unicode[STIX]{x1D705}_{\bot e}=4.66\unicode[STIX]{x1D70C}_{e}^{2}/\unicode[STIX]{x1D70F}_{e}$ . (The coefficient $4.66$ is derived in appendix D.)

5. (v) The ion heat flux is

   (3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{q}_{i}=-n_{e}\left(\unicode[STIX]{x1D705}_{\Vert i}\unicode[STIX]{x1D735}_{\!\Vert }T_{e}+\unicode[STIX]{x1D705}_{\bot i}\unicode[STIX]{x1D735}_{\!\bot }T_{e}+\frac{5}{2}\frac{v_{\text{t}i}^{2}}{\unicode[STIX]{x1D714}_{\text{c}i}}\widehat{\boldsymbol{b}}\times \unicode[STIX]{x1D735}T_{e}\right), & \displaystyle\end{eqnarray}$$
   where $\unicode[STIX]{x1D705}_{\Vert i}=3.9v_{\text{t}i}^{2}\unicode[STIX]{x1D70F}_{i}$ and $\unicode[STIX]{x1D705}_{\bot i}=2\unicode[STIX]{x1D70C}_{i}^{2}/\unicode[STIX]{x1D70F}_{i}$ .

6. (vi) The ion heat generation is

   (3.13) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{i}=Q_{\unicode[STIX]{x0394}}\doteq 3\left(\frac{m_{e}}{m_{i}}\right)\unicode[STIX]{x1D70F}_{e}^{-1}n_{e}(T_{e}-T_{i}). & \displaystyle\end{eqnarray}$$

7. (vii) The electron heat generation is

   (3.14) $$\begin{eqnarray}\displaystyle & \displaystyle Q_{e}=-\boldsymbol{R}\boldsymbol{\cdot }\boldsymbol{u}-Q_{\unicode[STIX]{x0394}}. & \displaystyle\end{eqnarray}$$

8. (viii) The stress tensor and viscosity coefficients are discussed in appendix E.

In addition to the basic mass-ratio and magnetic-field orderings, Braginskii’s equations have a restricted regime of validity. Even for linear response, they are correct only to first order in the gradients. Nonlinearly, as stressed by Mikhaǐlovskiǐ (1967), Mikhaǐlovskiǐ & Tsypin (1971, 1984) and Catto & Simakov (2004), Braginskii’s results correspond to a high-flow regime in which the mean flow velocity  $\boldsymbol{u}$ is of the order of the ion sound speed. When $\boldsymbol{u}$  is ordered smaller, additional terms must be retained. (These points are not made clear in Braginskii’s article; they are discussed in Part 2.)

3.2 The hydrodynamic projection for multispecies and magnetised plasma

The present goal is to rederive the linearized version of Braginskii’s closure by using projection methods. However, before proceeding with a hydrodynamic projection, one must decide whether to derive one-fluid or $S$ -fluid equations, where $S$  is the number of species. Because the null space of the collision operator is five-dimensional, it would be simplest to project into a five-dimensional, one-fluid hydrodynamic subspace. The resulting single-fluid equations would be identical in form to those derived earlier for the OCP except for the presence of the Lorentz force term. Unfortunately, such a one-fluid hydrodynamics disguises the important fact that when the mass ratio is small the physics of the electrons and the ions are quite different. Thus, following Braginskii, I shall derive a set of $S$ -fluid equations. However, this leads to technical complications because one now has $5S$  equations although the null space remains merely five-dimensional. A consequence is that the interspecies collisional coupling described by  $\boldsymbol{R}$ and  $Q$ results in some eigenvalues of the linearized problem being of the order of a collision frequency  $\unicode[STIX]{x1D708}$ (implying fast relaxation of some modes on the kinetic time scale) instead of being proportional to  $k^{2}$ (implying slow relaxation constrained by conservation laws).

To begin defining the relevant projection operator, let me introduce the natural scalar product, which is the generalization of (2.13c ) to include a species summation:

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A|\text{L}|B\rangle \doteq \mathop{\sum }_{\overline{s}}\int \!\text{d}\overline{\boldsymbol{v}}\mathop{\sum }_{\overline{s}^{\prime }}\int \!\text{d}\overline{\boldsymbol{v}}^{\prime }\,A_{\overline{s}}(\overline{\boldsymbol{v}})L_{\overline{s}\overline{s}^{\prime }}(\overline{\boldsymbol{v}},\overline{\boldsymbol{v}}^{\prime })B_{\overline{s}}(\overline{\boldsymbol{v}})f_{\text{M},\overline{s}^{\prime }}(\overline{\boldsymbol{v}}^{\prime }). & \displaystyle\end{eqnarray}$$

This is natural for several reasons:

1. (i) The linearized collision operator (times  $\overline{n}$ ) is self-adjoint with respect to it:

   (3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D713}|\overline{n}\widehat{\text{C}}|\unicode[STIX]{x1D712}\rangle =\langle \unicode[STIX]{x1D712}|\overline{n}\widehat{\text{C}}|\unicode[STIX]{x1D713}\rangle & \displaystyle\end{eqnarray}$$
   (see appendix B).

2. (ii) For $\boldsymbol{A}$  defined as in (2.25), the conservation properties of  $\widehat{\text{C}}$ take the simple form ²²

   (3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \overline{n}\boldsymbol{A}|\,\widehat{\text{C}}=0. & \displaystyle\end{eqnarray}$$

3. (iii) The first-order electric field follows as

   (3.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\boldsymbol{E}_{\boldsymbol{k}}=\langle \pmb{\mathbb{E}}_{\boldsymbol{k}}|\unicode[STIX]{x1D712}_{\boldsymbol{k}}\rangle , & \displaystyle\end{eqnarray}$$
   where
   (3.19) $$\begin{eqnarray}\displaystyle & \displaystyle \pmb{\mathbb{E}}_{\boldsymbol{k},s\overline{s}}(\boldsymbol{v},\overline{\boldsymbol{v}})\doteq (\overline{n}q)_{\overline{s}}\unicode[STIX]{x1D750}_{\boldsymbol{k}} & \displaystyle\end{eqnarray}$$
   (independent of  $s$ , $\boldsymbol{v}$ , and  $\overline{\boldsymbol{v}}$ ).

Although projections are often discussed in terms of scalar products, and the scalar product (3.15) is natural from several points of view, its use in defining an $S$ -species projection operation is somewhat problematical because of the implied species summation. If one insists on defining bras and kets in terms of a scalar product, that summation must sometimes be inhibited in one way or another in order that one end up with a species-dependent result. A definition that generalizes naturally to the $\unicode[STIX]{x1D6E4}$ -space theory discussed in Part 2 is the following. Subsume the species index into a generalized field index: $\boldsymbol{A}_{s}\equiv A_{s}^{\unicode[STIX]{x1D707}}\rightarrow A^{\unicode[STIX]{x1D707}}$ . (When a species index is written explicitly, then the superscript refers to just the usual field index.) Then define

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}\doteq |A^{\unicode[STIX]{x1D707}}\rangle M_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}^{\prime }}^{-1}\langle A^{\unicode[STIX]{x1D707}^{\prime }}|. & \displaystyle\end{eqnarray}$$

Also define

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}a^{\unicode[STIX]{x1D707}}=\langle A^{\unicode[STIX]{x1D707}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle . & \displaystyle\end{eqnarray}$$

Here one is treating  $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}$ as the collection of all  $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}_{s}$ values. Normally, the natural scalar product would require that the  $s$ be changed to  $\overline{s}$ and a summation over  $\overline{s}$ be done. However, $A^{\unicode[STIX]{x1D707}}$  depends on a specific species  $s_{\unicode[STIX]{x1D707}}$ , not the entire collection $A_{s}^{\unicode[STIX]{x1D707}}$ for all  $s$ . A consistent interpretation is that the presence of a specific species index inhibits the species summation in the scalar product. Effectively, $A_{s_{\unicode[STIX]{x1D707}}}^{\unicode[STIX]{x1D707}}$  behaves inside a scalar product as the $s$ -dependent quantity $\unicode[STIX]{x1D6FF}_{s_{\unicode[STIX]{x1D707}}(s)}A_{(s)}^{\unicode[STIX]{x1D707}}$ , where parentheses inhibit the summation convention. Then

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle \langle A^{\unicode[STIX]{x1D707}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\mathop{\sum }_{\overline{s}}\int \!\text{d}\overline{\boldsymbol{v}}\,\unicode[STIX]{x1D6FF}_{s_{\unicode[STIX]{x1D707}}\overline{s}}A_{\overline{s}}^{\unicode[STIX]{x1D707}}(\overline{\boldsymbol{v}})\unicode[STIX]{x0394}\unicode[STIX]{x1D712}_{\overline{s}}(\overline{\boldsymbol{v}})=\int \!\text{d}\overline{\boldsymbol{v}}\,A_{s_{\unicode[STIX]{x1D707}}}^{\unicode[STIX]{x1D707}}(\overline{\boldsymbol{v}})\unicode[STIX]{x0394}\unicode[STIX]{x1D712}_{s_{\unicode[STIX]{x1D707}}}(\overline{\boldsymbol{v}})\doteq \unicode[STIX]{x0394}a^{\unicode[STIX]{x1D707}}. & \displaystyle\end{eqnarray}$$

To extract all hydrodynamic field components of specific species  $s$ , I shall use the notation

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{a}_{s}=\langle \boldsymbol{A}1_{s}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle , & \displaystyle\end{eqnarray}$$

where the  $1_{s}$ inhibits the species summation in the scalar product.

Because $\boldsymbol{A}$  is defined in terms of purely kinetic quantities that do not couple species, in fact $M_{ss^{\prime }}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}^{\prime }}$ is diagonal in the species indices. It is also diagonal in the field indices because of the orthogonality built into the definition of  $\boldsymbol{A}$ (see (2.28)). Thus, $M^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D707}^{\prime }}=M^{(\unicode[STIX]{x1D707})}\unicode[STIX]{x1D6FF}_{(\unicode[STIX]{x1D707})\unicode[STIX]{x1D707}^{\prime }}$ ( $M^{\unicode[STIX]{x1D707}}\doteq M^{(\unicode[STIX]{x1D707})(\unicode[STIX]{x1D707})}$ ) and

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \text{P}=\mathop{\sum }_{\unicode[STIX]{x1D707}}|A^{\unicode[STIX]{x1D707}}\rangle M_{\unicode[STIX]{x1D707}}^{-1}\langle A^{\unicode[STIX]{x1D707}}|. & \displaystyle\end{eqnarray}$$

In Part 2, potential-energy contributions will be added to  $\boldsymbol{A}$ , $\unicode[STIX]{x1D648}$  will no longer be diagonal, and (a space-dependent generalization of) (3.20) will be used.

Define the magnetic-field operator by

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\widehat{\text{M}}\doteq c^{-1}\boldsymbol{v}\times \boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x2202}}=\unicode[STIX]{x1D714}_{\text{c}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}$  is the negative ²³ of the polar angle in velocity space. It is then a straightforward exercise to show that the kinetic equation linearized around an absolute Maxwellian can be written as

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle +\,\text{i}{\mathcal{L}}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =0, & \displaystyle\end{eqnarray}$$

where

(3.27a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}\doteq \text{L}+\text{L}_{\boldsymbol{E}}+\text{L}_{\text{M}},\quad \text{L}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}, & \displaystyle\end{eqnarray}$$

with $\text{L}_{\text{M}}\doteq \widehat{\text{M}}$ . The projections of the kinetic equation are formally the same as (2.43) and (2.45). Working out the components of the frequency matrix is straightforward. There are no surprises for $\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{v}}$ ; it leads to the same (species-dependent) Euler contributions as for the OCP. It is easy to show that $(\text{i}\unicode[STIX]{x1D734}_{\text{M}})_{\boldsymbol{p}}^{\boldsymbol{p}}$ gives rise to the usual Lorentz force, while all other magnetic contributions to the frequency matrix vanish. ²⁴ It is shown in appendix B that

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D734}_{\text{C}}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}=m_{s}\mathop{\sum }_{s^{\prime }}\unicode[STIX]{x1D708}_{ss^{\prime }}(\unicode[STIX]{x0394}\boldsymbol{u}_{s}-\unicode[STIX]{x0394}\boldsymbol{u}_{s^{\prime }})+3\mathop{\sum }_{s^{\prime }}\left(\frac{m_{s}}{M_{ss^{\prime }}}\right)\unicode[STIX]{x1D708}_{ss^{\prime }}(\unicode[STIX]{x0394}T_{s}-\unicode[STIX]{x0394}T_{s^{\prime }}), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D708}_{ss^{\prime }}$ is a generalized interspecies collision frequency defined by (B 30) and $M_{ss^{\prime }}\doteq m_{s}+m_{s^{\prime }}$ . At this point, we have obtained

(3.29a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{\unicode[STIX]{x0394}n}{\overline{n}}\right)_{s}=-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}_{s}, & \displaystyle\end{eqnarray}$$

(3.29b ) $$\begin{eqnarray}\displaystyle \displaystyle (\overline{n}m)_{s}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}\boldsymbol{u}_{s}}{\unicode[STIX]{x2202}t} & = & \displaystyle (\overline{n}q)_{s}(\unicode[STIX]{x0394}\boldsymbol{E}+c^{-1}\unicode[STIX]{x0394}\boldsymbol{u}_{s}\times \boldsymbol{B})-\unicode[STIX]{x1D735}\unicode[STIX]{x0394}p_{s}-(\overline{n}m)_{s}\mathop{\sum }_{s^{\prime }}\unicode[STIX]{x1D708}_{ss^{\prime }}(\unicode[STIX]{x0394}\boldsymbol{u}_{s}-\unicode[STIX]{x0394}\boldsymbol{u}_{s^{\prime }})\nonumber\\ \displaystyle & & \displaystyle -\,\langle \overline{n}\boldsymbol{P}^{\prime }1_{s}|\text{P}\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle ,\end{eqnarray}$$

(3.29c ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{3}{2}\overline{n}_{s}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}T_{s}}{\unicode[STIX]{x2202}t} & = & \displaystyle -p_{s}\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}_{s}-3\overline{n}_{s}\mathop{\sum }_{s^{\prime }}\left(\frac{m_{s}}{M_{ss^{\prime }}}\right)\unicode[STIX]{x1D708}_{ss^{\prime }}(\unicode[STIX]{x0394}T_{s}-\unicode[STIX]{x0394}T_{s^{\prime }})\nonumber\\ \displaystyle & & \displaystyle -\,\langle \overline{n}K^{\prime }1_{s}|\text{P}\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle .\end{eqnarray}$$

Obviously, the forms of the linearized fluid equations are beginning to emerge, with the parts involving $|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ to be determined by closure. Note that only $\text{L}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}$ enters those parts; it is easy to show that the magnetic-field operator does not couple the subspaces, so $\text{P}\text{i}\widehat{\text{M}}\text{Q}=\text{Q}\text{i}\widehat{\text{M}}\text{P}=0$ , and a similar result holds for  $\text{L}_{\boldsymbol{E}}$ . Also note that the frequency-matrix term involving $\unicode[STIX]{x0394}\boldsymbol{u}_{s}-\unicode[STIX]{x0394}\boldsymbol{u}_{s^{\prime }}$ in (3.29b ) is not, in general, the complete contribution to the interspecies momentum transfer. As is well known (Braginskii 1965), the effective collision frequency for the parallel momentum transfer differs by a numerical factor from the $\unicode[STIX]{x1D708}_{ss^{\prime }}$ defined by (B 30); for $Z=1$ , that factor is  $\unicode[STIX]{x1D6FC}=0.51$ . The physics is that perturbations to an absolute Maxwellian background do not in general merely produce a shifted Maxwellian; the $v^{-3}$ dependence of the electron–ion collision rate leads to a high-energy tail that enhances the parallel current for fixed electric field. This manifests as a reduction in the effective $\unicode[STIX]{x1D708}_{ss^{\prime },\Vert }$ . This effect is not seen in the frequency-matrix portion of the hydrodynamic projection, which only involves the perturbations of the quantities $n$ , $\boldsymbol{u}$ and  $T$ that would appear in a local Maxwellian distribution. Thus, the physics of the high-energy tail must be contained in the last, $|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ term of (3.29b ), as I shall now demonstrate.

3.3 Hydrodynamic closure

The straightforward generalization of (2.54) to the multispecies, magnetised case is

(3.30) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}\doteq \int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})\approx \langle A^{\unicode[STIX]{x1D707}}|\text{L}\text{Q}[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}\text{L}|A_{\unicode[STIX]{x1D708}}\rangle , & \displaystyle\end{eqnarray}$$

the only difference being the addition of the magnetic-field operator to  $\widehat{\text{C}}$ . ²⁵ A different way of representing the content of (3.30) is to rewrite the solution for the orthogonal projection,

(3.31) $$\begin{eqnarray}\displaystyle & \displaystyle |\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle \approx -[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}\text{i}\text{L}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle . & \displaystyle\end{eqnarray}$$

as the equation

(3.32) $$\begin{eqnarray}\displaystyle & \displaystyle (\text{i}\widehat{\text{M}}+\text{Q}\widehat{\text{C}})|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =-\text{Q}\text{i}\text{L}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle . & \displaystyle\end{eqnarray}$$

Here the result $\text{Q}\text{i}\widehat{\text{M}}\text{Q}=(1-\text{P})\text{i}\widehat{\text{M}}\text{Q}=\text{i}\widehat{\text{M}}\text{Q}$ was used. Unlike in the OCP, one cannot reduce $\text{Q}\widehat{\text{C}}\text{Q}\rightarrow \widehat{\text{C}}$ because the present hydrodynamic projection inhibits the species summation, so $\text{P}\widehat{\text{C}}\neq 0$ . Following Braginskii, I now restrict the calculation to a single species of ions. Then one finds

(3.33) $$\begin{eqnarray}\displaystyle \displaystyle \text{Q}\text{i}\text{L}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle & = & \displaystyle \frac{1}{2}\left|\frac{\boldsymbol{v}\,\boldsymbol{v}-(v^{2}/3)\unicode[STIX]{x1D644}}{v_{\text{t}}^{2}}\right\rangle \boldsymbol{ : }\unicode[STIX]{x1D652}[\unicode[STIX]{x0394}\boldsymbol{u}]+\left|\left(\frac{1}{2}\frac{v^{2}}{v_{\text{t}}^{2}}-\frac{5}{2}\right)\boldsymbol{v}\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{\unicode[STIX]{x0394}T}{T}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\text{Q}\widehat{\text{C}}\text{P}|\text{P}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle ,\end{eqnarray}$$

where the traceless tensor used by Braginskii is

(3.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D652}[\boldsymbol{u}]\doteq \unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}-{\textstyle \frac{2}{3}}(\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{u})\unicode[STIX]{x1D644}. & \displaystyle\end{eqnarray}$$

To simplify the last term of (3.33), use $\text{Q}\widehat{\text{C}}\text{P}=\widehat{\text{C}}\text{P}-\text{P}\widehat{\text{C}}\text{P}$ , where the last term is evaluated in § B.2 and is given by (B 29). The quantity $\widehat{\text{C}}\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ is evaluated in § B.3 for small mass ratio. For the electrons, the $\unicode[STIX]{x0394}T$  part of $\text{Q}\widehat{\text{C}}\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\!\rangle$ vanishes to lowest order in the mass ratio; for the ions, $\text{Q}\widehat{\text{C}}\text{P}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\!\rangle$ vanishes altogether to lowest order. Thus, with the results of appendix B, all pieces of (3.32) for $\text{Q}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ are known. To rearrange that equation into Braginskii’s form, use $\text{Q}\widehat{\text{C}}=\widehat{\text{C}}-\text{P}\widehat{\text{C}}$ and place the $\text{P}\widehat{\text{C}}$ terms on the right-hand side. For the electrons, one finds (cf. Braginskii’s equation (4.12))

(3.35) $$\begin{eqnarray}\displaystyle \displaystyle -\!(\text{i}\widehat{\text{M}}+\widehat{\text{C}}_{e})|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle & = & \displaystyle \frac{1}{2}\left|\frac{\boldsymbol{v}\,\boldsymbol{v}-(v^{2}/3)\unicode[STIX]{x1D644}}{v_{\text{t}e}^{2}}\right\rangle \boldsymbol{ : }\boldsymbol{W}[\unicode[STIX]{x0394}\boldsymbol{u}_{e}]+\left|\left(\frac{1}{2}\frac{v^{2}}{v_{\text{t}e}^{2}}-\frac{5}{2}\right)\boldsymbol{v}\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{\unicode[STIX]{x0394}T}{T}\right)_{e}\nonumber\\ \displaystyle & & \displaystyle +\frac{1}{v_{\text{t}e}^{2}\unicode[STIX]{x1D70F}_{e}}\left|\bigg[3\sqrt{\frac{\unicode[STIX]{x03C0}}{2}}\left(\frac{v_{\text{t}e}}{v}\right)^{3}-1\bigg]\boldsymbol{v}\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{u}-|\boldsymbol{v}\rangle \frac{1}{v_{\text{t}e}^{2}}\langle 1_{e}\boldsymbol{v}|\widehat{\text{C}}^{\text{Lor}}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle ,\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where the last term is the Lorentz approximation to $\text{P}\widehat{\text{C}}\text{Q}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$ ; only the momentum projection appears because the Lorentz collision operator conserves kinetic energy. The coefficient of that term is just the lowest-order approximation to the electron momentum transfer:

(3.36) $$\begin{eqnarray}\displaystyle & \displaystyle -|\boldsymbol{v}\rangle \frac{1}{v_{\text{t}e}^{2}}\langle 1_{e}\boldsymbol{v}|\widehat{\text{C}}^{\text{Lor}}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle =\frac{1}{\overline{n}T}|\boldsymbol{v}\rangle \boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{R}, & \displaystyle\end{eqnarray}$$

where (see (3.5b ))

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x0394}\boldsymbol{R}\doteq -\langle (\overline{n}m)_{e}\boldsymbol{v}1_{e}|\widehat{\text{C}}^{\text{Lor}}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle . & \displaystyle\end{eqnarray}$$

For the ions, one finds that the $\text{Q}\widehat{\text{C}}\text{P}$ term in (3.33) is negligible for small mass ratio, so (cf. Braginskii’s equation (4.15))

(3.38) $$\begin{eqnarray}\displaystyle & \displaystyle -(\text{i}\widehat{\text{M}}+\widehat{\text{C}}_{ii})|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\!\rangle =\frac{1}{2}\left|\frac{\boldsymbol{v}\,\boldsymbol{v}-(v^{2}/3)\unicode[STIX]{x1D644}}{v_{\text{t}i}^{2}}\right\rangle \boldsymbol{ : }\boldsymbol{W}[\unicode[STIX]{x0394}\boldsymbol{u}_{i}]+\left|\left(\frac{1}{2}\frac{v^{2}}{v_{\text{t}i}^{2}}-\frac{5}{2}\right)\boldsymbol{v}\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{\unicode[STIX]{x0394}T}{T}\right)_{i}; & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

this is identical in form to the correction equation for the OCP (see (A 18b )).

Thus, the projection-operator methodology restricted to first-order processes has reproduced Braginskii’s correction equations – as, of course, it must since physics content is invariant to mathematical representation. Although we have not obtained any new results, it is hoped that the use of projection operators clarifies the underlying structure of the transport equations, the key import of the null eigenspace, and the distinction between a perturbed local Maxwellian distribution and the true perturbed distribution that includes a high-energy, non-Maxwellian tail driven by the various thermodynamic forces.

From this point forward, the route to the final values of the transport coefficients, namely the evaluation of the matrix elements (3.30), follows that of Braginskii and other authors. In general, numerical or approximate analytical work is required; there is no need to repeat such analysis here. But as an illustration of the content of the correction equations and with the goal of providing further insight into the various orthogonal projections, I show in appendix D how to work out the perpendicular electron heat flow  $\boldsymbol{q}_{\bot ,e}$ in the limit of small $\unicode[STIX]{x1D708}_{e}/|\unicode[STIX]{x1D714}_{\text{c}e}|$ .

3.4 Onsager symmetries

Onsager’s symmetry theorem (Onsager 1931a , b ; Casimir 1945; Krommes & Hu 1993) is one of the deepest results in classical statistical physics. It is a statement about the relaxation of an arbitrarily coupled $N$ -body system slightly perturbed from a Gibbsian thermal equilibrium. For the transport matrix  $\boldsymbol{\unicode[STIX]{x1D702}}$ , it reads in the present covariant notation

(3.39) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\unicode[STIX]{x1D702}}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}(\boldsymbol{B})=\widehat{\unicode[STIX]{x1D702}}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D707}}(-\boldsymbol{B}). & \displaystyle\end{eqnarray}$$

Here $\widehat{\boldsymbol{\unicode[STIX]{x1D702}}}\doteq \unicode[STIX]{x1D640}\boldsymbol{\cdot }\boldsymbol{\unicode[STIX]{x1D702}}$ , where $\unicode[STIX]{x1D640}$  is the parity matrix such that under a time-reversal transformation $\boldsymbol{A}\rightarrow \unicode[STIX]{x1D640}\boldsymbol{\cdot }\boldsymbol{A}$ . (In a diagonal representation, $E_{(i)}^{(i)}=\pm 1$ depending on whether the $i$ th variable is even or odd under time reversal. For my choice of  $\boldsymbol{A}$ , $\unicode[STIX]{x1D640}=\text{diag}[1,\,-1,1]$ for each species.) Fundamentally, this symmetry is a consequence of the time reversibility of the microscopic dynamics. It is critical to observe that the theorem applies to the fully contravariant (or fully covariant) transport tensor, not the mixed tensor  $\boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}$ that appears naturally in the hydrodynamic equations. Failure to recognize this fact has led to confusion in the literature; a thorough discussion is given by Krommes & Hu (1993, § III).

As a consistency check, I shall sketch a proof that the present representation, involving weakly coupled dynamics represented by the Landau collision operator, possesses Onsager symmetry (as already discussed by Braginskii from a more traditional point of view). The fully contravariant version of (3.30) is

(3.40) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\langle A^{\unicode[STIX]{x1D707}}|\text{L}\text{Q}[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}\text{L}|A^{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

First suppose that instead of  $A^{\unicode[STIX]{x1D707}}$ and  $A^{\unicode[STIX]{x1D708}}$ one had generic functions  $\unicode[STIX]{x1D713}_{s}$ and  $\unicode[STIX]{x1D712}_{s}$ , where as usual the  $s$ is to be summed over in the standard scalar product. (Such functions would arise in a one-fluid hydrodynamics.) Then one could proceed by rearranging the scalar product with the aid of adjoint operators as follows:

(3.41) $$\begin{eqnarray}\displaystyle & \displaystyle F[\unicode[STIX]{x1D713},\unicode[STIX]{x1D712};\boldsymbol{B}]\doteq \langle \unicode[STIX]{x1D713}|\text{L}\text{Q}[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}\text{L}|\unicode[STIX]{x1D712}\rangle =\langle \unicode[STIX]{x1D712}|\text{L}^{\dagger }\text{Q}^{\dagger }{[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{\dagger }}^{-1}\text{Q}^{\dagger }\text{L}^{\dagger }|\unicode[STIX]{x1D713}\rangle . & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

With respect to the standard scalar product (which does not include complex conjugation), the operators $\text{Q}$ , $\overline{n}\widehat{\text{C}}$ , $\text{L}_{\boldsymbol{v}}\doteq -\text{i}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ and $\overline{n}\text{L}_{\text{C}}\doteq -\text{i}\overline{n}\widehat{\text{C}}$ are self-adjoint, whereas $\widehat{\text{M}}$  is anti-self-adjoint. ²⁶ Therefore,

(3.42) $$\begin{eqnarray}\displaystyle & \displaystyle F[\unicode[STIX]{x1D713},\unicode[STIX]{x1D712};\boldsymbol{B}]=\langle \unicode[STIX]{x1D712}|\text{L}\text{Q}[\text{Q}(-\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}\text{L}|\unicode[STIX]{x1D713}\rangle =F[\unicode[STIX]{x1D712},\unicode[STIX]{x1D713};-\boldsymbol{B}]. & \displaystyle\end{eqnarray}$$

This is a restricted form of Onsager’s symmetry. However, notice that if  $\unicode[STIX]{x1D713}$ and  $\unicode[STIX]{x1D712}$ were the hydrodynamic vector  $\boldsymbol{A}$ , then the properties $\langle \overline{n}\boldsymbol{A}|\widehat{\text{C}}=\mathbf{0}$ (conservation) and $\widehat{\text{C}}|\boldsymbol{A}\rangle =\mathbf{0}$ (null eigenvectors) remove the  $\widehat{\text{C}}$ from  $\text{L}$ and lead to the representation

(3.43) $$\begin{eqnarray}\displaystyle & \displaystyle F[\boldsymbol{A},\boldsymbol{A}^{\text{T}};\boldsymbol{B}]=\unicode[STIX]{x1D735}_{i}\langle \boldsymbol{A}|v_{i}\text{Q}[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}\text{Q}v_{j}|\boldsymbol{A}^{\text{T}}\rangle \unicode[STIX]{x1D735}_{j}. & \displaystyle\end{eqnarray}$$

Given the way $\boldsymbol{A}$ was constructed (its components are orthogonal), symmetry in velocity space implies that there is no cross-coupling between elements with opposite parity. Thus, Onsager’s symmetry (3.39) applies for the unhatted form of the transport matrix in this case.

In the more interesting case in which one projects onto a particular species, the transport matrix is constructed from the specific  $A^{\unicode[STIX]{x1D707}}$ and  $A^{\unicode[STIX]{x1D708}}$ , whose species indices  $s_{\unicode[STIX]{x1D707}}$ and  $s_{\unicode[STIX]{x1D708}}$ are not to be summed. Because the outer summations are inhibited, one cannot use the self-adjoint property of the  $\widehat{\text{C}}$ that appears in the $\text{L}$  operators, nor can one remove  $\widehat{\text{C}}$ from  $\text{L}$ by means of the species-summed conservation property; this implies the existence of non-trivial cross-terms in the transport matrix. I shall illustrate for the important special case of two species. One has

(3.44a ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D702}^{\boldsymbol{p}_{e}T_{e}} & = & \displaystyle -\langle \boldsymbol{P}_{e}^{\prime }|\widehat{\text{C}}_{ei}\text{Q}(\widehat{\text{D}}-1)_{ie}\text{Q}|K_{e}^{\prime }\boldsymbol{v}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735},\end{eqnarray}$$

(3.44b ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D702}^{T_{e}\boldsymbol{p}_{e}} & = & \displaystyle -\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\langle K_{e}^{\prime }\boldsymbol{v}|\text{Q}(\widehat{\text{D}}-1)_{ei}\text{Q}\widehat{\text{C}}_{ie}|\boldsymbol{P}_{e}^{\prime }\rangle ,\end{eqnarray}$$

where $\widehat{\text{D}}\doteq [\text{Q}(-\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}$ . The inverse of the matrix

(3.45) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D648}\doteq \left(\begin{array}{@{}cc@{}}\widehat{\text{A}} & \widehat{\text{B}}\\ \widehat{\text{C}} & \widehat{\text{D}}\end{array}\right), & \displaystyle\end{eqnarray}$$

where $\widehat{\text{A}}$ , $\widehat{\text{B}}$ , $\widehat{\text{C}}$ and  $\widehat{\text{D}}$ are non-commuting operators, is ²⁷

(3.46) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D648}^{-1}=\left(\begin{array}{@{}cc@{}}(\widehat{\text{A}}-\widehat{\text{B}}\widehat{\text{D}}-1\widehat{\text{C}})^{-1} & -(\widehat{\text{D}}\widehat{\text{B}}-1\widehat{\text{A}}-\widehat{\text{C}})^{-1}\\ -(\widehat{\text{A}}\widehat{\text{C}}-1\widehat{\text{D}}-\widehat{\text{B}})^{-1} & (\widehat{\text{D}}-\widehat{\text{C}}\widehat{\text{A}}-1\widehat{\text{B}})^{-1}\end{array}\right). & \displaystyle\end{eqnarray}$$

Because of the complicated form of (3.46), it is not yet obvious that (3.44a ) and (3.44b ) are equal to within a sign. To demonstrate that, use the momentum conservation property

(3.47) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \overline{n}_{e}\boldsymbol{P}_{e}^{\prime }|\widehat{\text{C}}_{ei}+\langle \overline{n}_{i}\boldsymbol{P}_{i}^{\prime }|\widehat{\text{C}}_{ie}=0 & \displaystyle\end{eqnarray}$$

and the result $\unicode[STIX]{x1D702}^{\text{T}_{e}\boldsymbol{P}_{e}}+\unicode[STIX]{x1D702}^{T_{e}\boldsymbol{P}_{i}}=0$ , which follows from $\widehat{\text{C}}|\boldsymbol{P}^{\prime }\rangle =0$ , to find

(3.48a ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}^{\boldsymbol{p}_{e}T_{e}}/T=\overline{n}_{e}^{-1}\langle \overline{n}_{i}\boldsymbol{P}_{i}^{\prime }|\widehat{\text{C}}_{ie}\text{Q}(\widehat{\text{D}}^{-1})_{ee}|\unicode[STIX]{x1D6FE}_{e}\boldsymbol{v}\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}, & \displaystyle\end{eqnarray}$$

(3.48b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}^{T_{e}\boldsymbol{p}_{e}}/T=\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\langle \unicode[STIX]{x1D6FE}_{e}\boldsymbol{v}|(\widehat{\text{D}}^{-1})_{ee}\text{Q}\widehat{\text{C}}_{ei}|\boldsymbol{P}_{i}^{\prime }\rangle , & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}(\boldsymbol{v})\doteq (1/2)mv^{2}/T-5/2$ arises from the calculation of $\text{Q}K^{\prime }\boldsymbol{v}$ . Since both expressions now involve the common matrix element ²⁸  ,where $\widehat{\text{C}}^{\prime }\doteq \text{i}\widehat{\text{M}}+\widehat{\text{C}}$ (cf. (3.46)), they can be easily compared. Upon referring to the form (B 6) of the linearized Landau operator, one sees that an integration by parts of $\langle \overline{n}_{i}\boldsymbol{P}_{i}^{\prime }|\widehat{\text{C}}_{ie}$ in the expression (3.48a ) introduces a minus sign and that the (3.48a,b ) are otherwise equal with $\boldsymbol{B}\rightarrow -\boldsymbol{B}$ . Thus, we have recovered (3.39). It is interesting to contemplate that the microscopic time reversibility used in Onsager’s original (and more general) derivation shows up in the above proof as the constraint of macroscopic momentum conservation.

Braginskii remarked upon the Onsager symmetry between the electron temperature-gradient contribution to the friction force and the flow-driven contribution to the electron heat flux. (Those effects are absent for the ions to lowest order in the mass ratio.) He failed to mention that the stress tensor  $\unicode[STIX]{x1D745}$ also affords an example of the symmetry. As discussed for the case of the OCP, $\unicode[STIX]{x1D745}$  can be written for infinitesimal perturbations as a fourth-order tensor  $\unicode[STIX]{x1D662}$ applied to $\unicode[STIX]{x1D735}\unicode[STIX]{x0394}\boldsymbol{u}$ . The ultimate effect in the momentum equation is $-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x1D745}=-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x1D662}\boldsymbol{ : }(\unicode[STIX]{x1D735}\unicode[STIX]{x0394}\boldsymbol{u})$ , which in Fourier space can be written as $(\boldsymbol{k}\boldsymbol{\cdot }\unicode[STIX]{x1D662}\boldsymbol{\cdot }\boldsymbol{k})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{u}\rightarrow \unicode[STIX]{x1D702}_{j}^{i}\unicode[STIX]{x0394}u^{j}=\unicode[STIX]{x1D702}^{ij}\unicode[STIX]{x0394}u_{j}$ ; here the lowering of the index just involves an index-independent normalization factor. As discussed in appendix E, $\unicode[STIX]{x1D662}$  is constructed from symmetrized tensor products of the matrices $\unicode[STIX]{x1D63D}\doteq \widehat{\boldsymbol{b}}\,\widehat{\boldsymbol{b}}$ , $\unicode[STIX]{x1D739}^{\bot }\doteq \unicode[STIX]{x1D644}-\widehat{\boldsymbol{b}}\,\widehat{\boldsymbol{b}}$ , and $\unicode[STIX]{x1D737}\doteq \widehat{\boldsymbol{b}}\times$ . The contributions to  $\boldsymbol{\unicode[STIX]{x1D702}}$ that do not involve  $\unicode[STIX]{x1D737}$ are easily seen to be symmetric and invariant under a change of sign of  $\boldsymbol{B}$ . The remaining terms (i.e. the gyroviscous stresses), involve either $\{\unicode[STIX]{x1D739}^{\bot }\unicode[STIX]{x1D737}\}$ or $\{\unicode[STIX]{x1D63D}\,\unicode[STIX]{x1D737}\}$ , where the symmetrization is denoted by the braces. Thus, the gyroviscous contributions to  $\boldsymbol{\unicode[STIX]{x1D702}}$ involve $\boldsymbol{k}\boldsymbol{\cdot }\{\unicode[STIX]{x1D739}^{\bot }\unicode[STIX]{x1D737}\}\boldsymbol{\cdot }\boldsymbol{k}$ or $\boldsymbol{k}\boldsymbol{\cdot }\{\unicode[STIX]{x1D63D}\,\unicode[STIX]{x1D737}\}\boldsymbol{\cdot }\boldsymbol{k}$ . These tensors are antisymmetric because of the factor of  $\unicode[STIX]{x1D737}$ , but since the gyroviscous terms are proportional to one power of the signed gyrofrequency, symmetry is restored under the replacement $\boldsymbol{B}\rightarrow -\boldsymbol{B}$ . Therefore, all contributions to  $\unicode[STIX]{x1D702}^{ij}$ obey the Onsager symmetry (3.39).

4 Generalized Langevin equation for the hydrodynamics of magnetised plasmas

The hydrodynamic equations developed in the previous sections describe the ensemble-averaged behaviour of the macroscopic observables. Underlying those equations is a system of generalized Langevin equations for fluctuating quantities whose means are the hydrodynamic variables. Langevin descriptions (Langevin 1908) are fundamental to the theory of both classical and turbulent transport, as they neatly capture and generalize the basic intuition associated with Brownian motion (Wang & Uhlenbeck 1945), namely, the competition between random excitation and coherent damping. In this section I shall develop a generalized Langevin equation for classical transport. ²⁹ It will be seen that the transport coefficients are the time integrals of the two-time correlations of the fluctuating forces that appear on the right-hand side of the Langevin equation. Such a representation is fundamental and a core topic in transport theory.

In the previous sections I used the Schrödinger representation, in which the state vector $|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle$ changes with time while the hydrodynamic observables  $\langle \boldsymbol{A}|$ are time independent. In this section I shall instead use the Heisenberg representation, in which averages are taken with the initial state $|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle$ while the observables become time dependent, to develop an evolution equation for the random hydrodynamic observables $\langle \boldsymbol{A}(t)|$ . The mean of that equation (its contraction with $|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle$ ) reduces to the usual fluid equations, but the random Langevin equation also contains fluctuating forces, analogous to the Langevin theory for classical Brownian motion.

4.1 Heisenberg versus Schrödinger representations

While the Heisenberg representation is familiar from quantum mechanics, there are some technical differences in the present application that need to be appreciated; therefore, I digress for a brief review. In quantum mechanics, the Schrödinger equation

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\hbar \,\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713}=\text{H}\unicode[STIX]{x1D713} & \displaystyle\end{eqnarray}$$

can be written as

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713}=-\text{i}{\mathcal{L}}\unicode[STIX]{x1D713}, & \displaystyle\end{eqnarray}$$

where ${\mathcal{L}}\doteq \text{H}/\hbar$ . For time independent  $\text{H}$ , the solution is given by

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D713}(t)=\text{G}(t)\unicode[STIX]{x1D713}(0), & \displaystyle\end{eqnarray}$$

where $\text{G}(t)\doteq \text{e}^{-\text{i}{\mathcal{L}}t}$ . Because $\text{H}$  is self-adjoint with respect to the usual complex-valued scalar product, $\text{G}$  is a unitary operator: $\text{G}\text{G}^{\dagger }=1$ .

In statistical mechanics, the $N$ -particle PDF  $P_{N}(\unicode[STIX]{x1D6E4},t)$ , where $\unicode[STIX]{x1D6E4}$  is the set of all phase-space coordinates, obeys the Liouville equation

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}P_{N}(\unicode[STIX]{x1D6E4},t)=-\text{i}{\mathcal{L}}P_{N}, & \displaystyle\end{eqnarray}$$

where ${\mathcal{L}}$  is the Liouville operator. Thus, the state evolves as

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle P_{N}(\unicode[STIX]{x1D6E4},t)=\text{e}^{-\text{i}{\mathcal{L}}t}P_{N}(\unicode[STIX]{x1D6E4},0). & \displaystyle\end{eqnarray}$$

Since ${\mathcal{L}}$  is anti-self-adjoint with respect to a real-valued scalar product, time dependence can be transferred to observables (functions of  $\unicode[STIX]{x1D6E4}$ that are to be averaged) according to

(4.6a ) $$\begin{eqnarray}\displaystyle \displaystyle \langle A(\unicode[STIX]{x1D6E4})\rangle & \doteq & \displaystyle \int \!\text{d}\unicode[STIX]{x1D6E4}\,A(\unicode[STIX]{x1D6E4})P_{N}(\unicode[STIX]{x1D6E4},t)\end{eqnarray}$$

(4.6b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \int \!\text{d}\unicode[STIX]{x1D6E4}\,A(\unicode[STIX]{x1D6E4})\text{e}^{-\text{i}{\mathcal{L}}t}P_{N}(\unicode[STIX]{x1D6E4},0)\end{eqnarray}$$

(4.6c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \int \!\text{d}\unicode[STIX]{x1D6E4}\,[\text{e}^{\text{i}{\mathcal{L}}t}A(\unicode[STIX]{x1D6E4})]P_{N}(\unicode[STIX]{x1D6E4},0)\end{eqnarray}$$

(4.6d ) $$\begin{eqnarray}\displaystyle & = & \displaystyle \langle A(t;\unicode[STIX]{x1D6E4})\rangle _{0},\end{eqnarray}$$

where $A(t;\unicode[STIX]{x1D6E4})\doteq \text{e}^{\text{i}{\mathcal{L}}t}A(\unicode[STIX]{x1D6E4})$ and the average is now with respect to the initial PDF. Thus, if the states are evolved with  $\text{G}(t)\doteq \text{e}^{-\text{i}{\mathcal{L}}t}$ , the trajectories evolve with  $\text{G}(-t)$ . This well-known result is a consequence of the fact that the microscopic dynamics is time reversible.

In the present situation governed by the linearized Landau kinetic equation, the state $\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(\boldsymbol{v},t)$ again evolves according to an equation of the form (4.2), where ${\mathcal{L}}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}+\text{L}_{\boldsymbol{E}}+\text{L}_{\text{M}}$  is given by (3.27). But ${\mathcal{L}}$ has no definite symmetry. The operators $\text{L}_{\boldsymbol{v}}\doteq \boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{v}$ and  $\text{L}_{\text{C}}\doteq -\text{i}\widehat{\text{C}}$ are self-adjoint with respect to the natural (real-valued) scalar product, $\text{L}_{\text{M}}$  is readily shown to be anti-self-adjoint, and the electric-field operator $\text{L}_{\boldsymbol{E}}\propto |\boldsymbol{v}\rangle \!\langle \!1|$ has no symmetry. Thus, the best one can do is to transfer the time dependence from the state to the observables according to

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{A}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(t)\rangle =\langle \boldsymbol{A}(t)|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle , & \displaystyle\end{eqnarray}$$

where

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{A}(t)\doteq \text{G}^{\dagger }(t)\boldsymbol{A}(0), & \displaystyle\end{eqnarray}$$

with

(4.9a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{G}(t)\doteq \text{e}^{-\text{i}{\mathcal{L}}t},\quad \text{G}^{\dagger }(t)\doteq \text{e}^{-\text{i}{\mathcal{L}}^{\dagger }t}. & \displaystyle\end{eqnarray}$$

As a consistency check, note that the magnetic-field operator is a special case of the Liouville operator and possesses the same (anti)symmetry as is demonstrated by (4.6).

4.2 Derivation of the generalized Langevin equation

To derive the generalized Langevin equation, consider the time evolution of the hydrodynamic variables:

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\langle \boldsymbol{A}(t)|=-\text{i}\langle {\mathcal{L}}^{\dagger }\boldsymbol{A}(t)|=-\text{i}\langle {\mathcal{L}}^{\dagger }\text{G}^{\dagger }\boldsymbol{A}(0)|=-\text{i}\langle \text{G}^{\dagger }{\mathcal{L}}^{\dagger }\boldsymbol{A}(0)|, & \displaystyle\end{eqnarray}$$

the last result following since ${\mathcal{L}}$  commutes with  $\text{G}$ (the latter being constructed from powers of  ${\mathcal{L}}$ ). As in previous manipulations, this result will be manipulated by a judicious insertion of the identity $\text{P}+\text{Q}=1$ . If that were done directly in the last form, virtually all of the symbols in the resulting expressions would be adorned with daggers. That could be avoided by working with the adjoint of (4.10). Alternatively, one can write formally

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\langle \text{G}(t)^{\dagger }{\mathcal{L}}^{\dagger }\boldsymbol{A}(0)|=-\text{i}\langle \boldsymbol{A}(0)|{\mathcal{L}}\text{G}(t), & \displaystyle\end{eqnarray}$$

anticipating that this bra will ultimately be combined with the Heisenberg state $|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\!\rangle$ . Proceeding similarly to the manipulations in the Schrödinger-picture projection, I rewrite this as

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\langle \boldsymbol{A}(0)|\,{\mathcal{L}}\text{G}=-\text{i}\langle \boldsymbol{A}(0)|\,{\mathcal{L}}(\text{P}+\text{Q})\text{G}. & \displaystyle\end{eqnarray}$$

The $\text{P}$  part of this becomes

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\langle \boldsymbol{A}(0)|{\mathcal{L}}|\boldsymbol{A}(0)\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}\boldsymbol{\cdot }\langle \boldsymbol{A}(0)|\text{G}=-\text{i}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\langle \boldsymbol{A}(t)|. & \displaystyle\end{eqnarray}$$

For the $\text{Q}$  part, Mori, Zwanzig, and others have shown that it is useful to express the final  $\text{G}$ in (4.12) in terms of the modified propagator  $\text{G}_{\text{Q}}$ defined by (2.47). To do so, consider the Fourier transform of $\text{G}(\unicode[STIX]{x1D70F})$ ,

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{G}(\unicode[STIX]{x1D714})=\int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\text{e}^{\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}}\text{e}^{-\text{i}{\mathcal{L}}\unicode[STIX]{x1D70F}}=[-\text{i}(\unicode[STIX]{x1D714}-{\mathcal{L}}+\text{i}\unicode[STIX]{x1D716})]^{-1}, & \displaystyle\end{eqnarray}$$

and use the identity, valid for arbitrary non-commuting operators  $\text{A}$ and  $\text{B}$ (assuming that $\text{A}^{-1}$  is defined),

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle (\text{A}+\text{B})^{-1}=\text{A}^{-1}-\text{A}^{-1}\text{B}(\text{A}+\text{B})^{-1}, & \displaystyle\end{eqnarray}$$

with

(4.16a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \text{A}\doteq -\text{i}(\unicode[STIX]{x1D714}-\text{Q}{\mathcal{L}}\text{Q}+\text{i}\unicode[STIX]{x1D716}),\quad \text{B}\doteq \text{i}({\mathcal{L}}-\text{Q}{\mathcal{L}}\text{Q}). & \displaystyle\end{eqnarray}$$

Thus, ³⁰

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{G}(\unicode[STIX]{x1D714})=\widehat{G}_{\text{Q}}(\unicode[STIX]{x1D714})-\widehat{G}_{\text{Q}}(\unicode[STIX]{x1D714})[\text{i}({\mathcal{L}}-\text{Q}{\mathcal{L}}\text{Q})]\widehat{G}(\unicode[STIX]{x1D714}). & \displaystyle\end{eqnarray}$$

Now

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}-\text{Q}{\mathcal{L}}\text{Q}={\mathcal{L}}-[(1-\text{P}){\mathcal{L}}(1-\text{P})]=\text{P}{\mathcal{L}}+{\mathcal{L}}\text{P}-\text{P}{\mathcal{L}}\text{P}. & \displaystyle\end{eqnarray}$$

One requires $\text{Q}\text{G}$ for use in (4.12). Because $\text{Q}\text{G}_{\text{Q}}=\text{G}_{\text{Q}}\text{Q}$ and $\text{Q}\text{P}=0$ , the first and last terms of (4.18) do not contribute to (4.17). Therefore, upon noting that $\text{L}_{\boldsymbol{E}}$ and  $\text{L}_{\text{M}}$ do not contribute to  $\text{Q}{\mathcal{L}}$ , one finds

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \text{Q}\text{G}=\text{Q}\text{G}_{\text{Q}}-\text{Q}\text{G}_{\text{Q}}\text{i}\text{L}\text{P}\text{G}. & \displaystyle\end{eqnarray}$$

Upon inserting the explicit form of  $\text{P}$ into (4.19), one can rewrite the last term of (4.12) as

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\langle \boldsymbol{A}(0)|\text{L}\text{Q}\text{G}_{\text{Q}}(t)-\langle \boldsymbol{A}(0)|\text{L}\text{Q}\text{G}_{\text{Q}}(t)\text{L}|\boldsymbol{A}^{\text{T}}(0)\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}\boldsymbol{\ast }\langle \boldsymbol{A}(t)|, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{\ast }$  denotes time convolution as well as dot product. The first term of (4.20) can be written as a random force $\langle \boldsymbol{f}(t)|$ , where

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle |\,\boldsymbol{f}^{\text{T}}(t)\rangle \doteq -\text{i}\text{G}_{\text{Q}}^{\dagger }(t)\text{Q}{\mathcal{L}}^{\dagger }|\boldsymbol{A}^{\text{T}}(0)\rangle =\text{Q}\text{G}_{\text{Q}}^{\dagger }(t)|\dot{\boldsymbol{A}}^{\text{T}}(0)\rangle . & \displaystyle\end{eqnarray}$$

The last term of (4.20) can be written as

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle -\langle \boldsymbol{A}(0)|\text{L}\text{Q}\text{G}_{\text{Q}}(t)\text{L}|\boldsymbol{A}^{\text{T}}(0)\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}\boldsymbol{\ast }\langle \boldsymbol{A}(t)|=-\unicode[STIX]{x1D72E}\boldsymbol{\ast }\langle \boldsymbol{A}(t)|, & \displaystyle\end{eqnarray}$$

where, upon recalling (4.21),

(4.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D72E}(t)\doteq \langle \boldsymbol{f}(t)\boldsymbol{f}^{\text{T}}(0)\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D648}^{-1}. & \displaystyle\end{eqnarray}$$

In summary, we have found the exact generalized Langevin equation ³¹

(4.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\langle \boldsymbol{A}(t)|+\text{i}\unicode[STIX]{x1D734}\boldsymbol{\cdot }\langle \boldsymbol{A}(t)|+\int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}(\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\langle \boldsymbol{A}(t-\overline{\unicode[STIX]{x1D70F}})|=\langle \boldsymbol{f}(t)|. & \displaystyle\end{eqnarray}$$

To demonstrate compatibility with the previous results, one may apply $|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\!\rangle$ to (4.24), thus performing the statistical average. One needs

(4.25) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \boldsymbol{f}(t)|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle =-\text{i}\langle \boldsymbol{A}(0)|\text{L}\text{G}_{\text{Q}}(t)\text{Q}|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle . & \displaystyle\end{eqnarray}$$

This generates the same contribution to the $\text{P}$  equation from the initial condition that one would have found by retaining the first term of (2.46). When the system is prepared in the hydrodynamic subspace, one has $|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle =0$ and the contribution from the random force vanishes. The resulting equation,

(4.26) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{t}+\text{i}\unicode[STIX]{x1D734}+\unicode[STIX]{x1D72E}\boldsymbol{\ast })\langle \boldsymbol{A}(t)|\unicode[STIX]{x0394}\unicode[STIX]{x1D712}(0)\rangle =0, & \displaystyle\end{eqnarray}$$

is identical to that previously derived from the Schrödinger representation (see (2.48)).

4.3 Fluctuating hydrodynamics and transport coefficients

While (4.24) has the form of a generalized Langevin equation, it must not be assumed that it is always justifiable to treat $\langle \boldsymbol{f}(t)|$ as being white noise, as is often done in simple models (for example, see the discussion of the Brownian test particle in § G.3). The random noise involves the modified propagator  $\text{G}_{\text{Q}}$ , which encapsulates complicated details of the dynamics. Generalized Langevin equations can be derived for projections into essentially any subspace whatsoever, and the properties of  $\langle \boldsymbol{f}|$ depend on the dimensionality of the subspace and the choice of variables  $\boldsymbol{A}$ that is made. (For some important caveats relating to the choice of projection operators, see appendix G.) The issue is particularly clear when one follows Mori (1965) and projects the Liouville equation. Then (4.24) merely describes an exact rearrangement of the $N$ -particle dynamics, with both (some) linear and nonlinear physics being buried in  $\boldsymbol{f}$ . Note that the precise way in which physics content is apportioned between  $\unicode[STIX]{x1D734}$ , $\unicode[STIX]{x1D72E}$ and  $\boldsymbol{f}$ depends on the choice of the projection operator. In particular, for arbitrary  $\text{P}$ there is a part of  $\boldsymbol{f}$ that lives partly in the hydrodynamic subspace and whose mean does not vanish. However, Zwanzig (2001, p. 156) shows that provided that one chooses  $\text{P}$ as I have done (using the standard scalar product), that mean vanishes to linear order. Furthermore, the specific choice of the hydrodynamic variables  $\boldsymbol{A}$ that I have used to build  $\text{P}$ ensures that the long-wavelength limit of  $\unicode[STIX]{x1D72E}$ is well behaved for the evolution of the conserved quantities. Thus, the exact generalized Langevin equation (4.24) is useful for the treatment of linear perturbations from thermal equilibrium, to which this paper is restricted. (In Part 2, I show how to generalize the procedure to include nonlinear effects.)

Indeed, for the standard hydrodynamic projection, several classical results for neutral fluids, as well as their extensions to magnetised plasmas, readily follow from the previous results. The topic of hydrodynamic fluctuations and their relation to transport coefficients has a long history that I shall not attempt to fully review here. In brief: Landau & Lifshitz (1957, 1987) argued that the transport coefficients of a classical fluid are intimately related to the two-time correlation functions of certain fluctuating forces; for example, the thermal conductivity is related to the autocorrelation of a random heat flow. Kadanoff & Martin (1963) stressed the importance of the double (ordered) limit $\lim _{\unicode[STIX]{x1D714}\rightarrow 0}\lim _{k\rightarrow 0}$ in extracting transport coefficients from certain response formulas. (See the discussion of the plateau phenomenon in § G.3.) The Landau–Lifshitz formulas were derived more systematically from kinetic theory by Bixon & Zwanzig (1969), whose work was slightly generalized by Hinton (1970). A review article that provides useful background is by Fox (1978).

4.3.1 Transport coefficients as current–current correlations

To tie those discussions of hydrodynamic fluctuations to the present formalism, compare (2.49) with formula (4.21), which defines the fluctuating force. One readily sees that

(4.27) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})=\langle \,f^{\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F})f_{\unicode[STIX]{x1D708}}(0)\rangle . & \displaystyle\end{eqnarray}$$

To obtain the Markovian transport matrix  $\boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}$ , (3.30), one takes $k,\unicode[STIX]{x1D714}\rightarrow 0$ in  $\text{G}_{\text{Q}}$ (the order of the limits is immaterial). To illustrate, first consider the unmagnetised OCP. Then only $\text{L}_{\boldsymbol{v}}$ contributes to formulas (3.30) and (4.21). With $\unicode[STIX]{x1D735}\rightarrow \text{i}\boldsymbol{k}$ , one finds ³²

(4.28) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}\rightarrow \boldsymbol{k}\,\boldsymbol{k}\boldsymbol{ : }\langle A^{\unicode[STIX]{x1D707}}|\boldsymbol{v}\text{Q}\widehat{\text{C}}-1\text{Q}\boldsymbol{v}|A_{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

If one writes $f^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\widehat{\boldsymbol{J}}^{\unicode[STIX]{x1D707}}$ for generalized (subtracted) currents  $\widehat{\boldsymbol{J}}^{\unicode[STIX]{x1D707}}$ , then one has

(4.29) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}=\boldsymbol{k}\,\boldsymbol{k}\boldsymbol{ : }\langle \widehat{\boldsymbol{J}}^{\unicode[STIX]{x1D707}}|\widehat{\text{C}}-1|\widehat{\boldsymbol{J}}_{\unicode[STIX]{x1D708}}\rangle . & \displaystyle\end{eqnarray}$$

By the symmetry of the unmagnetised system, the expectation must be proportional to the unit tensor,

(4.30) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \widehat{\boldsymbol{J}}^{\unicode[STIX]{x1D707}}|\widehat{\text{C}}-1|\widehat{\boldsymbol{J}}_{\unicode[STIX]{x1D708}}\rangle =D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D644}, & \displaystyle\end{eqnarray}$$

so one can obtain the generalized transport coefficients  $D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}$ by

(4.31) $$\begin{eqnarray}\displaystyle & \displaystyle D_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}=k^{-2}\boldsymbol{\unicode[STIX]{x1D702}}_{\unicode[STIX]{x1D708}}^{\unicode[STIX]{x1D707}}. & \displaystyle\end{eqnarray}$$

As an example, the thermal conductivity follows as

(4.32) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D705}=\langle \widehat{J}^{\text{T}}|\widehat{\text{C}}-1|\widehat{J}_{T}\rangle , & \displaystyle\end{eqnarray}$$

where

(4.33) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{J}^{\text{T}}\doteq \left(\frac{1}{2}\frac{v^{2}}{v_{\text{t}}^{2}}-\frac{5}{2}\right)v_{z}. & \displaystyle\end{eqnarray}$$

Note that  $(5/2)nT$ is the ideal-gas value of the enthalpy. The role of the enthalpy subtraction and the thermodynamic interpretation of  $\widehat{J}^{\text{T}}$ as a heat current is discussed by Kadanoff & Martin (1963, p. 441).

4.3.2 Fluctuating forces and collision-driven fluxes

In the multispecies case, $\widehat{\text{C}}$  also contributes to the  $\text{L}$ in  $f^{\unicode[STIX]{x1D707}}$ . That gives rise to a fluctuating friction force  $\unicode[STIX]{x1D6FF}\boldsymbol{R}$ and a fluctuating thermal force  $\unicode[STIX]{x1D6FF}\boldsymbol{q}$ . The autocorrelation of  $\unicode[STIX]{x1D6FF}\boldsymbol{R}$ leads to the non-hydrodynamic part of the friction force. The cross-correlation of  $\unicode[STIX]{x1D6FF}\boldsymbol{R}$ and  $\unicode[STIX]{x1D6FF}\boldsymbol{q}$ leads to the temperature-gradient-driven part of the momentum transfer and, by Onsager symmetry, to the flow-driven part of the heat flow.

The existence of all of these effects was well known to Braginskii, who interpreted the systematic Chapman–Enskog mathematics with simple physical pictures. Those arguments are entirely correct, and I have nothing to add to the physics. However, since Braginskii does not explicitly mention fluctuating forces in the sense of the present formalism, it is useful to understand the connection between the various approaches. As an example, consider the temperature-gradient contribution to the electron momentum transfer. This arises from

(4.34) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{T}^{\boldsymbol{p}}=\int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\langle \,f^{\boldsymbol{p}}(\unicode[STIX]{x1D70F})f_{T}(0)\rangle =\int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\langle \,f^{\boldsymbol{p}}(0)|\text{G}_{\text{Q}}(\unicode[STIX]{x1D70F})|f_{T}(0)\rangle & \displaystyle\end{eqnarray}$$

when $f^{\boldsymbol{p}}$  is evaluated with  $\text{L}_{\text{C}}$ and $f_{T}$  is evaluated with  $\text{L}_{\boldsymbol{v}}$ .

The streaming contribution to the fluctuating heat flow is

(4.35a ) $$\begin{eqnarray}\displaystyle \displaystyle |\,f_{T}(0)\rangle & = & \displaystyle -\text{i}\text{Q}\text{L}_{\boldsymbol{v}}|A_{T}(0)\rangle\end{eqnarray}$$

(4.35b ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\text{Q}\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\left(\frac{3}{2}T\right)^{-1}\left|\frac{1}{2}\frac{v^{2}}{v_{\text{t}}^{2}}-\frac{3}{2}\right\rangle\end{eqnarray}$$

(4.35c ) $$\begin{eqnarray}\displaystyle & = & \displaystyle -\!\left(\frac{3}{2}T\right)^{-1}\left|\boldsymbol{v}\left(\frac{1}{2}\frac{v^{2}}{v_{\text{t}}^{2}}-\frac{5}{2}\right)\!\right\rangle \boldsymbol{\cdot }\unicode[STIX]{x1D735}.\end{eqnarray}$$

This describes the fact that a microscopic velocity stream carries with it the ideal-gas value of the enthalpy, which must be subtracted from the kinetic-energy flux to give the gradient-driven heat flow.

Next, one has

(4.36) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{0}^{\infty }\!\text{d}\unicode[STIX]{x1D70F}\,\lim _{k\rightarrow 0}\text{G}_{\text{Q}}(\unicode[STIX]{x1D70F})|\,f_{T}(0)\rangle =[\text{Q}(\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}|\,f_{T}(0)\rangle . & \displaystyle\end{eqnarray}$$

For the parallel physics, this states that the characteristic autocorrelation time of the fluctuations is the collision time, and it introduces the collisional mean free path  $\unicode[STIX]{x1D706}_{\text{mfp}}$ as the characteristic characteristic length. In Braginskii’s discussion, the macroscopic temperature profile is expanded in the small ratio $\unicode[STIX]{x1D706}_{\text{mfp}}/L_{\Vert }$ , $L_{\Vert }$  being a macroscopic parallel scale length. For perpendicular motions, the characteristic time scale is the gyroperiod, the characteristic extent of the interactions is the gyroradius $\unicode[STIX]{x1D70C}\doteq v_{\text{t}}/|\unicode[STIX]{x1D714}_{\text{c}}|$ , and in the limit of $\unicode[STIX]{x1D708}/|\unicode[STIX]{x1D714}_{\text{c}}|\ll 1$ the net autocorrelation time is the gyroperiod reduced by the small ratio ³³   $\unicode[STIX]{x1D708}/|\unicode[STIX]{x1D714}_{\text{c}}|$ , namely $\unicode[STIX]{x1D70F}_{\text{ac}}=(\unicode[STIX]{x1D708}/|\unicode[STIX]{x1D714}_{\text{c}}|)|\unicode[STIX]{x1D714}_{\text{c}}|^{-1}$ .

The microscopic velocity stream mentioned above suffers the fluctuating friction force

(4.37) $$\begin{eqnarray}\displaystyle & \displaystyle |\,f_{\boldsymbol{p}}(0)\rangle =-\text{Q}\widehat{\text{C}}|A_{\boldsymbol{p}}\rangle . & \displaystyle\end{eqnarray}$$

For the electrons, one has

(4.38) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\text{C}}|A_{\boldsymbol{p}}\rangle =\widehat{\text{C}}_{ei}|T^{-1}\boldsymbol{v}\rangle . & \displaystyle\end{eqnarray}$$

The distinction here between the contravariant component $A^{\boldsymbol{p}}=m\boldsymbol{v}$ and covariant component $A_{\boldsymbol{p}}=A^{\boldsymbol{p}}/N_{\boldsymbol{p}}=\boldsymbol{v}/T$ (see (2.28) and (2.29)) is important: the contravariant component contains a mass, whereas the covariant one does not. For the latter, this means that the conventional orderings in the mass ratio may be used, so $\widehat{\text{C}}_{ei}\approx \widehat{\text{C}}^{\text{Lor}}$ and

(4.39) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\text{C}}_{ei}|T^{-1}\boldsymbol{v}\rangle \approx 2T^{-1}\unicode[STIX]{x1D708}(v)|\boldsymbol{v}\rangle . & \displaystyle\end{eqnarray}$$

Sans the temperature factor, this can be interpreted as the velocity-dependent friction on a microscopic velocity stream, which is one of the principal ingredients in Braginskii’s heuristic pictures (cf. Braginskii’s discussion of his figure 1). Note that upon applying the  $\text{Q}$ that is required in (4.37), one obtains a ket that is orthogonal to  $\langle \boldsymbol{v}|$ :

(4.40) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{v_{\text{t}e}^{2}\unicode[STIX]{x1D70F}_{e}}\left|\bigg[3\sqrt{\frac{\unicode[STIX]{x03C0}}{2}}\left(\frac{v_{\text{t}e}}{v}\right)^{3}-1\bigg]\boldsymbol{v}\right\rangle . & \displaystyle\end{eqnarray}$$

This is, in fact, exactly the ket that multiplies  $m\unicode[STIX]{x0394}\boldsymbol{u}$ in the second line of (3.35); it describes the non-hydrodynamic part of the flow-driven tail on the perturbed distribution function.

The net frictional effect on the microscopic heat flow is given by the cross-correlation between the fluctuating friction force (4.40) and the fluctuating heat flow (4.36). It is easy to see that that correlation gives rise to the same matrix element calculated by Braginskii for the off-diagonal contribution to the heat flow.

5 Discussion

The purpose of this paper has been to describe the application of projection-operator methods to classical plasma transport for the special case of linear response and the Braginskii (or Navier–Stokes) transport coefficients.

In general, there are two routes to the derivation of irreversible transport coefficients: (i) first derive an irreversible kinetic equation from the reversible Liouville equation, then project into the hydrodynamic subspace; (ii) project the Liouville equation, then perform a $\unicode[STIX]{x1D6E4}$ -space ensemble average in order to obtain the irreversible decay of correlation functions whose time integrals are the transport coefficients. In the present paper, route (i) was followed: projection was done on the (linearized) irreversible kinetic equation. (In Part 2, I shall instead follow route (ii).) It is useful to compare method (i) with the traditional Chapman–Enskog approach, which is reviewed in appendix A for the special case of the one-component plasma. Obviously, both that method and the projection-operator approach capture the same physics and make compatible predictions when their regimes of validity overlap. The traditional approach allows for a background zeroth-order flow, so it contains nonlinear advective derivatives. Those are absent in the linear-response formalism (developed via either projection operators or in any other way), where perturbations are made to an absolute Maxwellian distribution. However, both methods predict the same hydrodynamic fluxes to first order in the gradients.

At the level of linear response, the principal difference in the formalisms is the way in which the solvability constraints are satisfied. In the projection-operator method, the frequency operator $\text{P}{\mathcal{L}}\text{P}$ leads to the Euler part of the hydrodynamic equations, and use of the orthogonal projector  $\text{Q}$ in the correction terms replaces the traditional Chapman–Enskog substitution of the partial time derivatives in the Euler equations by spatial gradients (see (A 18a )). The projection-operator method provides an optimally concise representation of that algebra, which leads to the subtracted fluxes.

It must be emphasized, however, that the methods are equivalent only when the proper hydrodynamic projector is used. For classical transport, the natural projection operator is clear; it is built from the null eigenvectors of the linearized collision operator. However, one can project into any subspace whatsoever. Since the physics is invariant to the mathematical representation, the same results must ensue in the long-wavelength, low-frequency limit regardless of the choice of  $\text{P}$ . However, one must be extremely cautious because if the projection is chosen inaptly the Markovian approximation will not be satisfied. This issue is explored in §§G.1–G.3. However, a simple example given in § G.4 shows that provided that one projects at least into all of the null subspace of the collision operator, a higher-dimensional Markovian projection can also be used if one desires information additional to that contained in the natural transport equations.

For the purposes of introduction and simplicity, the calculations in this paper were restricted to linear perturbations and first-order dissipative processes, and they assumed the validity of the Landau collision operator. That is not always adequate; for example, it precludes consideration of convective amplification of locally unstable fluctuations (Kent & Taylor 1969), even when the system is globally stable. Non-local wave-induced transport (Rosenbluth & Liu 1976) is also omitted from the present calculations. A few further words on those topics are given in Part 2.

Furthermore, the various approximations made in this paper may obscure some deep conceptual issues and certain general properties of hydrodynamic transport theory. For example, the transport matrix  $\boldsymbol{\unicode[STIX]{x1D702}}$ was shown to be the infinite-time integral of the two-time correlation function of certain fluctuating forces – but as defined in this paper, those forces were not exact, being built from the Landau operator. In fact, a generalization of these techniques defines the true  $\boldsymbol{\unicode[STIX]{x1D702}}$ in terms of a certain exact two-time correlation function. This is done in Part 2, where it is also shown how to recover the present results in the limit of weak coupling.

A related point is the possible existence and construction of an exact hydrodynamic invariant manifold embedded in the kinetic many-body system, as discussed by Gorban & Karlin (2014). If that exists, it would be an attractor from which an appropriate projection leads to an certain set of exact hydrodynamic equations. For neutral fluids, this is apparently the case in the limit of vanishing Mach number and Knudsen number ³⁴  (Gorban & Karlin 2014, and references therein), and this implies that the plasma equations that were developed in the present paper are likely to be good descriptions of physical reality. Calculation of Burnett effects (see Part 2) is the next step in the asymptotic construction of such a manifold. However, an invariant hydrodynamic manifold may not exist at all due to entanglement between the hydrodynamic and non-hydrodynamic modes. The pursuit of such issues goes well beyond the scope of this paper, but it is an interesting area for future research.

In conclusion, the projection-operator approach to the derivation of linearized fluid equations is intuitive and technically efficient. It embeds the classical plasma derivations of transport equations into more general and modern formulations of statistical dynamics. Although projection is a linear operation, the methodology is useful even for nonlinear response, as Brey et al. (1981) have shown. That topic is addressed in Part 2, where it is shown how to obtain nonlinear fluid equations and the second-order (Burnett) corrections to the classical transport coefficients. It should also be clear that the formalism is not restricted to classical transport; one can contemplate applications to neoclassical theory and to gyrokinetics (Krommes 2012), for example. Projection-operator methods should be in the toolbox of every serious plasma theorist.