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  $O$ .Footnote 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 Reference Krommes2015) 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 ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{3}{2}n\frac{\text{d}T(\boldsymbol{x},t)}{\text{d}t}=-p\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{u}-\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\boldsymbol{q}-(\unicode[STIX]{x1D735}\boldsymbol{u})\boldsymbol{ : }\unicode[STIX]{x1D745}. & \displaystyle\end{eqnarray}$$

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

$\boldsymbol{w}\doteq \boldsymbol{v}-\boldsymbol{u}$

$p\doteq nT$

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

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

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

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

$\unicode[STIX]{x1D707}_{0}$

$\unicode[STIX]{x1D705}_{0}$

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

whereFootnote 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 kernelFootnote 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  $\boldsymbol{k}$ .Footnote 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 areFootnote 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.Footnote 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 matrixFootnote 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,Footnote 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 Reference Krommes2002, Reference Krommes2015, 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 notationFootnote 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.Footnote 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) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)+\text{i}\unicode[STIX]{x1D734}_{\boldsymbol{k}}\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t)+\int _{0}^{t}\!\text{d}\overline{\unicode[STIX]{x1D70F}}\,\unicode[STIX]{x1D72E}_{\boldsymbol{k}}(\overline{\unicode[STIX]{x1D70F}})\boldsymbol{\cdot }\unicode[STIX]{x0394}\boldsymbol{a}_{\boldsymbol{k}}(t-\overline{\unicode[STIX]{x1D70F}})=0, & \displaystyle\end{eqnarray}$$

whereFootnote 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.

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

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

$\unicode[STIX]{x1D745}$

$\boldsymbol{R}$

$\boldsymbol{q}$

$Q$

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

$\boldsymbol{w}_{s}\doteq \boldsymbol{v}-\boldsymbol{u}_{s}$

$\boldsymbol{R}\equiv \boldsymbol{R}_{e}$

$\sum _{i}\boldsymbol{R}_{i}=-\boldsymbol{R}$

$\boldsymbol{R}$

$Q$

(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 (Reference Mikhaǐlovskiǐ1967), Mikhaǐlovskiǐ & Tsypin (Reference Mikhaǐlovskiǐ and Tsypin1971, Reference Mikhaǐlovskiǐ and Tsypin1984) and Catto & Simakov (Reference Catto and Simakov2004), 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 formFootnote 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 negativeFootnote 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.Footnote 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 ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{3}{2}\overline{n}_{s}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x0394}T_{s}}{\unicode[STIX]{x2202}t} & = & \displaystyle -p_{s}\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\unicode[STIX]{x0394}\boldsymbol{u}_{s}-3\overline{n}_{s}\mathop{\sum }_{s^{\prime }}\left(\frac{m_{s}}{M_{ss^{\prime }}}\right)\unicode[STIX]{x1D708}_{ss^{\prime }}(\unicode[STIX]{x0394}T_{s}-\unicode[STIX]{x0394}T_{s^{\prime }})\nonumber\\ \displaystyle & & \displaystyle -\,\langle \overline{n}K^{\prime }1_{s}|\text{P}\text{i}\text{L}\text{Q}|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle .\end{eqnarray}$$

$|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$

$\text{L}\doteq \text{L}_{\boldsymbol{v}}+\text{L}_{\text{C}}$

$\text{P}\text{i}\widehat{\text{M}}\text{Q}=\text{Q}\text{i}\widehat{\text{M}}\text{P}=0$

$\text{L}_{\boldsymbol{E}}$

$\unicode[STIX]{x0394}\boldsymbol{u}_{s}-\unicode[STIX]{x0394}\boldsymbol{u}_{s^{\prime }}$

$\unicode[STIX]{x1D708}_{ss^{\prime }}$

$Z=1$

$\unicode[STIX]{x1D6FC}=0.51$

$v^{-3}$

$\unicode[STIX]{x1D708}_{ss^{\prime },\Vert }$

$n$

$\boldsymbol{u}$

$T$

$|\text{Q}\unicode[STIX]{x0394}\unicode[STIX]{x1D712}\rangle$

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  $\widehat{\text{C}}$ .Footnote 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 Reference Onsager1931a ,Reference Onsager b ; Casimir Reference Casimir1945; Krommes & Hu Reference Krommes and Hu1993) 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 (Reference Krommes and Hu1993, § 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.Footnote 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 ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D702}^{T_{e}\boldsymbol{p}_{e}} & = & \displaystyle -\unicode[STIX]{x1D735}\,\boldsymbol{\cdot }\,\langle K_{e}^{\prime }\boldsymbol{v}|\text{Q}(\widehat{\text{D}}-1)_{ei}\text{Q}\widehat{\text{C}}_{ie}|\boldsymbol{P}_{e}^{\prime }\rangle ,\end{eqnarray}$$

$\widehat{\text{D}}\doteq [\text{Q}(-\text{i}\widehat{\text{M}}+\widehat{\text{C}})\text{Q}]^{-1}$

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

$\unicode[STIX]{x1D6FE}(\boldsymbol{v})\doteq (1/2)mv^{2}/T-5/2$

$\text{Q}K^{\prime }\boldsymbol{v}$

$\widehat{\text{C}}^{\prime }\doteq \text{i}\widehat{\text{M}}+\widehat{\text{C}}$

$\langle \overline{n}_{i}\boldsymbol{P}_{i}^{\prime }|\widehat{\text{C}}_{ie}$

$\boldsymbol{B}\rightarrow -\boldsymbol{B}$

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).

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

$A(t;\unicode[STIX]{x1D6E4})\doteq \text{e}^{\text{i}{\mathcal{L}}t}A(\unicode[STIX]{x1D6E4})$

$\text{G}(t)\doteq \text{e}^{-\text{i}{\mathcal{L}}t}$

$\text{G}(-t)$

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,Footnote 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 equationFootnote 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 (Reference Mori1965) 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 (Reference Zwanzig2001, 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 (Reference Landau and Lifshitz1957, Reference Landau and Lifshitz1987) 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 (Reference Kadanoff and Martin1963) 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 (Reference Bixon and Zwanzig1969), whose work was slightly generalized by Hinton (Reference Hinton1970). A review article that provides useful background is by Fox (Reference Fox1978).

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 findsFootnote 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 (Reference Kadanoff and Martin1963, 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 ratioFootnote 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.