Quasilinear theory for inhomogeneous plasma
Abstract
This paper presents quasilinear theory (QLT) for classical plasma interacting with inhomogeneous turbulence. The particle Hamiltonian is kept general; for example, relativistic, electromagnetic, and gravitational effects are subsumed. A Fokker–Planck equation for the dressed ‘oscillation-center’ distribution is derived from the Klimontovich equation and captures quasilinear diffusion, interaction with the background fields, and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (i.e. not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum, and energy, and it also satisfies the -theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, the theory conserves the momentum and energy of the wave–plasma system. The action of nonresonant waves is also conserved, unlike in the standard version of QLT. Dewar’s oscillation-center QLT of electrostatic turbulence (1973, Phys. Fluids 16, 1102) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed.
1 Introduction
1.1 Background
Electromagnetic waves are present in plasmas naturally, and they are also launched into plasmas using external antennas, for example, for plasma heating and current drive (Stix, 1992; Fisch, 1987; Pinsker, 2001). Nonlinear effects produced by these waves are often modeled within the quasilinear (QL) approximation, meaning that the nonlinearities are retained in the low-frequency (‘average’) dynamics but neglected in the high-frequency dynamics. Two separate paradigms exist within this approach.
In the first paradigm, commonly known as ‘the’ QL theory (QLT), the focus is made on resonant interactions. Nonresonant particles are considered as a background that is homogeneous in spatial (Vedenov et al., 1961; Drummond & Pines, 1962; Kennel & Engelmann, 1966; Rogister & Oberman, 1968, 1969) or generalized coordinates (Kaufman, 1972; Eriksson & Helander, 1994; Catto et al., 2017); then the oscillating fields can be described in terms of global modes. This approach has the advantage of simplicity, but its applications are limited in that real plasmas are never actually homogeneous in any predefined variables (and, furthermore, tend to exhibit nonlinear instabilities in the presence of intense waves). The ‘ponderomotive’ dynamics determined by the gradients of the wave and plasma parameters is lost in this approach; then, spurious effects can emerge and have to be dealt with (Lee et al., 2018).
The second paradigm successfully captures the ponderomotive dynamics by introducing effective Hamiltonians for the particle average motion (Gaponov & Miller, 1958; Motz & Watson, 1967; Cary & Kaufman, 1981; Kaufman, 1987; Dodin, 2014). But as usual in perturbation theory (Lichtenberg & Lieberman, 1992), those Hamiltonians are by default singular for resonant interactions. Thus, such models have limited reach as well, and remarkable subtleties are still found even in basic QL problems. For example, it is still debated (Ochs & Fisch, 2021a; Ochs, 2021) to which extent the QL effects that remove resonant particles while capturing their energy (Fisch & Rax, 1992) also remove charge along with the resonant particles thereby driving plasma rotation (Fetterman & Fisch, 2008). This state of affairs means, arguably, that a clear comprehensive theory of QL wave–plasma interactions remains to be developed — a challenge that must be faced.
The first framework that subsumed both resonant and nonresonant interactions in inhomogeneous plasmas was proposed by Dewar (1973) for electrostatic turbulence in nonmagnetized plasma and is known as ‘oscillation-center’ (OC) QLT. It was later extended by McDonald et al. (1985) to nonrelativistic magnetized plasma. However, both of these models are partly heuristic and limited in several respects. For example, they are bounded by the limitations of the variational approach used therein, and they separate resonant particles from nonresonant particles somewhat arbitrarily (see also (Ye & Kaufman, 1992)). Both models also assume specific particle Hamiltonians and require that waves be governed by a QL wave-kinetic equation (WKE), i.e. be only weakly dissipative, or ‘on-shell’. (Somewhat similar formulations were also proposed, independently and without references to the OC formalism, in (Weibel, 1981; Yasseen, 1983; Yasseen & Vaclavik, 1986).) This means that collisions and microscopic fluctuations are automatically excluded. Attempts to merge QLT and the WKE with theory of plasma collisions were made (Rogister & Oberman, 1968; Schlickeiser & Yoon, 2014; Yoon et al., 2016) but have not yielded a local theory applicable to inhomogeneous plasma. In particular, the existing models rely on global-mode decompositions and treat complex frequencies heuristically. Thus, the challenge stands.
Related problems are also of interest in the context of gravitostatic interactions (Chavanis, 2012; Hamilton, 2020; Magorrian, 2021), where inhomogeneity of the background fields cannot be neglected in principle (Binney & Tremaine, 2008). (To our knowledge, OC QLT analogs have not been considered in this field.) Similar challenges also arise in QLT of dispersive gravitational waves (Garg & Dodin, 2021a, 2020). Hence, one cannot help but wonder whether a specific form of the particle Hamiltonian really matters for developing QLT or it is irrelevant and therefore should not be assumed. Since basic theory of linear waves is independent of Maxwell’s equations (Tracy et al., 2014; Dodin & Fisch, 2012; Dodin et al., 2017), a general QLT might be possible too, and it might be easier to develop than a zoo of problem-specific models.
1.2 Outline
Here, we propose a general QLT that allows for plasma inhomogeneity and is not restricted to any particular Hamiltonian or interaction field. By starting with the Klimontovich equation, we derive a model that captures QL diffusion, interaction with background fields, and ponderomotive effects simultaneously. The local diffusion coefficient in this model is manifestly positive-semidefinite. Waves are allowed to be off-shell, and a collision integral of the Balescu–Lenard type emerges for general Hamiltonian interactions. This operator conserves particles, momentum, and energy, and it also satisfies the -theorem, as usual. As a spin-off, a general expression for the spectrum of microscopic fluctuations of the interaction field is derived. For on-shell waves governed by the WKE, the theory conserves the momentum and energy of the wave–plasma system. The action of nonresonant waves is also conserved, unlike in the standard version of QLT.111The standard QLT (as in, for example, (Drummond & Pines, 1962)) does not properly conserve energy–momentum either, even though it is formally conservative (see section 7.3.2). Dewar’s OC QLT of electrostatic turbulence (Dewar, 1973) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and QLT for gravitational waves is proposed. Overall, our formulation interconnects many known results that in the past were derived independently and reproduces them within a unifying framework.
This progress is made by giving up the traditional Fourier–Laplace approach. The author takes the stance that the global-mode language is unnatural for inhomogeneous-plasma problems (i.e. all real-plasma problems). A fundamental theory must be local. Likewise, the variational approach that is used sometimes in QL calculations is not universally advantageous, especially for describing dissipation. Instead of those methods, we use operator analysis and the Weyl symbol calculus, as has also been proven fruitful in other recent studies of ponderomotive effects and turbulence (Ruiz, 2017; Ruiz & Dodin, 2017b; Zhu & Dodin, 2021) and linear-wave theory (Dodin et al., 2019). No logical leaps are made in this paper other than assuming the QL approximation per se and a certain ordering.222We treat the traditional QL approximation as a given mathematical model. We seek to push this model to its limits rather than to examine its validity, which is a separate issue. For discussions on the validity of the QL approximation, see (Besse et al., 2011; Escande et al., 2018; Crews & Shumlak, 2022). In a nutshell, we treat the commonly known QL-diffusion coefficient as a nonlocal operator, and we systematically approximate it using the Weyl symbol calculus. It is the nonlocality of this operator that gives rise to ponderomotive effects and ensures the proper conservation laws. The existing concept of ‘adiabatic diffusion’ (Galeev & Sagdeev, 1985; Stix, 1992) captures some of that, but systematic application of operator analysis yields a more general, more accurate, and more rigorous theory.
The author hopes not that this paper is an entertaining read. However, the paper was intended as self-contained, maximally structured, and easily searchable, so readers interested in specific questions could find and understand answers without having to read the whole paper. The text is organized as follows. In section 2, we present a primer on the Weyl symbol calculus and the associated notation. In section 3, we formulate our general model. In section 4, we introduce the necessary auxiliary theorems. In section 5, we derive a QL model for plasma interacting with prescribed waves. The waves may or may not be on-shell or self-consistent. (Their origin and dynamics are not addressed in section 5.) In section 6, we consider interactions with self-consistent waves. In particular, we separate out microscopic fluctuations, calculate their average distribution, and derive the corresponding collision operator. In section 7, we assume that the remaining macroscopic waves are on-shell, rederive the WKE, and show that our QL model combined with the WKE is conservative. In section 8, we discuss the general properties that our model predicts for plasmas in thermal equilibrium. In section 9, we show how to apply our theory to nonrelativistic electrostatic interactions, relativistic electromagnetic interactions, Newtonian gravity, and relativistic gravity. In section 10, we summarize our results. Auxiliary calculations are presented in appendices A–D, and appendix G summarizes our notation. This notation is extensive and may not be particularly intuitive. Thus, readers are encouraged to occasionally scout section 9 for examples even before fully absorbing the preceding sections.
An impatient reader can also skip calculations entirely and consult only the summaries of the individual sections (2.3, 3.4, 4.4, 5.6, 6.9, 7.6, 8.5; they are mostly self-contained) and then proceed to the examples in section 9. However, the main point of this work is not just the final results per se (surely, some readers will find them obvious) but also that they are derived with minimal assumptions and rigorously, which makes them reliable. A reader may also notice that we rederive some known results along the way, for example, basic linear-wave theory and the WKE. This is done for completeness and, more importantly, with the goal to present all pieces of the story within a unified notation.
2 A math primer
Here, we summarize the machinery to be used in the next sections. This machinery is not new, but a brief overview is in order at least to introduce the necessary notation. A more comprehensive summary, with proofs, can be found in (Dodin et al., 2019, supplementary material). For extended discussions, see (Tracy et al., 2014; Ruiz, 2017; McDonald, 1988; Littlejohn, 1986).
2.1 Weyl symbol calculus on spacetime
2.1.1 Basic notation
We denote the time variable as , space coordinates as , spacetime coordinates as , where and . The symbol denotes definitions, and Latin indices from the middle of the alphabet () range from 1 to unless specified otherwise. We assume the spacetime-coordinate domain to be .333This excludes periodic boundary conditions, albeit not entirely (section 3.1). Other than that, the spacetime metric can still be non-Euclidean, as illustrated by an application to relativistic gravity in section 9.4. See also the footnotes on pages 5 and 25. Functions on form a Hilbert space with an inner product that we define as
| (1) |
The symbol ∗ denotes complex conjugate,
| (2) |
(a similar convention is assumed also for other multi-dimensional variables used below), and integrals in this paper are taken over unless specified otherwise. Operators on will be denoted with carets, and we will use indexes and to denote their Hermitian and anti-Hermitian parts. For a given operator , one has ,
| (3) |
where † denotes the Hermitian adjoint with respect to the inner product (1). The case of a more general inner product is detailed in (Dodin et al., 2019, supplementary material).
2.1.2 Vector fields
For multi-component fields (our ⊺ denotes the matrix transpose), or ‘row vectors’ (actually, tuples), we define the dual ‘column vectors’ via . The matrix is assumed to be real, diagonal, invertible, and constant; other than that, it can be chosen as suits a specific problem. (For example, a unit matrix may suffice.) This induces the standard rule of index manipulation
| (4) |
where are elements of and are elements of . Summation over repeating indices is assumed. The rules of matrix multiplication apply to row and column vectors as usual. Then, for and , the quantity is a matrix with elements , is a matrix with elements and is its scalar trace:444A common notation is , but we reserve the dot-product notation for a scalar product of different quantities (section 2.1.3).
| (5) |
(Similarly, for and , is a matrix with elements .) We use the (5) to define a Hilbert space of -dimensional vector fields on , specifically, by adopting the inner product
| (6) |
Below, the distinction between and will be assumed but not emphasized. Also note that (5) yields
| (7) |
for any and . Thus, dyadic matrices of the form are positive-semidefinite, even though may be negative (when is not positive-definite).
For general matrices, the indices can be raised and lowered using and as usual. The Hermitian adjoint for a given matrix is defined such that for any and , which means
| (8) |
The Hermitian and anti-Hermitian parts are defined as
| (9) |
so . For one-dimensional matrices (scalars), one has ,
| (10) |
where and denote the real part and the imaginary part, respectively. We also define matrix operators as matrices of the corresponding operators . Because is constant, index manipulation applies as usual. Also as usual, one has
| (11) |
and , where † is the Hermitian adjoint with respect to the inner product (6).
2.1.3 Bra–ket notation
Let us define the following operators that are Hermitian under the inner product (1):
| (12) |
where and . Accordingly,
| (13) |
are understood as the spacetime-position operator and the corresponding wavevector operator, which will also be expressed as follows:
| (14) |
Also note the commutation property, where is the Kronecker symbol:555Spaces with periodic boundary conditions require a different approach (Rigas et al., 2011), so they are not considered here (yet see section 3.1). That said, for a system that is large enough, the boundary conditions are unimportant; then the toolbox presented here is applicable as is.
| (15) |
The eigenvectors of the operators (14) will be denoted as ‘kets’ and :666More precisely, is the ket that is an eigenvector of each with the corresponding eigenvalues being . A similar comment applies to .
| (16) |
and we assume the usual normalization:
| (17) |
where is the Dirac delta function. Both sets and , where , form a complete basis on , and the eigenvalues of these operators form an extended real phase space , where
| (18) |
The notation will be assumed for any , and . In particular, for any and constant , one has
| (19) |
as seen from comparing the Taylor expansions of the latter two expressions. (A generalization of this formula is discussed in section 4.1.) Also,
| (20) |
and
| (21) |
Here, ‘bra’ is the one-form dual to , is the one-form dual to , and is the unit operator. Any field on can be viewed as the representation (‘coordinate representation’) of , i.e. the projection of an abstract ket vector on :
| (22) |
Similarly, is the representation (‘spectral representation’) of , or the Fourier image of :
| (23) |
2.1.4 Wigner–Weyl transform
For a given operator and a given field , can be expressed in the integral form
| (24) |
where is a function of . This is called the representation (‘coordinate representation’) of . Equivalently, can be given a phase-space, or Weyl, representation, i.e. expressed through a function of the phase-space coordinates, :777Analytic continuation to complex arguments is possible, but by default, and are real.
| (25) |
The function , called the Weyl symbol (or just ‘symbol’) of , is given by
| (26) |
The and phase-space representations are connected by the Fourier transform:
| (27) |
This also leads to the following notable properties of Weyl symbols:
| (28) |
An operator unambiguously determines its symbol, and vice versa. We denote this isomorphism as . The mapping is called the Wigner transform, and is called the Weyl transform. For uniformity, we call them the direct and inverse Wigner–Weyl transform. The isomorphism is natural in that it has the following properties:
| (29) |
where is any function and is any operator. The product of two operators maps to the so-called Moyal product, or star product, of their symbols (Moyal, 1949):
| (30) |
which is associative:
| (31) |
Here, , and the arrows indicate the directions in which the derivatives act. For example, is just the canonical Poisson bracket on :
| (32) |
These formulas readily yield
| (33) |
also , etc. Another notable formula to be used below, which flows from (28) and (31), is
| (34) |
The Moyal product is particularly handy when . Such is often called the geometrical-optics parameter. Since , one can express the Moyal product as an asymptotic series in powers of :
| (35) |
2.1.5 Weyl expansion of operators
Operators can be approximated by approximating their symbols (Dodin et al., 2019; McDonald, 1988). If is approximately local in (i.e. if is determined by values only with small enough ), its symbol can be adequately represented by the first few terms of the Taylor expansion in :
| (36) |
Application of to this formula leads to
| (37) |
where . One can also rewrite (37) using the commutation property
| (38) |
In the representation, this leads to
| (39) |
The effect of a nonlocal operator on eikonal (monochromatic or quasimonochromatic) fields can be approximated similarly. Suppose , where the dependence of and on is slower than that of by factor . Then, , where , and the symbol of can be approximated as follows:
| (40) |
By expanding this in and applying , one obtains
| (41) |
where . Neglecting the corrections in this formula leads to what is commonly known as the geometrical-optics approximation (Dodin et al., 2019).
2.1.6 Wigner functions
Any ket generates a dyadic . In quantum mechanics, such dyadics are known as density operators (of pure states). For our purposes, though, it is more convenient to define the density operator in a slightly different form, namely, as
| (42) |
The symbol of this operator, , is a real function called the Wigner function. It is given by
| (43) |
which is manifestly real and can be understood as the (inverse) Fourier image of
| (44) |
Any function bilinear in and can be expressed through . Specifically, for any operators and , one has
| (45) |
where and are the corresponding symbols and (28) was used along with (31). As a corollary, and as also seen from (28), one has
| (46) |
As a reminder, and is the Fourier image of (23), so and can be loosely understood as the densities of quanta (associated with the field ) in the -space and the -space, respectively. Because of (46), is commonly attributed as a quasiprobability distribution of wave quanta in phase space. (The prefix ‘quasi’ is added because can be negative.) In case of real fields, which satisfy , one also has
| (47) |
Of particular importance are Wigner functions averaged over a sufficiently large phase-space volume . The average Wigner function is a local property of the field (as opposed to, say, the field’s global Fourier spectrum) and satisfies (appendix A)
| (48) |
2.1.7 Generalization to vector fields
In case of vector (tuple) fields , kets are column vectors, , and bras are row vectors, . The operators acting on such kets and bras are matrices of operators. The Weyl symbol of a matrix operator is defined as the matrix of the corresponding symbols. As a result, the symbol of a Hermitian adjoint of a given operator is the Hermitian adjoint of the symbol of that operator:
| (49) |
and as a corollary, the symbol of a Hermitian matrix operator is a Hermitian matrix.
In particular, the density operator of a given vector field is a matrix operator
| (50) |
The symbol of this operator, , is a Hermitian matrix function888By construction, is a matrix with mixed indices, . In sections 5.1 and 5.2, we also operate with a Wigner matrix that has two upper indices. Because the field of interest is real there, the dagger † in (51) is assumed omitted in that case.
| (51) |
called the Wigner matrix. (It is also called the ‘Wigner tensor’ when is a true vector rather than a tuple.) It can be understood as the (inverse) Fourier image of
| (52) |
The analog of (45) is (appendix B.1)
| (53a) | |||
| The Wigner matrix averaged over a sufficiently large phase-space volume is a local property of the field, and it is positive-semidefinite (appendix A). | |||
2.2 Weyl symbol calculus on phase space
2.2.1 Notation
Consider a Hamiltonian system with coordinates and canonical momenta . Together, these variables comprise the phase-space coordinates , i.e.
| (54) |
Components of will be denoted with Greek indices ranging from 1 to .999However, the index is reserved as a tag for individual particles and waves.
Hamilton’s equations for can be written as , or equivalently, as
| (55) |
Here, is a Hamiltonian, ,
| (56) |
is the Poisson bracket on , is the canonical Poisson structure:
| (57) |
is an -dimensional zero matrix, and is an -dimensional unit matrix. The corresponding equation for the probability distribution is
| (58) |
Solutions of (58) and other functions of the extended-phase-space coordinates can be considered as vectors in the Hilbert space with the usual inner product101010Note that the inner product (59) is different from (1). Still, we use the same notation assuming it will be clear from the context which inner product is used in each given case.
| (59) |
Assuming the notation , one has
| (60) |
Let us introduce the position operator on ,
| (61) |
and the momentum operator on ,
| (62) |
where but ; that is, , , and
| (63) |
Then, much like in section 2.1, one can also introduce the position and momentum operators on the extended phase space :
| (64) |
Assuming the convention that Latin indices from the beginning of the alphabet range from 0 to , and , one can compactly express this as
| (65) |
The eigenvectors of these operators will be denoted and :
| (66) |
and we assume the usual normalization:
| (67) |
Both sets and form a complete basis on , and the eigenvalues of these operators form a real extended phase space , where
| (68) |
Particularly note the following formula, which will be used below:
| (69) |
2.2.2 Wigner–Weyl transform
One can construct the Weyl symbol calculus on the extended phase space just like it is done on spacetime in section 2.1, with an obvious modification of the notation. The Wigner–Weyl transform is defined as
| (70) | |||
| (71) |
(Notice the change in the font and in the index compared to (26) and (25).) The corresponding Moyal product is denoted (as opposed to introduced earlier):
| (72) |
where can be expressed as follows:
| (73) |
If an operator is local in , its representation and representation satisfy
| (74) |
and therefore the Weyl symbol of is the same irrespective of whether the operator is considered on or on . In this case, we will use a unifying notation instead of and .
2.2.3 Wigner functions and Wigner matrices
The density operator of a given scalar field is given by
| (75) |
The symbol of this operator, , is a real function called the Wigner function. It is given by
| (76) |
which can be understood as the (inverse) Fourier image of
| (77) |
In particular, one has
| (78) |
where the right-hand side is given by (43), with treated as a parameter. Also, for real fields,
| (79) |
The density operator of a given vector field is a matrix operator
| (80) |
The symbol of this operator, or the Wigner matrix, is a Hermitian matrix function
| (81) |
which can be understood as the (inverse) Fourier image of
| (82) |
In particular, one has
| (83) |
where the right-hand side is given by (51), with treated as a parameter. Also, for real fields,
| (84) |
Like those on , the Wigner matrices (Wigner functions) on become positive-semidefinite (non-negative), and characterize local properties of the corresponding fields, when averaged over a sufficiently large phase-space volume .
2.3 Summary of section 2
In summary, we have introduced a generic -dimensional physical space , the dual -dimensional wavevector space , the corresponding -dimensional () spacetime , and the dual -dimensional wavevector space . We have also introduced an -dimensional momentum space , the corresponding -dimensional phase space , the -dimensional () extended space , and the dual -dimensional wavevector space , where is the -dimensional wavevector space dual to . We have also introduced the -dimensional phase space . Each of the said variables has a corresponding operator associated with it, which is denoted with a caret. For example, is the operator of position in the space, and is the corresponding wavevector operator.
Functions on form a Hilbert space , and the corresponding bra-ket notation is introduced as usual. Any operator on can be represented by its Weyl symbol . The correspondence between operators and their symbols, , is determined by the Wigner–Weyl transform and is natural in the sense that (29) is satisfied. In particular, , where is the Moyal product on . When the geometrical-optics parameter is negligible (), one has and the Moyal product becomes the usual product. Similarly, functions on form a Hilbert space , the corresponding bra-ket notation is also introduced as usual, any operator on can be represented by its Weyl symbol , and . An operator that is local in has the same symbol irrespective of whether it is considered on or on .
Any given field generates the corresponding density operator and its symbol called the Wigner function (Wigner matrix, if the field is a vector). If the density operator is considered on , it is denoted and the corresponding Wigner function is denoted . If the density operator is considered on , it is denoted and the corresponding Wigner function is denoted . The two Wigner functions are related via , where enters as a parameter, if at all. If averaged over a sufficiently large phase-space volume, the Wigner functions (matrices) are non-negative (positive-semidefinite) and characterize local properties of the corresponding fields.
3 Model
Here, we introduce the general assumptions and the key ingredients of our theory.
3.1 Basic assumptions
3.1.1 Ordering
Let us consider particles governed by a Hamiltonian such that
| (85) |
In other words, serves as a small perturbation to the leading-order Hamiltonian . The system will be described in canonical variables . Let us also assume that the system is close to being homogeneous in . This includes two conditions. First, we require that the external fields are weak (yet see section 3.1.2), meaning
| (86) |
where is a small parameter, and are the characteristic inverse scales in the and spaces, respectively, and the bar denotes local averaging.111111An exception will be made for eikonal waves, specifically, for quantities evaluated on the local wavevector . Hence, the particle momenta are close to being local invariants. Second, the statistical properties of are also assumed to vary in slowly. These properties can be characterized using the density operator of the perturbation Hamiltonian,
| (87) |
and its symbol, the (real) Wigner function, as in (43):
| (88) |
Specifically, we will use the average Wigner function, , which represents the Fourier spectrum of the symmetrized autocorrelation function of :
| (89) |
The averaging is performed over sufficiently large volume of to eliminate rapid oscillations and also over phase-space volumes , which guarantees to be non-negative and local (section 2.2.3). The function can be understood as a measure of the phase-space density of wave quanta when the latter is well defined (section 7).
We will assume121212As a reminder, the notation does not rule out the possibility that is small. Also note that the terms ‘’ and ‘of order’ in this paper mean the same as ‘’.
| (90) |
That said, we will also allow (albeit not require) for oscillations to be constrained by a dispersion relation. In this case, , so (90) per se is not satisfied; then we assume a similar ordering for instead. Also note that in application to the standard QLT of homogeneous turbulence (Stix, 1992, chapter 16), is understood as the geometrical-optics parameter characterizing the smallness of the linear-instability growth rates. (We discuss the ordering further in the end of section 3.3.)
3.1.2 Quasilinear approximation
The particle-motion equations can be written as
| (91) |
where and are understood as the unperturbed phase-space velocity and the perturbation to the phase-space velocity, respectively:
| (92) |
The notation (with ) will also be used for the spatial part of the phase-space velocity , i.e. for the true velocity per se. Likewise, will be used to denote either the phase-space velocity vector or the spatial velocity vector depending on the context. Also note that a slightly different definition of will be used starting from section 5.6.
The corresponding Klimontovich equation for the particle distribution is
| (93) |
(If collisions are not of interest, (93) can as well be understood as the Vlasov equation. Also, a small collision term can be included ad hoc; see the comment in the end of section 3.3.) Let us search for in the form
| (94) |
The equations for and are obtained as the average and oscillating parts of (93), and we neglect the nonlinearity in the equation for , following the standard QL approximation (Stix, 1992, chapter 16). Then, one obtains
| (95) | |||
| (96) |
A comment is due here regarding plasmas in strong fields and magnetized plasmas in particular. Our formulation can be applied to such plasmas in canonical angle–action variables . For fast angle variables, the ordering (86) is not satisfied and the Weyl symbol calculus is inapplicable as is (see the footnote on p. 5). Such systems can be accommodated by representing the distribution function as a Fourier series in and treating the individual-harmonic amplitudes separately as slow functions of the remaining coordinates. Then, our averaging procedure subsumes averaging over , so the averaged quantities are -independent and (86) is reinstated. In particular, magnetized plasmas can be described using guiding-center variables. Although not canonical by default (Littlejohn, 1983), they can always be cast in a canonical form, at least in principle (Littlejohn, 1979). Examples of canonical guiding-center variables are reviewed in (Cary & Brizard, 2009). To make the connection with the homogeneous-plasma theory, one can also order the canonical pairs of guiding-center variables such that they would describe the gyromotion, the parallel motion, and the drifts separately (Wong, 2000). This readily leads to results similar to those in (Catto et al., 2017). Further discussions on this topic are left to future papers.
3.2 Equation for
Let us consider solutions of (96) as a subclass of solutions of the more general equation
| (97) |
Here, we have introduced an auxiliary second ‘time’ , the operator
| (98) |
(here and further, denotes a placeholder), and is the unperturbed velocity in the space. Note that
| (99) |
due to the incompressibility of the phase flow. Hence, , so is anti-Hermitian.
Let us search for a solution of (97) in the form131313Using the auxiliary variable allows us to express the propagator as a regular exponential, rather than ordered exponential, even for -dependent , because is independent of .
| (100) |
Then, , so and therefore
| (101) |
where . Hence, one obtains
| (102) |
or equivalently, using ,
| (103) |
Here, is a solution of , specifically,
| (104) |
and we have also introduced
| (105) |
Because is anti-Hermitian, the operator is unitary, and comparison with (19) shows that it can be recognized as a shift operator. For further details, see section 4.1.
Using , one can express (103) as
| (106) |
where is the Green’s operator understood as the right inverse of the operator , or on the space of -independent functions, . Let us rewrite this operator as , where
| (107) |
and is a positive constant. Note that is well defined at , meaning that is well defined for any physical (bounded) field .141414Unlike classic plasma-wave theory, this approach does not involve spectral decomposition, so there is no need to consider fields that are exponential in time on the whole interval . Thus, so is . Let us take and then take . (Here, denotes that must remain positive, i.e. the upper limit is taken.) Then, (106) can be expressed as
| (108) |
Here, we introduced an ‘effective’ Green’s operator , i.e.
| (109) |
This operator will be discussed in section 4.2, and will be discussed in section 4.3. Meanwhile, note that because is just an auxiliary variable, we will be interested in solutions independent of . In particular, this means that , so , so (104) leads to
| (110) |
3.3 Equation for
Using (106), one can rewrite (96) for as follows:
| (111) |
Notice that
| (112) |
and also
| (113) |
The field enters here as a multiplication factor and can be considered as an operator:
| (114) |
Then, (113) can be compactly represented as
| (115) |
We will also use the notation
| (116) |
This leads to the following equation for :
| (117) |
where we introduced the following average quantities:
| (118) |
Our goal is to derive explicit approximate expressions for the quantities (118) and to rewrite (117) in a more tractable form using the assumptions introduced in section 3.1. We will use151515Starting with section 5.6, we will assume instead.
| (119) |
and we will keep terms of order , , and in the equation for , while terms of order , , and higher will be neglected. This implies the ordering
| (120) |
As a reminder, is a linear measure of the characteristic amplitude of oscillations, and is the geometrical-optics parameter, which is proportional to the inverse scale of the plasma inhomogeneity in spacetime. As usual then, linear dissipation is assumed to be of order . This model implies the assumption that collisionless dissipation is much stronger than collisional dissipation, which is to emerge as an effect quadratic in (section 6). Furthermore, the inverse plasma parameter161616By the plasma parameter we mean the number of particles within the Debye sphere. will be assumed to be of order , so the collision operator for (section 6.8) will be of order . Within the assumed accuracy, this operator must be retained, while the dynamics of is considered linear and therefore collisionless. Alternatively, one can switch from the Klimontovich description to the Vlasov–Boltzmann description and introduce an ad hoc order- collision operator directly in (93). This will alter the Green’s operator, but the conceptual formulation would remain the same, so it will not be considered separately in detail.
3.4 Summary of section 3
Our QL model is defined as usual, except: (i) we allow for a general particle Hamiltonian ; (ii) we use the Klimontovich equation rather than the Vlasov equation to retain collisions; (iii) we use local averaging (denoted with overbar) and allow for weak inhomogeneity of all averaged quantities; (iv) we retain the initial conditions for the oscillating part of the distribution function (defined as in (108) but yet to be calculated explicitly). Then, the average part of the distribution function satisfies
| (121) |
where , , is the wave-driven perturbation of the phase-space velocity (see (92)), is the same quantity considered as an operator on (see (114)), and is the ‘effective’ Green’s operator given by
| (122) |
Also, , and is the Poisson bracket on the particle phase space . The equation for used in the standard QLT is recovered from (121) by neglecting and the spatial gradients (in particular, the whole Poisson bracket) and also by approximating the operator with a local function of .
4 Preliminaries
Before we start calculating the functions in (121) explicitly, let us get some preliminaries out of the way. In this section, we discuss the shift operators (section 4.1), approximate the operator (section 4.2), and develop a model for the function that encodes the initial conditions for (section 4.3).
4.1 Shift operator
Here, we derive some properties of the shift operator introduced in section 3.2.
4.1.1 as a shift
Here, we formally prove (an admittedly obvious fact) that
| (123) |
where the ‘characteristics’ solve171717In terms of , (124) has a more recognizable form , with .
| (124) |
and thus can be Taylor-expanded in as
| (125) |
As the first step to proving (123), let us Taylor-expand around a fixed point :
| (126) |
where . If one neglects the first and higher derivatives of , one obtains
| (127) |
By taking the limit , which corresponds to , one obtains
| (128) |
Similarly, if one neglects the second and higher derivatives of , one obtains181818We use the Zassenhaus formula
| (129) |
In the limit , when and , one obtains
| (130) |
In conjunction with (125), equations (128) and (130) show agreement with the sought result (125) within the assumed accuracy. One can also retain derivatives of and derive the corresponding approximations similarly. Then the error will be .
4.1.2 Symbol of
Using the bra-ket notation, (123) can be written as
| (131) |
Thus, , so
| (132) |
Using (70), one obtains the Weyl symbol of in the form
| (133) |
From (125), one has
| (134) |
where we introduced a matrix , or explicitly,
| (135) |
Let us express the term in (134) as . Then,
| (136) |
Because , the well-known formula yields . But by (99), so
| (137) |
The last term is insignificant and can be neglected right away, so (133) leads to
| (138) |
where we have introduced the following notation:
| (139) |
By definition, is a polynomial of with coefficients that are of order and therefore small. But because can be large, and because is under the exponent, this makes potentially sensitive to this term, so we retain it (for now).
4.2 Effective Green’s operator
The effective Green’s operator (109) can be understood as the right inverse of the operator (cf. section 3.2)
| (140) |
so we denote it also as (which is admittedly abuse of notation). Because has real representation by definition, the representation of is real too. In particular, is real, hence
| (141) |
by definition of the Weyl symbol (70). As a corollary, the derivative of with respect to the th component of the whole second argument, denoted , satisfies
| (142) |
Also note that can be expressed as
| (143) |
(the notation ‘’ will also be shortened as ‘’), whence
| (144) |
Due to (138), the leading-order approximation of the symbol of the operator (109) is , where
| (145) |
and the (standard) notation is defined as follows:
| (146) |
This means, in particular, that for any , one has
| (147) |
where is a principal-value integral. Also usefully, and
| (148) |
where the notation is defined, for any and , as follows:
| (149) |
Now let us reinstate the term in (138). It is readily seen (appendix B.2) that although may significantly affect per se, its effect on is small, namely,
| (150) |
Below, we apply this formulation to , in which case (150) becomes . Such corrections are negligible within our model, so from now on we adopt
| (151) |
4.3 Initial conditions
Consider the function from (108). Using (110), the latter can be written as follows:
| (152) |
Because is smooth and is rapidly oscillating, the second term in the external parenthesis is an oscillatory function of with the average negligible at . But the whole expression in these parenthesis is independent of at large (section 3.2). Thus, it can be replaced with its own average over , denoted . Because there is no -dependence left in this case, one can also omit . That gives
| (153) |
Using
| (154) |
where the sum is taken over individual particles, one can write191919Taylor-expanding delta functions is admittedly a questionable procedure, but here it is understood as a shorthand for Taylor-expanding integrals of .
| (155) |
where are the -driven small deviations from the particle unperturbed trajectories . Then, , and the linearized perturbation is given by
| (156) |
By definition, the unperturbed trajectories satisfy , where as in (98); thus,
| (157) |
Also, , because are oscillatory functions of that is slowly evolved by . Hence, is the microscopic part of the unperturbed distribution function:
| (158) |
This indicates that the term defined in (118) is due to collisional effects. We postpone discussing these effects until section 6, so will be ignored for now.
4.4 Summary of section 4
The main result of this section is that the Weyl symbol of the effective Green’s operator can be approximated within the assumed accuracy as follows:
| (159) |
Here, is the unperturbed velocity in the space, so , where is the unperturbed velocity in the space, and
| (160) |
We also show that the term defined in (118) is due to collisional effects. We postpone discussing these effects until section 6, so will be ignored for now.
5 Interaction with prescribed fields
In this section, we explore the effect of the diffusion operator . The oscillations will be described by as a prescribed function, so they are allowed (yet not required) to be ‘off-shell’, i.e. do not have to be constrained by a dispersion relation. Examples of off-shell fluctuations include driven near-field oscillations, evanescent waves, and microscopic fluctuations (see also section 6). We will first derive the symbol of and, using this symbol, approximate the diffusion operator with a differential operator (section 5.1). Then, we will calculate the coefficients in the approximate expression for (sections 5.2 and 5.3). Finally, we will introduce the concept of the OC distribution (section 5.4) and summarize and simplify the resulting equations (section 5.6).
5.1 Expansion of the dispersion operator
The (effective) Green’s operator can be represented through its symbol using (71):
| (161) |
The corresponding representation of is even simpler, because the symbol of is independent of :202020One can also derive (162) formally from (71).
| (162) |
Let us also introduce the Wigner matrix of , denoted , and its inverse Fourier transform as in section 2.2.3. Using these together with (67), one obtains
| (163) |
Then, by taking of (163), one finds that the symbol of is a convolution of and (appendix B.3):
| (164) |
Let us Taylor-expand the symbol (164) in :
| (165) |
As a reminder, denotes the derivative of with respect to (the th component of) the whole second argument, , and
| (166) |
Upon application of , gets replaced (roughly) with and gets replaced (also roughly) with . By (144), the last term in (166) is of order too. This means that the contribution of the whole term to the equation for is of order . The standard QLT neglects this contribution entirely, i.e. adopts , in which case the diffusion operator becomes just a local function of phase-space variables, . In this work, we retain corrections to the first order in , i.e. keep the second term in (165) as well, while neglecting the higher-order terms as usual.
Within this model, one can rewrite (165) as follows:
| (167) |
Here, we used (141) and introduced
| (168) | |||
| (169) |
which satisfy (appendix B.4)
| (170) |
The first-order Weyl expansion of is obtained by applying to (167). Namely, for any , one has (cf. section 2.1.5)
| (171) |
What remains now is to calculate the functions and explicitly.
5.2 Wigner matrix of the velocity oscillations
To express through the Wigner function of the perturbation Hamiltonian (section 3.1.1), we need to express through . Recall that is the symbol of the density operator (section 2.2.3). By definition (92), one has , where (section 2.2.1). Then,
| (172) |
where is the density operator whose symbol is . By applying , one obtains
| (173) |
where is the Moyal product (72). Using formulas analogous to (33) in the space, one obtains
| (174) |
Hence, and are connected via the following exact formula:
| (175) |
5.3 Nonlinear potentials
Due to (170), one has . Using this together with (168), (175), (151), and (145), one obtains
| (176) |
with notation as in (10). This can be written as , where
| (177) |
and we also introduced
| (178) | |||
| (179) |
As shown in appendix B.5, the contributions of these two functions to (117) are
| (180) |
Thus, must be retained and must be neglected, which leads to
| (181) |
The function can be written as follows:
where we introduced
| (182) |
and is defined as in (149). Then finally, one can rewrite (171) as follows:
| (183) |
where we used (99). With some algebra (appendix B.6), and assuming the notation
| (184) |
one finds that (183) leads to
| (185) |
Hence, (117) becomes (to the extent that is negligible; see section 6.7)
| (186) |
The functions , , and that determine the coefficients in this equation are fundamental and, for the lack of a better term, will be called nonlinear potentials.
5.4 Oscillation-center distribution
Let us introduce
| (187) |
Then, using (185), one can rewrite (186) as212121The difference between and is related to the concept of so-called adiabatic diffusion (Galeev & Sagdeev, 1985; Stix, 1992), which captures some but not all adiabatic effects.
| (188) |
where corrections have been neglected and we introduced . As a reminder, the nonlinear potentials in (188) are as follows:
| (189) | ||||
| (190) | ||||
| (191) |
Equations (187)–(191) form a closed model that describes the evolution of the average distribution in turbulence with prescribed . In particular, (188) can be interpreted as a Liouville-type equation for as an effective, or ‘dressed’, distribution. The latter can be understood as the distribution of ‘dressed’ particles called OCs. Then, serves as the OC Hamiltonian, is the phase-space diffusion coefficient, is the ponderomotive energy, , and . Within the assumed accuracy, one can redefine to be the OC velocity rather than the particle velocity; specifically,222222The advantage of the amended definition (192) is that it will lead to exact conservation laws of our theory, as to be discussed in section 7.5.
| (192) |
Then, the presence of in (189) signifies that OCs diffuse in phase space in response to waves they are resonant with. Below, we use the terms ‘OCs’ and ‘particles’ interchangeably except where specified otherwise.
That said, the interpretation of OCs as particle-like objects is limited. Single-OC motion equations are not introduced in our approach. (They would have been singular for resonant interactions.) Accordingly, the transformation (187) of the distribution function is not derived from a coordinate transformation but rather is fundamental. As a result, particles and OCs live in the same phase space, but the ‘dynamics of OCs’ can be irreversible (section 5.5). This qualitatively distinguishes our approach from the traditional OC theory (Dewar, 1973) and from the conceptually similar gyrokinetic theory (Littlejohn, 1981; Cary & Brizard, 2009), where coordinate transformations are central.
5.5 -theorem
Because is non-negative (section 2.1.6), is positive-semidefinite; that is,
| (193) |
for any real . This leads to the following theorem. Consider the OC entropy defined as
| (194) |
According to (188), satisfies
| (195) |
The first integral vanishes due to . The second integral is non-negative due to (193). Thus,
| (196) |
which is recognized as the -theorem (Lifshitz & Pitaevskii, 1981, section 4) for QL OC dynamics.
5.6 Summary of section 5
From now on, we assume that the right-hand side of (188) scales not as but as , either due to the scarcity of resonant particles or, for QL diffusion driven by microscopic fluctuations (section 6), due to the plasma parameter’s being large. Also, the spatial derivatives can be neglected within the assumed accuracy in the definition of (187) and on the right-hand side of (188). Using this together with (69), and with (56) for the Poisson bracket, our results can be summarized as follows.
QL evolution of a particle distribution in a prescribed wave field is governed by232323Remember that here we neglect (118), which is a part of the collision operator to be reinstated in section 6.
| (197) |
The OC distribution is defined as
| (198) |
so the density of OCs is the same as the locally averages density of the true particles:
| (199) |
The function is understood as the OC Hamiltonian. It is given by
| (200) |
where is the average Hamiltonian, which may include interaction with background fields, and is the ponderomotive potential. The nonlinear potentials that enter (197) can be calculated to the zeroth order in and are given by242424See section 9 for examples and section 6.6 for the explanation on how is related to , which is yet to be introduced. Also note that in combination with (200), equation (203) generalizes the related results from (Kentwell, 1987; Fraiman & Kostyukov, 1995; Dodin & Fisch, 2014).
| (201) | ||||
| (202) | ||||
| (203) |
where is a dyadic matrix with two lower indices, and the same conventions apply as in section 2.1.2. Also, is hereby redefined as the OC spatial velocity, namely,
| (204) |
The function is defined as
| (205) |
where is the average Wigner function (88) of the perturbation Hamiltonian, i.e. the spectrum of its symmetrized autocorrelation function (89). Due to (78), it can be understood as the average of (where is defined in (87)), i.e. as the Wigner function of the perturbation Hamiltonian with treated as a parameter. As such, is non-negative, so is positive-semidefinite. This leads to an -theorem (proven similarly to (196)) for the entropy density :
| (206) |
6 Interaction with self-consistent fields
Here, we explain how to calculate the function in the presence of microscopic fluctuations (nonzero ). In particular, we reinstate the term that was omitted in section 5. We also show that a collision operator of the Balescu–Lenard type emerges from our theory within a general interaction model. This calculation can be considered as a generalization of that in (Rogister & Oberman, 1968) for homogeneous plasmas. Another related calculation was proposed in (Chavanis, 2012) in application to potential interactions in inhomogeneous systems using action–angle variables, with global averaging over the angles. (See also (Mynick, 1988) for a related calculation in action–angle variables based on the Fokker–Planck approach.) In contrast, our model holds for any Hamiltonian interactions via any vector fields and allows for weak inhomogeneities in canonical coordinates.
6.1 Interaction model
Let us assume that particles interact via an -component real field . It is treated below as a column vector; hence the index ⊺. (A complex field can be accommodated by considering its real and imaginary parts as separate components.) We split this field into the average part and the oscillating part . The former is considered given. For the latter, we assume the action integral of this field without plasma in the form
| (208) |
(see section 9 for examples), where is a Hermitian operator252525The field action often has the form , where is a spacetime metric, , and is Hermitian with respect to the inner product . Using and , one can cast this action in the form (208), with that is Hermitian with respect to the inner product (6). and is a row vector dual to . Plasma is allowed to consist of multiple species, henceforth denoted with index . Because is assumed small, the generic Hamiltonian for each species can be Taylor-expanded in and represented in a generic form
| (209) |
(see section 9 for examples), which can be considered as a second-order Taylor expansion of the full Hamiltonian in . Here, is independent of , is a column vector whose elements are linear operators on , and the dagger is added so that could be understood as a row vector whose elements act on the individual components of the field; i.e. . We let be nonlocal in and (for example, can be a spacetime derivative or a spacetime integral), and we also let depend on parametrically, so
| (210) |
The matrix operators and and their symbols and are understood similarly.
The Lagrangian density of the oscillating-field–plasma system is
| (211) |
where the sum is taken over individual particles. Note that
| (212) |
so the -dependent part of the system action can be written as with
| (213) | |||
| (214) |
(The contribution of to the second term in (213) has been omitted because it averages to zero at integration over spacetime and thus does not contribute to .) This ‘abridged’ action is not sufficient to describe the particle motion, but it is sufficient to describe the dynamics of at given , as discussed below. The operator can be considered Hermitian without loss of generality, because its anti-Hermitian part does not contribute to . Also, we assume that unless either of and is zero, the high-frequency field has no three-wave resonances, so terms cubic in can be neglected in ;262626This is tacitly assumed already in (209), where cubic terms are neglected. Also note that three-wave interactions that involve resonances between low-frequency oscillations of and two high-frequency waves, like Raman scattering (Balakin et al., 2016), are still allowed. then,
| (215) |
Using the same assumption, one can also adopt
| (216) |
because in the absence of three-wave resonances, the oscillating part of contributes only terms to the equation for .
6.2 Field equations
The Euler–Lagrange equation for derived from (213) is
| (217) |
Then, to the extent that the linear approximation for is sufficient (see below), one finds that the oscillating part of the field satisfies
| (218) |
where we used (108). Note that the right-hand side of (218) is determined by microscopic fluctuations (section 4.3). Equation (218) can also be expressed as
| (219) |
where is understood as the plasma dispersion operator and is given by
| (220) |
where is a placeholder. The general solution of (219) can be written as
| (221) |
Here, is the right inverse of (meaning yet ) such that vanishes at zero .272727Most generally, the problem of finding is the standard problem of calculating the field produced by a given radiation source. The rest of the solution, , is the macroscopic field that satisfies
| (222) |
In the special case when the dispersion operator is Hermitian (), (222) also flows from the ‘adiabatic’ macroscopic part of the action , namely,
| (223) |
Because we have assumed a linear model for in (218), is decoupled from , and hence the dynamics of turns out to be collisionless. This is justified, because collisional dissipation is assumed to be much slower that collisionless dissipation (section 3.3). One can reinstate collisions in (222) by modifying ad hoc, if necessary. Alternatively, one can avoid separating and and, instead, derive an equation for the average Wigner matrix of the whole (McDonald, 1991). However, this approach is beyond QLT, so it is not considered in this paper.
6.3 Dispersion matrix
As readily seen from the definition (220), the operator can be expressed as
| (224) |
The corrections caused by nonzero and in this formula will be insignificant for our purposes, so they will be neglected. In particular, this means that can be adopted in the form independent of (section 4.2):
| (225) |
Then, can be considered as an operator on with as a parameter, and . Also,
| (226) |
This readily yields the ‘dispersion matrix’ :
| (227) | |||
| (228) |
(see section 9 for examples). Here, is a dyadic matrix, and the arguments and are henceforth omitted for brevity. Also, we introduced the operators and their symbols as
| (229) |
The appearance of in the denominator in (227) is related to the Landau rule. (Remember that as arguments of Weyl symbols, and are real by definition.) The Hermitian and anti-Hermitian parts of the dispersion matrix are
| (230) | ||||
| (231) |
Assuming the notation , the inverse dispersion matrix can be expressed as
| (232) |
Because , this leads to the following formulas, which we will need later:
| (233) |
6.4 Spectrum of microscopic fluctuations
Other objects to be used below are the density operators of the oscillating fields:
| (234) |
and the corresponding average Wigner matrices on . The former, , is readily found by definition (51), and the latter, , is calculated as follows. Let us consider as a ket in , with as a parameter. Then, (221) readily yields
| (235) |
By applying to this, one obtains
| (236) |
where most arguments are omitted for brevity and (appendix B.7)
| (237) |
assuming corrections due to inter-particle correlations are negligible. Then, (236) gives
| (238) |
where . It is readily seen from (238) that is positive-semidefinite. One can also recognize (238) as a manifestation of the dressed-particle superposition principle (Rostoker, 1964). Specifically, (238) shows that the contributions of individual particles to are additive and affected by the plasma collective response, i.e. by the difference between and the vacuum dispersion matrix .
Using (238), one can also find other averages quadratic in the field via (cf. (53a))
| (239) |
where and are any linear operators and and are their symbols; for example,
| (240) |
Because of this, we loosely attribute as the spectrum of microscopic oscillations, but see also section 8.2, where an alternative notation is introduced and a fluctuation–dissipation theorem is derived from (238) for plasma in thermal equilibrium. See also section 9 for specific examples.
6.5 Nonlinear potentials
From (221), the oscillating part of the Hamiltonian (216) can be split into the macroscopic part and the microscopic part as , , and
| (241) |
Here, is an operator on given by
| (242) |
with the symbol
| (243) |
(see section 9 for examples). The corresponding average Wigner functions on are , where the index ‘m’ stands for ‘macroscopic’ and the index ‘’ stands for ‘microscopic’. Because the dependence on and is slow, one can approximate them as follows:
| (244a) | |||
| (244b) | |||
The matrix is positive-semidefinite as an average Wigner tensor (section 2.1.7), and so is (section 6.4). Hence, both and are non-negative. Using (238), one can also rewrite the Wigner function of more compactly as
| (245) |
Now we can represent the nonlinear potentials (201)–(203) as
| (246) |
Here, the index denotes contributions from and the index (μ) denotes contributions from . Specifically,
| (247) | ||||
| (248) | ||||
| (249) |
Here, is a non-negative function (244a), so is positive-semidefinite and leads to an -theorem similar to (206). One also has
| (250) | ||||
| (251) | ||||
| (252) |
The functions and scale as , i.e. as (section 3.3). Their contribution to (197) is of order and , respectively, so it scales as and therefore is negligible within our model. In contrast, must be retained alongside with . This is because although weak, macroscopic fluctuations can resonate with particles from the bulk distribution, while the stronger macroscopic fluctuations are assumed to resonate only with particles from the tail distribution, which are few.
6.6 Oscillation-center Hamiltonian
Within the assumed accuracy, the OC Hamiltonian is , and is given by (216). Combined with the general theorem (53c), the latter readily yields , where
| (253) |
and the contribution of has been neglected. Because both and are quadratic in and enter only in the combination , it is convenient to attribute the latter as the ‘total’ ponderomotive energy. Using (249) in combination with (244a), one can express it as follows:
| (254) |
(see section 9 for examples). Notably,
| (255) |
where denotes a functional derivative and is the adiabatic action defined in (223). Equation (255) is a generalization of the well-known ‘– theorem’ (Kaufman & Holm, 1984; Kaufman, 1987). Loosely speaking, it says that the coefficient connecting with is proportional to the linear polarizability of an individual particle of type (Dodin et al., 2017; Dodin & Fisch, 2010a). (‘’ in the name of this theorem is the same as our , and ‘’ is the linear susceptibility.) Also, the OC Hamiltonian and the OC velocity can be expressed as
| (256) |
6.7 Polarization drag
Within the assumed accuracy, the OC distribution can be expressed as
| (257) |
and (197) becomes
| (258) | |||
| (259) |
where we have reinstated the term introduced in section 3.3. As a collisional term, is needed only to the zeroth order in , so
| (260) |
Correlating with is only the microscopic part of , so using (241) one obtains
| (261) |
Next, let us use (28) and to express this result as follows:
| (262) |
where we have approximated with the usual product and substituted (237). Hence,
| (263) |
where can be interpreted as the polarization drag (i.e. the average force that is imposed on an OC by its dress) and is given by
| (264) |
Using (243), one also rewrite this as follows:
| (265a) | ||||
| (265b) | ||||
where we have substituted (233) for . With (231) for , this yields
| (266) |
The product of the last two lines equals . Hence,
| (267) |
6.8 Collision operator
By combining (263) for with (250) for , one can express as
| (268) |
where is given by (243). One can recognize this as a generalization of the Balescu–Lenard collision operator (Krall & Trivelpiece, 1973, section 11.11) to interactions via a general multi-component field . Specific examples can be found in section 9.
It is readily seen that conserves particles, i.e.
| (269) |
and vanishes in thermal equilibrium (section 8.1). Other properties of are determined by the properties of the coupling coefficient , which are as follows. Note that
| (270) |
where we introduced
| (271) | ||||
| (272) |
To calculate , note that (243) yields
| (273) |
whence one obtains
| (274) |
The operators , , and (for all ) have been introduced for real fields, so their matrix elements in the coordinate representation are real. Then, the corresponding symbols satisfy , where is any of the three symbols. This gives
| (275) |
Because the rest of the integrand in (6.8) is even in , (275) signifies that does not contribute to . Thus, in (6.8) can as well be replaced with :
| (276) |
In this representation, the coupling coefficient in is manifestly symmetric,
| (277) |
which readily leads to momentum and energy conservation (appendix C):282828Remember that is defined as the OC velocity in the above formulas (section 5.6). If is treated as the particle velocity instead, then in (278) should be replaced with . Both options are admissible within the assumed accuracy, but the former option is preferable because it leads to other conservation laws that are exact within our model (section 7.5).
| (278) |
The collision operator also satisfies the -theorem (appendix C.3):
| (279) |
where the entropy density is defined as
| (280) |
and . Note that these properties are not restricted to any particular . Also note that if applied in proper variables (section 3.1.2), our formula (6.8) can describe collisions in strong background fields. This topic, including comparison with the relevant literature, is left to future work.
6.9 Summary of section 6
Let us summarize the above general results (for examples, see section 9). We consider species governed by a Hamiltonian of the form
| (281) |
where is a real oscillating field (of any dimension ), which generally consists of a macroscopic part and a microscopic part . The term is independent of , and the operators , , and may be nonlocal in and and may depend on the momentum parametrically. The dynamics of this system averaged over the fast oscillations can be described in terms of the OC distribution function
| (282) |
(the index is henceforth omitted for brevity), which is governed by the following equation of the Fokker–Planck type:
| (283) |
Here, is the OC Hamiltonian, is the dressing function, and is the total ponderomotive energy (i.e. the part of the OC Hamiltonian that is quadratic in ), so is the OC velocity. Specifically,
| (284a) | ||||
| (284b) | ||||
| (284c) | ||||
Here, is a scalar function, the average Wigner matrix is understood as the Fourier spectrum of the symmetrized autocorrelation matrix of the macroscopic oscillations:
| (285) |
with . Also, the vector is the Weyl symbol of as defined in (26), , and are the Weyl symbols of and , respectively, and denotes the Hermitian part. The matrix is positive-semidefinite and satisfies an -theorem of the form (206). Also, satisfies the ‘– theorem’:
| (286) |
The matrix characterizes the collective plasma response to the field and is given by
| (287) |
Here, the arguments are omitted for brevity, is a dyadic matrix, and is the symbol of the Hermitian dispersion operator that governs the field in the absence of plasma. Specifically, is defined such that the field Lagrangian density without plasma is .
The spectrum of microscopic fluctuations (specifically, the spectrum of the symmetrized autocorrelation function of the microscopic field ) is a positive-semidefinite matrix function and given by
| (288) |
where . (The dependence on and is assumed too but not emphasized.) The microscopic fluctuations give rise to a collision operator of the Balescu–Lenard type:
| (289) |
where the coupling coefficient is given by
| (290) | |||
| (291) |
The operator satisfies the -theorem and conserves particles, momentum, and energy:
7 Interaction with on-shell waves
Here, we discuss QL interaction of plasma with ‘on-shell’ waves, i.e. waves constrained by dispersion relations. To motivate the assumptions that will be adopted, and also to systematically introduce our notation, we start with briefly overviewing theory of linear waves in dispersive media (Tracy et al., 2014; Whitham, 1974), including monochromatic waves (section 7.1), conservative eikonal waves (section 7.2), general eikonal waves (section 7.3), and general broadband waves described by the WKE (section 7.4). After that, we derive conservation laws for the total momentum and energy, which are exact within our model (section 7.5). All waves in this section are considered macroscopic, so we adopt a simplified notation and the index will be omitted.
7.1 Monochromatic waves
Conservative (nondissipative) waves can be described using the least-action principle . Assuming the notation as in section 6.2, the action integral can be expressed as with the Lagrangian density given by
| (292) |
First, let us assume a homogeneous stationary medium, so . Because we assume real fields,292929A complex field can be accommodated by considering its real and imaginary parts as separate components. is real for all , one also has
| (293) |
where the latter equality is due to .
Because is Hermitian, it has orthonormal eigenvectors :
| (294) |
Here are the corresponding eigenvalues, which are real and satisfy
| (295) |
Due to (293), one has
| (296) |
where stands for the th eigenvalue. Using this together with (293), one obtains from (294) that
| (297) |
whence
| (298) |
Let us consider a monochromatic wave of the form
| (299) |
with real frequency , real wavevector , and complex amplitude . For such a wave, the action integral can be expressed as , where the average Lagrangian density is given by303030Here we use that for any oscillating and , one has and that is real because is Hermitian.
| (300) |
Let us decompose in the basis formed by the eigenvectors , that is, as
| (301) |
Then, (300) becomes
| (302) |
The real and imaginary parts of the amplitudes can be treated as independent variables. This is equivalent to treating and as independent variables, so one arrives at the following Euler–Lagrange equations:
| (303) |
Hence the th mode with a nonzero amplitude satisfies the dispersion relation
| (304) |
Equation (304) determines a dispersion surface in the space where the waves can have nonzero amplitude. This surface is sometimes called a shell, so waves constrained by a dispersion relation are called on-shell. Also note that combining (304) with (294) yields that on-shell waves satisfy
| (305) |
which are two mutually adjoint representations of the same equation.
Below, we consider the case when (304) is satisfied only for one mode at a time, so summation over and the index itself can be omitted. (A more general case is discussed, for example, in (Dodin et al., 2019).) Then, ,
| (306) |
and is connected with via , where is the function that solves . Also importantly, (305) ensures that
| (307) |
where can be replaced with any variable.
7.2 Conservative eikonal waves
7.2.1 Basic properties
In case of a quasimonochromatic eikonal wave and, possibly, inhomogeneous non-stationary plasma, one can apply the same arguments as in section 7.1 except the above equalities are now satisfied up to . For a single-mode wave, one has
| (308) |
where the local frequency and the wavevector,
| (309) |
are slow functions of , and so is , which satisfies (294). Then,
| (310) |
Within the leading-order theory, the term is neglected.313131Corrections to the lowest-order dispersion relation produce the so-called spin Hall effect; see (Dodin et al., 2019; Ruiz & Dodin, 2017a) for an overview and (Bliokh et al., 2015; Ruiz & Dodin, 2015a; Oancea et al., 2020; Andersson et al., 2021) for examples. These corrections are beyond the accuracy of the model considered, so they will be ignored. Then, the least action principle
| (311) |
leads to the same (but now local) dispersion relation as for monochromatic waves, . This shows that quasimonochromatic waves are also on-shell, and thus they satisfy (307) as well. Also notice that the dispersion relation can now be understood as a Hamilton–Jacobi equation for the eikonal phase :
| (312) |
Like in the previous section, let us introduce the function that solves
| (313) |
and therefore satisfies
| (314) |
Differentiating (313) with respect to , , and leads to
| (315a) | |||
| (315b) | |||
| (315c) | |||
where the derivatives of are evaluated at . In particular, (315c) gives
| (316) |
for the group velocity , whose physical meaning is to be specified shortly.
Because is now an additional dynamical variable, one also obtains an additional Euler–Lagrange equation:
| (317) |
where is called the action density and is the action flux density:
| (318) | |||
| (319) |
where we used (307) and the derivatives are evaluated on . Using (316), one can also rewrite (319) as
| (320) |
(The arguments will be omitted from now on for brevity. We will also use as a shorthand for where applicable.) Then, (317) becomes
| (321) |
which can be a understood as a continuity equation for quasiparticles (‘photons’ or, more generally, ‘wave quanta’) with density and fluid velocity (see also section 7.2.2). Thus, if an eikonal wave satisfies the least-action principle, its total action (‘number of quanta’) is an invariant. This conservation law can be attributed to the fact that the wave Lagrangian density depends on derivatives of but not on per se.
Also notice the following. By expanding (310) in around , which is satisfied on any solution, one obtains
| (322) |
where we used that due to (312). Then, one arrives at the canonical form of the action integral (Hayes, 1973)
| (323) |
From here, yields the dispersion relation in the Hamilton–Jacobi form , and yields the action conservation (321).
7.2.2 Ray equations
By (309), one has the so-called consistency relations:
| (324) |
These lead to
| (325) |
and similarly,
| (326) |
where we used (325). Using the convective derivative associated with the group velocity,
| (327) |
one can rewrite these compactly as
| (328) |
One can also represent (328) as ordinary differential equations for and , where are the ‘ray trajectories’ governed by
| (329) |
Specifically, together with (329), equations (328) become Hamilton’s equations also known as the ray equations:
| (330) |
where is the coordinate, is the momentum, is the energy, is the Hamiltonian, and the constant factor can be anything. If is chosen to be the Planck constant, then (330) can be interpreted as the motion equations of individual wave quanta, for example, photons. Hamilton’s equations for ‘true’ particles, such as electrons and ions, are also subsumed under (330) in that they can be understood as the ray equations of the particles considered as quantum-matter waves in the semiclassical limit.
7.2.3 Wave momentum and energy
Using (321) and (327), one arrives at the following equality for any given field :
| (331) |
For and , (331) yields, respectively,
| (332a) | ||||
| (332b) | ||||
where we used (328) and introduced the following notation:
| (333) |
When a medium is homogeneous along , (332a) yields . Likewise, when a medium is stationary, (332b) yields . Hence, by definition, and are the densities of the wave canonical momentum and energy, at least up to a constant factor .323232Therefore, in a zero-dimensional wave, where can be omitted, conservation of the total action implies conservation of , which is a well-known adiabatic invariant of a discrete harmonic oscillator with a slowly varying frequency (Landau & Lifshitz, 1976, section 49). A proof that can be found, for example, in (Dodin & Fisch, 2012). In section 7.5, we will show this using different arguments.
7.3 Non-conservative eikonal waves
In a medium with nonzero , where waves are non-conservative, the wave properties are defined as in the previous section but the wave action evolves differently. The variational principle is not easy to apply in this case (however, see (Dodin et al., 2017)), so a different approach will be used to derive the action equation. A more straightforward but less intuitive approach can be found in (Dodin et al., 2019; McDonald, 1988).
7.3.1 Monochromatic waves
First, consider a homogeneous stationary medium and a ‘monochromatic’ (exponentially growing at a constant rate) wave field in the form
| (334) |
where the constants and are, as usual, the real frequency and wavenumber, and is the linear growth rate, which can have either sign. Then, (222) becomes
| (335) |
where we assume that is a smooth function of and also that both and are . Like in section 7.2.1, we adopt , where the polarization vector is the relevant eigenvector of . Then, by projecting (335) on , one obtains
| (336) |
where is the corresponding eigenvalue of and we used (307). Let us neglect , divide (336) by , and consider the real and imaginary parts of the resulting equation separately:
| (337) |
The former is the same dispersion relation for as for conservative waves, and the latter yields , where
| (338) |
Because , one can write the amplitude equation as
| (339) |
One can also define the action density as in section 7.2.1 and rewrite (339) in terms of that. Because , one obtains
| (340) |
7.3.2 Non-monochromatic waves
When weak inhomogeneity and weak dissipation coexist, their effect on the action density is additive, so (321) and (340) merge into a general equation
| (341) |
(A formal derivation of (341), which uses the Weyl expansion (41) and projection of the field equation on the polarization vector, can be found in (Dodin et al., 2019).) Then, (331) is modified as follows:
| (342) |
and the equations (332) for the wave momentum and energy (333) become
| (343a) | ||||
| (343b) | ||||
A comment is due here regarding the relation between (341) and the amplitude equation (339) that is commonly used in the standard QLT for homogeneous plasma (for example, see (2.21) in (Drummond & Pines, 1962)). In a nutshell, the latter is incorrect, even when . Because is time-dependent, waves do not grow or decay exponentially. Rather, they can be considered as geometrical-optics (WKB) waves, and unlike in section 7.3.1, the ratio generally evolves at a rate comparable to . The standard QLT remains conservative only because it also incorrectly replaces (103) with its stationary-plasma limit () and the two errors cancel each other. These issues are less of a problem for waves in not-too-hot plasmas (\egLangmuir waves), because in such plasmas, changing the distribution functions does not significantly affect the dispersion relations and thus does in fact approximately remain constant. See also the discussion in section 9.1.4.
7.4 General waves
Let us now discuss a more general case that includes broadband waves. The evolution of such waves can be described statistically in terms of their average Wigner matrix . This matrix also determines the function that is given by (244a) and enters the nonlinear potentials (284). Below, we derive the general form of in terms of the phase-space action density and the governing equation for (sections 7.4.1–7.4.3). Then, we also express the function through (section 7.4.4). Related calculations can also be found in (McDonald & Kaufman, 1985; Ruiz, 2017).
7.4.1 Average Wigner matrix of an eikonal wave field
Let us start with calculating the average Wigner matrix of an eikonal field of the form (308) (see also appendix A.2). Using , it can be readily expressed through the average Wigner functions of the complexified field333333Field complexification is discussed, for example, in (Brizard et al., 1993). and of its complex conjugate:
| (344) |
For , where the arguments are omitted for brevity, one has
| (345) |
where we neglected the dependence of and on because it is weak compared to that of . By Taylor-expanding , one obtains
For , which can also be written as due to (298), the result is the same up to replacing and . Also notice that
| (346) |
so one can rewrite as follows:
| (347) |
where . Thus finally,
| (348) |
7.4.2 Average Wigner matrix of a general wave
Assuming the background medium is sufficiently smooth, a general wave field can be represented as a superposition of eikonal fields:
| (349) |
As a quadratic functional, its average Wigner matrix equals the sum of the average Wigner matrices of the individual eikonal waves:
| (350) |
where and are the average Wigner matrices of and , respectively:
| (351) |
Equation (350) can also be expressed as
| (352) |
where are the average Wigner matrices of and , respectively:
| (353) |
Because , the matrix can also be written as follows:
| (354) |
where is given by
| (355) |
This shows that for broadband waves comprised of eikonal waves, has the same form as for an eikonal wave except is not necessarily delta-shaped.
7.4.3 Phase-space action density and the wave-kinetic equation
The wave equation for the complexified field can be written in the invariant form as . Multiplying it by from the right leads to
| (356) |
This readily yields an equation for the Wigner matrix: . Let us integrate this equation over to make the left-hand side a smooth function of . Let us also take the trace of the resulting equation to put it in a scalar form:
| (357) |
As usual, we assume with for generic . The integrand in (357) can be written as , and its expansion in the differential operator (32) contains derivatives of all orders. High-order derivatives on are not negligible per se, because for on-shell waves this function is delta-shaped. However, using integration by parts, one can reapply all derivatives with respect to to and take the remaining derivatives (with respect to , , and ) outside the integral. Then it is seen that each power of in the expansion of contributes to the integral. Let us neglect terms with and use (353). Hence, one obtains343434McDonald & Kaufman (1985) first Taylor-expand and then integrate over . Strictly speaking, that is incorrect (because is not smooth), but the final result is the same.
| (358) |
Let us also re-express this as follows, using (295) and (338):
| (359) |
Clearly,
| (360) |
To simplify the remaining terms, we proceed as follows. As a Hermitian matrix, can be represented in terms of its eigenvalues and eigenvectors as . For , let us use (353) again, where is one of the vectors , say, . (Accordingly, .) Then, for any , one has
| (361) |
where we used and, in particular, . Then, (359) can be written as
| (362) |
where
| (363) |
The real part of (362) gives , which is the dispersion relation. The imaginary part of (362) gives . To understand this equation, let us rewrite as
| (364) |
Here, we introduced
| (365) |
which, according to (315), satisfy
| (366a) | ||||
| (366b) | ||||
| (366c) | ||||
Note that using (355), one can also express as
| (367) |
where are the action densities (318) of the individual eikonal waves that comprise the total wave field (section 7.4.2). In particular, , which is the total action density. Therefore, the function can be interpreted as the phase-space action density. In terms of , the equation can be written as
| (368) |
This equation, called the WKE, serves the same role in QL wave-kinetic theory as the Vlasov equation serves in plasma kinetic theory.353535The term ‘WKE’ is also used for the equation that describes nonlinear interactions of waves in statistically homogeneous media, or ‘wave–wave collisions’ (Zakharov et al., 1992). That is not what we consider here. Inhomogeneities are essential in our formulation, and the QL WKE is linear (in ) by definition of the QL approximation. That said, the Weyl symbol calculus that we use can facilitate derivations of wave–wave collision operators as well (Ruiz et al., 2019). Unlike the field equation used in the standard QLT (Drummond & Pines, 1962), (368) exactly conserves the action of nonresonant waves, i.e. those with . Also note that (341) for eikonal waves can be deduced from (368) as a particular case by assuming the ansatz
| (369) |
and integrating over . In other words, eikonal-wave theory can be understood as the ‘cold-fluid’ limit of wave-kinetic theory.
7.4.4 Function in terms of
Here we explicitly calculate the function (244a) that determines the nonlinear potentials (284). Using (354), one obtains
| (370) |
where . By definition of , the function is real for all and , so by definition of the Weyl symbol (26). Together with (298), this gives , so
| (371a) | |||
| and similarly, | |||
| (371b) | |||
This also means that . Then finally, using (365), one can express this function through the phase-space action density:
| (372) | |||
| (373) |
7.5 Conservation laws
Let us rewrite (368) together with (283) in the ‘divergence’ form:
| (374) | ||||
| (375) |
Using (353), the diffusion matrix can be represented as follows:
| (376) |
Also, by substituting (231) into (338), one finds
| (377) |
Together with (366), these yield the following notable corollaries. First of all, if , , and are independent of ,363636Having -dependence in , , or would signify interaction with external fields not treated self-consistently. Such fields could exchange momentum with the wave–plasma system, so the momentum of the latter would not be conserved. A similar argument applies to the temporal dependence of these coefficients vs. energy conservation considered below. one has for each that (appendix D.1)
| (378) |
This can be viewed as a momentum-conservation theorem, because at , one has
| (379) |
Also, the ponderomotive force on a plasma is readily found from (378) as the sum of the terms quadratic in the wave amplitude (after has been expressed through ). Similarly, if , , and are independent of , one has (appendix D.2)
| (380) |
This can be viewed as an energy-conservation theorem, because at , one has
| (381) |
Related equations are also discussed in (Dodin & Fisch, 2012; Dewar, 1977).
The individual terms in (378) and (380) can be interpreted as described in table 1. The results of section 7.2.3 are reproduced as a particular case for the eikonal-wave ansatz (369).373737There is no ambiguity in the definition of the wave momentum and energy in this case (i.e. ), because (379) and (381) connect those with the momentum and energy of particles (OCs), which are defined unambiguously. In particular, note that electrostatic waves carry nonzero momentum density just like any other waves, even though the electrostatic field of these waves carries no momentum. The momentum is stored in the particle motion in this case (section 9.1.3), and it is pumped there via either temporal dependence (Liu & Dodin, 2015, section II.2) or spatial dependence (Ochs & Fisch, 2021b, 2022) of the wave amplitude. This shows that homogeneous-plasma models that ignore ponderomotive effects cannot adequately describe the energy–momentum transfer between waves and plasma even when resonant absorption per se occurs in a homogeneous-plasma region. The OC formalism presented here provides means to describe such processes rigorously, generally, and without cumbersome calculations.
| Quantity | Notation | Interpretation |
|---|---|---|
| OC momentum density | ||
| OC energy density | ||
| OC momentum flux density | ||
| OC energy flux density | ||
| wave momentum density | ||
| wave energy density | ||
| wave momentum flux density | ||
| wave energy flux density |
7.6 Summary of section 7
In summary, we have considered plasma interaction with general broadband single-mode on-shell waves (for examples, see section 9). Assuming a general response matrix , these waves have a dispersion function and polarization determined by
| (382) |
where the normalization is assumed. Specifically for given by (287), one has
| (383) |
where the arguments are omitted for brevity. (Some notation is summarized in section 6.9.) The wave frequency satisfies
| (384) |
and , where is a real function at real arguments. The wave local linear growth rate , which is assumed to be small in this section, is
| (385) |
or explicitly,
where and . The nonlinear potentials (284) are expressed through the scalar function
| (386) |
where is evaluated at ; see also (373). The function is the phase-space action density governed by the WKE:
| (387) |
where is the group velocity. Collisional dissipation is assumed small compared to collisionless dissipation, so it is neglected in (387) but can be reintroduced by an ad hoc modification of (section 6.2). Unlike the field equation used in the standard QLT, (387) exactly conserves the action of nonresonant waves, i.e. those with . The WKE must be solved together with the QL equation for the OC distribution ,
| (388) |
because determines the coefficients in (387) and determines the coefficients in (388). When and are independent of and , (387) and (388) conserve the total momentum and energy of the system; specifically,
| (389) | ||||
| (390) |
Here, the notation is as in table 1, or see (378) and (380) instead.
8 Thermal equilibrium
In this section, we discuss, for completeness, the properties of plasmas in thermal equilibrium.
8.1 Boltzmann–Gibbs distribution
As discussed in section 6.8, collisions conserve the density of each species, the total momentum density, and the total energy density, while the plasma total entropy density either grows or remains constant. Let us search for equilibrium states in particular. At least one of the states in which remains constant is the one that maximizes the entropy density at fixed , , and . This ‘state of thermal equilibrium’ can be found as an extremizer of
| (391) |
considered as a functional of all , where , , and are Lagrange multipliers. Using (280), one finds that extremizers of satisfy
| (392) |
whence
| (393) |
The pre-exponential constant is determined by the given density of species , while and can be expressed through the densities of the plasma momentum and energy stored in the whole distribution. Because
| (394) |
the matrix is positive-definite, so the entropy is maximal (as opposed to minimal) at the extremizer (393).
The distribution (393) is known as the Boltzmann–Gibbs distribution, with being the temperature (common for all species). Also, let us introduce a new, rescaled Lagrange multiplier via . Then,
| (395) |
where is independent of . Correspondingly,
| (396) |
where we used (204). From (396), one obtains
| (397) |
where . Then, (6.8) yields that the collision operator vanishes on the Boltzmann–Gibbs distribution, and thus, expectedly, . One can also show that the Boltzmann–Gibbs distribution is the only distribution (strictly speaking, a class of distributions parameterized by and ) for which the entropy density is conserved (appendix E).
The property (396) of the thermal-equilibrium state also leads to other notable results that we derive below. In doing so, we will assume the reference frame where , so the Boltzmann–Gibbs distribution has a better known form
| (398) |
(For isotropic in , this is the frame where the plasma total momentum density is zero.) The generalizations to arbitrary are straightforward.
8.2 Fluctuation–dissipation theorem
Let us describe microscopic fluctuations in equilibrium plasmas in terms of , i.e.
| (399) |
which can also be represented in terms of the Fourier image of the microscopic field :
| (400) |
For statistically homogeneous fields that persist on time within volume , the Fourier transform is formally divergent; hence the appearance of the factors and in (400).383838To make (400) look more physical (local), one can absorb the global factors and in the definition of the Fourier transform; cf. section 9.1.5. Also, as seen from (239), any quadratic function of the microscopic field can be expressed through via
| (401) |
where and are any linear operators and and are their symbols.
From (238), one finds that, in general,
| (402) |
For a thermal distribution in particular, which satisfies (398), one can rewrite (231) as follows:
| (403) |
By comparing this with (402), one also finds that
| (404) |
Due to (233), this leads to the fluctuation–dissipation theorem in the following form:
| (405) |
For examples of for specific systems, see section 9.
8.3 Kirchhoff’s law
Consider the power deposition via polarization drag:
| (406) |
Using (265a) for , (403) for , and (405) for , this can also be expressed as follows:
| (407) |
Thus, the spectral density of the power deposition via polarization drag is given by
| (408) |
which is a restatement of Kirchhoff’s law (Krall & Trivelpiece, 1973, section 11.4). For examples of for specific systems, see section 9.
8.4 Equipartition theorem
As flows from section 7.5, the energy of on-shell waves of a field in a homogeneous -dimensional plasma of a given volume can be written as
| (409) |
To apply this to microscopic fluctuations, one can replace with and substitute . Then, the total energy of a mode with given wavevector and polarization can be expressed as
| (410) |
where the arguments are omitted for brevity. For thermal equilibrium, one can substitute (405) for ; then,
| (411) |
The integrand peaks at , where the mode eigenvalue is small. Due to damping, the actual zero of is slightly below the real axis in the complex-frequency space. Then, at infinitesimally small damping, can be approximated near as
| (412) |
This leads to the well-known equipartition theorem:
| (413) |
Note that according to (413), the sum is divergent. This indicates that not all modes can be classical and on-shell (weakly damped) simultaneously.
8.5 Summary of section 8
In thermal equilibrium, when all species have Boltzmann–Gibbs distributions with common temperature , the collision operator vanishes, the entropy is conserved, and the spectrum of microscopic fluctuations (399) satisfies the fluctuation–dissipation theorem:
| (414) |
where is the dispersion matrix (287) and denotes the anti-Hermitian part (or the imaginary part in case of scalar fields). From here, it is shown that the spectral density of the power deposition via polarization drag is given by , which is a restatement of Kirchhoff’s law. For on-shell waves, (414) reduces to the equipartition theorem, which says that the energy per mode equals . Applications to specific systems are discussed in section 9.
9 Examples
In this section, we show how to apply our general formulation to nonrelativistic electrostatic interactions (section 9.1), relativistic electromagnetic interactions (section 9.2), Newtonian gravity (section 9.3), and relativistic gravity, including gravitational waves (section 9.4).
9.1 Nonrelativistic electrostatic interactions
9.1.1 Main equations
Let us show how our general formulation reproduces (and generalizes) the well-known results for electrostatic turbulence in nonmagnetized nonrelativistic plasma. In this case,
| (415) |
where is the electric charge, is the electrostatic potential, and and are its average and oscillating parts, respectively. Then, , , , and , so . The matrix (285) is a scalar (Wigner function) given by
| (416) |
(Underlining denotes the macroscopic part, , and the arguments will be omitted from now on.) Correspondingly,
| (417) | ||||
| (418) | ||||
| (419) |
and also
| (420) |
The Lagrangian density of a free electrostatic field is
| (421) |
The first term on the right-hand side does not contribute to the field action and thus can be ignored. The second term is of the form (208) with , (section 2.1.2), and , so , where . Then, (227) gives
| (422) |
where the arguments and are omitted for brevity and is the parallel permittivity:
| (423) |
9.1.2 Collisions and fluctuations
By (402), the spectrum of microscopic oscillations of is a scalar given by
| (424) |
where we substituted for three-dimensional plasma. For thermal equilibrium, (405) leads to the well-known formula (Lifshitz & Pitaevskii, 1981, section 51)
| (425) |
The spectrum of charge-density fluctuations is found using Poisson’s equation , whence . Fluctuations of other fields are found similarly. Also, (243) leads to
| (426) |
Then, (6.8) yields the standard Balescu–Lenard collision operator:
| (427) |
(As a reminder, the distribution functions are normalized such that is the local average density of species (199).)
9.1.3 On-shell waves
For on-shell waves, (354) gives , where is determined by the dispersion relation
| (428) |
and is given by
| (429) |
The phase-space density of the wave action, defined in (365), is
| (430) |
and the dressing function (418) is given by
| (431) |
Using these, one obtains (appendix F.1.1)
| (432) |
so the conserved quantity (379) is the average momentum of the plasma (while the electrostatic field carries no momentum, naturally). Also (appendix F.1.2),
| (433) |
so, expectedly, the conserved quantity (381) is the average particle energy plus the energy of the electrostatic field. In combination with our equations for and (section 7.6), these results can be considered as a generalization and concise restatement of the OC QLT by Dewar (1973), which is rigorously reproduced from our general formulation as a particular case.
9.1.4 Eikonal waves
As a particular case, let us consider an eikonal wave
| (434) |
which may or may not be on-shell. As seen from section 7.4.1,
| (435) |
For nonresonant particles, the dressing function is well defined is found as follows:
| (436) |
Similarly, the ponderomotive energy for nonresonant particles is
| (437) |
in agreement with (Dewar, 1972; Cary & Kaufman, 1977). One can also express these functions in terms of the electric-field envelope :
| (438) |
For on-shell in particular, one can use (430) together with (cf. (355)) to obtain the well-known expression for the wave action density :
| (439) |
For non-too-hot plasma, one has , where is the plasma frequency. The corresponding waves are Langmuir waves. Their dispersion relation is , so (and accordingly, the wave energy density is for either sign). Remember, though, that this expression is only approximate. Using it instead of (439) can result in violation of the exact conservation laws of QLT. Conservation of the Langmuir-wave action in non-stationary plasmas beyond the cold-plasma approximation is also discussed in (Dodin et al., 2009; Dodin & Fisch, 2010b; Schmit et al., 2010).
9.1.5 Homogeneous plasma
In homogeneous -dimensional plasma of a given volume , the Wigner function (416) has the form . The function is readily found using (207):
| (440) |
Then,
| (441) |
This coincides with the well-known formula for the QL-diffusion coefficient in homogeneous electrostatic plasma.393939See, for example, equation (16.17) in (Stix, 1992). The extra mass factor appears there because QL diffusion is considered in the velocity space instead of the momentum space. The functions and are also important in homogeneous turbulence in that they ensure the proper energy–momentum conservation; for example, see (Stix, 1992, section 16.3) and (Liu & Dodin, 2015, section II.2). These functions can be expressed through too. However, they have a simpler representation in terms of the Wigner function , as in (418) and (419), respectively. This is because is a local property of the field, which makes it more fundamental than the amplitudes of global Fourier harmonics commonly used in the literature.
9.2 Relativistic electromagnetic interactions
9.2.1 Main equations
Let us extend the above results to relativistic electromagnetic interactions. In this case,
| (442) |
where is the speed of light and is the vector potential. Let us adopt the Weyl gauge for the oscillating part of the electromagnetic field () and Taylor-expand to the second order in . This leads to
| (443) | |||
| (444) |
(although plasma is assumed nonmagnetized, a weak average magnetic field is allowed, so can be order-one and thus generally must be retained), where
| (445) |
and . In the equations presented below, (where is the OC velocity) is a sufficiently accurate approximation. Also, can be interpreted as the relativistic-mass tensor.
Let us choose the field of our general theory to be the oscillating electric field ; then (cf. (209)),
| (446) |
(Other ways to identify and are also possible and lead to the same results.) Then,
| (447) |
The average Wigner matrix of is
| (448) |
(the arguments and are henceforth omitted), and the nonlinear potentials are
| (449) | ||||
| (450) | ||||
| (451) |
When plasma is nonrelativistic and the field is electrostatic (so , where is scalar), (449) gives the same as (417) and (451) gives the same as (419). For , the equivalence between (450) and (418) should not be expected because is a part of a distribution function, which is not gauge-invariant. (Canonical momenta in the Weyl gauge are different from those in the electrostatic gauge.) But it is precisely the dressing function (450) that leads to the correct expressions for the momentum and energy stored in the OC distribution (section 9.2.3).
The Lagrangian density of a free electromagnetic field is
| (452) |
From Faraday’s law, one has .404040Here, the oscillating field has the same dimension as , so the standard vector notation (including the dot product and the cross product) is naturally extended to . Then, can be represented as follows (up to a divergence, which is insignificant):
| (453) |
Then, the vacuum dispersion operator can be written as (cf. (208))
| (454) |
The total dispersion matrix is readily found to be
| (455) |
where (not to be confused with the small parameter that we introduced earlier) is the dielectric tensor:
| (456) |
Here, is the squared relativistic plasma frequency, which is a matrix, because the ‘masses’ are matrices:
| (457) |
9.2.2 Collisions and fluctuations
By (402), the spectrum of microscopic oscillations of is a matrix given by
| (458) |
In the electrostatic limit, one can replace with , where is the relativistic generalization of (423); then (458) leads to (424) as a particular case. For thermal equilibrium, one can also use (405) and the following form of for isotropic plasma:
| (459) |
where is the (scalar) transverse permittivity. Also, (243) leads to
| (460) |
Then the collision operator (6.8) is obtained in the form
| (461) |
which is in agreement with (Hizanidis et al., 1983; Silin, 1961). Replacing with leads to the standard Balescu–Lenard operator (427) as a particular case.
9.2.3 On-shell waves
Electromagnetic on-shell waves satisfy
| (462) |
where is the complex envelope vector parallel to the polarization vector ; also,
| (463) |
This yields (see (453))
| (464) |
Then, the phase-space density of the wave action (365) can be cast in the form
| (465) |
(cf. (Dodin et al., 2019)), and the dressing function (450) is given by
| (466) |
Using these, one obtains (appendix F.2.1)
| (467) |
where is the average density of the plasma kinetic (up to ) momentum,
| (468) |
the functions are the distributions of kinetic (up to ) momenta, and the second term in (467) is the well-known average momentum of electromagnetic field. Similarly (appendix F.2.2),
| (469) |
where is given by
| (470) |
In other words, the total momentum and energy of the system in the OC–wave representation are the same as those in the original particle–field variables.
9.2.4 Eikonal waves
9.3 Newtonian gravity
For Newtonian interactions governed by a gravitostatic potential , one has
| (475) |
where is the gravitational constant. This system is identical to that considered in section 9.1 for nonrelativistic electrostatic interactions up to coefficients. Specifically, are replaced with , a factor appears in , and the dispersion matrix becomes
| (476) |
Thus, is replaced with , where is the gravitostatic permittivity given by
| (477) |
This readily yields, for example, kinetic theory of the Jeans instability (Trigger et al., 2004), whose dispersion relation is given by (modulo the usual analytic continuation of the permittivity to modes with ).
9.4 Relativistic gravity
9.4.1 Main equations
The dynamics of a relativistic neutral particle with mass in a spacetime metric with signature is governed by a covariant Hamiltonian (see, for example, (Garg & Dodin, 2020))
| (478) |
Here, , and , as usual. Also, is the index-free notation for the four-momentum , is the inverse metric, is the index-free notation for , the units are such that , and the species index is omitted.414141This section uses notation different from that used in the rest of the paper. In particular, the Greek indices span from 0 to 3, and the standard rules of index manipulations apply. The corresponding Hamilton’s equations, with the proper time, are
| (479) |
This dynamics is constrained to the shell , where is the (negative) solution of
| (480) |
This means that the particle distribution in the space is delta-shaped and thus does not satisfy (119). Hence, we will consider particles in the six-dimensional space instead. The corresponding dynamics is governed by the Hamiltonian
| (481) |
This is seen from the fact that
| (482) |
where , so Hamilton’s equations obtained from (481) are equivalent to (479):
| (483) |
Let us decompose the metric into the average part and oscillations, , and approximate the inverse metric to the second order in :
| (484) |
where the indices of are manipulated using the background metric . This gives
| (485) |
The Hamiltonian (481) is expanded in as follows:
| (486) |
where and the derivatives on the right-hand side are evaluated on . To calculate these derivatives, let us differentiate (480) and use (485) for . This gives
| (487) |
where the derivatives with respect to the oscillating metric are taken at fixed and at , and ; thus,
| (488) |
Similarly, differentiating (480) twice gives
whence
Then, (486) yields
| (489) |
where we introduced , , and
| (490) |
9.4.2 Nonlinear potentials
Let us treat as a 16-dimensional vector (Garg & Dodin, 2021b), so serves as and serves as . (Because these operators happen to be local in the representation, here we do not distinguish them from their symbols.) Let us also introduce
| (491) |
and notice that (see (483)), so and . Then, one finds from (284) that (appendix B.8)
| (492) | ||||
| (493) | ||||
| (494) |
Equation (494) (where one takes after the differentiation) is in agreement with the result that was obtained for quasimonochromatic waves in (Garg & Dodin, 2020). The derivation of the dispersion matrix for relativistic gravitational interactions in matter is cumbersome, so it is not presented here, but see (Garg & Dodin, 2022). The collision integral and fluctuations for relativistic gravitational interactions are straightforward to obtain from the general formulas presented in sections 6.9 and 8. This can be used to describe QL interactions of gravitational waves, including not only the usual vacuum modes424242Vacuum gravitational waves satisfy . Hence, satisfying the resonance condition requires , which requires particle speeds not smaller than the speed of light (remember that in our units). For massive particles, this cannot be satisfied, so vanishes for vacuum gravitational waves. However, such waves can still produce adiabatic ponderomotive effects determined by (Garg & Dodin, 2020). but also waves coupled with matter, for example, the relativistic Jeans mode.
Also notice that the OC Hamiltonian can be put in a covariant form as follows. Like in the original system (section 9.4.1), determines the ponderomotively modified shell via . On one hand, the covariant OC Hamiltonian vanishes on this shell,434343The covariant Hamiltonian is the dispersion function of particles as quantum waves in the semiclassical limit (Garg & Dodin, 2020). so it can be Taylor-expanded as follows:
| (495) |
On the other hand, it can also be represented as (here is the ponderomotive term yet to be found) and Taylor-expanded around the unperturbed shell as
| (496) |
By comparing (495) with (496), one finds that . Because , this leads to the following covariant Hamiltonian for OCs:
| (497) | |||
| (498) |
9.4.3 Gauge invariance
As shown in (Garg & Dodin, 2021a, b) adiabatic QL interactions via gravitational waves (i.e. those determined by and ) can be formulated in a form invariant with respect to gauge transformations
| (499) |
where is the covariant derivative associated with the background metric , is an arbitrary vector field, and . Let us show that this also extends to resonant interactions. Recall that within the assumed accuracy the nonlinear potentials are supposed to be calculated only to the zeroth order in the geometrical-optics parameter. Then, the modification of the average Wigner matrix of the metric oscillations under the transformation (499) can be written as
| (500) |
where and is the average Wigner matrix of . The corresponding change of is
Then, the difference in the diffusion coefficients (492) is
| (501) |
because . In particular, this rules out QL diffusion via ‘coordinate waves’.
9.4.4 Lorenz gauge and effective metric
Let us consider gravitational waves in the Lorenz gauge, . In this case,
| (502) |
and thus . Then,
| (503) |
This simplifies the expression (494) for and (497) for . Furthermore, if the waves are not significantly affected by matter, so the vacuum dispersion can be assumed, the term (503) vanishes completely. Then, (497) becomes
| (504) |
and QL diffusion disappears, because particles cannot resonate with waves. This shows that the only average QL effect of vacuum gravitational waves on particles is the modification of the spacetime metric by . For quasimonochromatic waves, this effect is discussed in further detail in (Garg & Dodin, 2020).
10 Summary
In summary, we have presented quasilinear theory for classical plasma interacting with inhomogeneous turbulence in the presence of background fields. Because we use the Weyl symbol calculus, global-mode decomposition is not invoked, so our formulation is local and avoids the usual issues with complex-frequency modes. Also, the particle Hamiltonian is kept general, so the results are equally applicable to relativistic, electromagnetic, and even non-electromagnetic (for example, gravitational) interactions. Because our approach is not bounded by the limitations of variational analysis either, effects caused by collisional and collisionless dissipation are also included naturally.
Our main results are summarized in sections 5.6, 6.9, 7.6, 8.5 and are as follows. Starting from the Klimontovich equation, we derive a Fokker–Planck model for the dressed oscillation-center distribution. This model captures quasilinear diffusion, interaction with the background fields, and ponderomotive effects simultaneously. The local diffusion coefficient is manifestly positive-semidefinite. Waves are allowed to be off-shell (not constrained by a dispersion relation), and a collision integral of the Balescu–Lenard type emerges in a form that is not restricted to any particular Hamiltonian. This operator conserves particles, momentum, and energy, and it also satisfies the -theorem, as usual. As a spin-off, a general expression for the spectrum (average Wigner matrix) of microscopic fluctuations of the interaction field is derived. For on-shell waves, which satisfy a quasilinear wave-kinetic equation, our theory conserves the momentum and energy of the wave–plasma system. Dewar’s oscillation-center quasilinear theory of electrostatic turbulence (Dewar, 1973) is proven formally as a particular case and given a concise formulation. Also discussed as examples are relativistic electromagnetic and gravitational interactions, and quasilinear theory for gravitational waves is proposed.
Aside from having the aesthetic appeal of a rigorous local theory, our formulation can help, for example, better understand and model quasilinear plasma heating and current drive. First of all, it systematically accounts for the wave-driven evolution of the nonresonant-particle distribution and for the ponderomotive effects caused by plasma inhomogeneity in both time and space. As discussed above (section 7.5), this is generally important for adequately calculating the energy–momentum transfer between waves and plasma even when resonant absorption per se occurs in a homogeneous-plasma region. Second, our formulation provides general formulas that equally hold in any canonical variables and for any Hamiltonians that satisfy our basic assumptions (section 3.1). Therefore, our results can be applied to various plasma models immediately. This eliminates the need for ad hoc calculations, which can be especially cumbersome beyond the homogeneous-plasma approximation. Discussing specific models of applied interest, however exciting, is beyond the scope of this paper and is left to future work.
Funding
This work was supported by the U.S. DOE through Contract DE-AC02-09CH11466. It is also based upon the work supported by National Science Foundation under the grant No. PHY 1903130.
Declaration of interests
The author reports no conflict of interest.
Appendix A Average Wigner matrices
A.1 Positive semidefinitness
As known from (Cartwright, 1976), the average Wigner function of any scalar field on the real axis is non-negative if the averaging is done over a sufficiently large phase-space volume. Here, we extend this theorem to average Wigner matrices of vector fields in a multi-dimensional space, , and show that such matrices are positive-semidefinite.
For any given function , we define its local phase-space average as the following convolution integral:444444This ensures that , as readily seen from (505) using integration by parts.
| (505) |
with a Gaussian window function
| (506) |
and positive constants and yet to be specified. Unlike in section 2.1.1, the following notation will be assumed for the ‘scalar product’ for variables with upper, lower, and mixed indices:
| (507) |
(The Latin indices in this appendix range from 0 to , and are unit matrices, and summation over repeating indices is assumed.) In particular, note that and must not be confused with the squared spacetime interval, which can have either sign. Likewise, must not be confused with .
The average Wigner matrix of any given vector field is given by
| (508) |
The integral over can be readily taken:
| (509) |
Then, using the variables , one can rewrite (508) as follows:
| (510) | |||
| (511) |
The function can also be expressed as , where
| (512) |
and . Then, using , one obtains from (510) that
| (513) |
By Taylor-expanding , one obtains
| (514) |
where . Note that
| (515) |
where the summation is performed over all combinations of integers such that . Thus,
| (516) |
Because each is positive-semidefinite, the Wigner matrix is positive-semidefinite when , or equivalently, when . This condition is assumed to be satisfied for the phase-space averaging of used in the main text. Loosely, this means that the averaging is done over the phase-space volume .
A.2 Invariant limit for eikonal fields
For eikonal fields (308), one has
| (517) |
Here, , , ‘c.c.’ stands for complex conjugate, we used the linear approximation , with . Then,
| (518) |
Let us adopt , where is the least characteristic scale of , , and . Then,
| (519) |
Here, are normalized Gaussians that can be replaced with delta functions if is small compared to any scale of interest in the space:
| (520) |
Also, the function
| (521) |
can be made exponentially small by adopting .454545Even though has been assumed small compared to , the smallness of the geometrical-optics parameter allows choosing in the interval . In this limit, the average Wigner matrix of an eikonal field is independent of and :
| (522) |
This is also Hermitian and positive-semidefinite (in agreement with the general theory from section A.1), because so are and . The same properties pertain to the Wigner matrix of an ensemble of randomly phased eikonal fields, because it equals the sum of the Wigner matrices of the individual components (see also section 7.4).
Appendix B Auxiliary proofs
B.1 Proof of (53)
Like in the case of (45), one finds that
| (523) |
This proves (53a). At , when becomes the usual product, (523) gives
| (524) |
and in particular, taking the trace of (524) yields
| (525) |
Here, , and we used that for any matrices and .
For real fields, one can also replace the integrand with
| (526) |
where we used , , , and, again, , respectively. In summary then,
| (527) |
so the anti-Hermitian part of does not contribute to the integrals. Thus,
| (528) |
Because and are arbitrary, they can as well be swapped; then one obtains (53c).
B.2 Proof of (150)
B.3 Proof of (164)
Here, we show that
| (530) |
B.4 Proof of (170)
B.5 Proof of (180)
Let us estimate
| (533) |
where has the form
| (534) |
First, notice that
| (535) |
Because and are , one has , where is the characteristic inverse scale along the th phase-space axis. Thus,
| (536) |
where we used (see (69) and (86))
| (537) |
The first part of (180) is obtained by considering and using (536).
Let us also estimate
| (538) |
where has the form
| (539) |
First, note that
| (540) |
Next, note that
| (541) |
Assuming the notation
| (542) |
one can then rewrite as follows:
Each term on the right-hand side of this equation scales as , so
| (543) |
where we again used (537). The second part of (180) is obtained by considering and using (543).
B.6 Proof of (185)
Using (183) and assuming the notation , one finds that
| (544) |
Because and , the first term on the right-hand side of (544) is negligible. Also note that due to (99), the factor in the last term on the right-hand side of (544) commutes with . Hence, one obtains
| (545) |
Next, notice that
| (546) |
Assuming the notation
| (547) |
one can rewrite (546) compactly as follows:
| (548) |
Notice also that . Hence, for introduced in (545), one has
| (549) |
where we used (92) for and the anti-symmetry of . Therefore,
| (550) |
and accordingly,
| (551) |
Here, , which is equivalent to (184). From (551) and the fact that due to the anti-symmetry of , one has
| (552) |
B.7 Proof of (237)
The correlation function
| (553) |
can be readily expressed as
Here, is another (in addition to overbar) notation for averaging used in this appendix, the dependence of on is neglected, and ‘’ denotes that excluded are the terms that have and simultaneously. Aside from this, the summations over are taken over all particles of type , and the summations over are taken over all particles of type . Also,
| (554) |
To the leading order, pair correlations can be neglected. Then,
| (555) |
Let us also use . Then,
| (556) |
Next, notice that
| (557) |
Hence,
| (558) |
where we used . Therefore,
| (559) |
B.8 Proof of (494)
Using the symmetry , one readily obtains from (254) that
| (560) | |||
| (561) |
where and the prime in denotes that is considered as a function of at differentiation. One can also write this as follows:
| (562) |
As shown in (Garg & Dodin, 2020, appendix B), the following equality is satisfied:
| (563) |
Also notice that
| (564) |
where we used Hamilton’s equation . Therefore,
| (565) |
The first and the last terms can be merged; then, one obtains
| (566) |
Appendix C Properties of the collision operator
Here, we prove the properties of the collision operator discussed in section 6.8. To shorten the calculations, we introduce two auxiliary functions,
| (567) |
which have the following properties:
| (568) |
C.1 Momentum conservation
C.2 Energy conservation
Energy conservation is proven similarly, using that and the fact that and are interchangeable due to the presence of in :
| (571) |
Like in the previous case, the third and the fifth lines are minus each other, whence .
C.3 -theorem
Appendix D Conservation laws for on-shell waves
Here, we prove the momentum-conservation theorem (378) and the energy-conservation theorem (380) for QL interactions of plasmas with on-shell waves.
D.1 Momentum conservation
Let us multiply (374) by and integrate over . Then, one obtains
| (578) |
Similarly, multiplying (375) by and integrating over yields
| (579) |
Let us sum up (579) over species and also add it with (578). The contribution of the collision integral disappears due to (278), so one obtains
| (580) |
Next, notice that
| (581) |
Also, assuming that , , and are independent of and using (371), one gets
| (582) |
where we also used (366b). Substituting (581) and (582) into (580) leads to (378).
D.2 Energy conservation
Let us multiply (374) by and integrate over . Then, one obtains
| (583) |
Similarly, multiplying (375) by and integrating over yields
| (584) |
Let us sum up (584) over species and also add it with (583). The contribution of the collision integral disappears due to (278), so one obtains
| (585) |
Next, notice that
| (586) |
Also, assuming that , , and are independent of and using (371), one gets
| (587) |
where we also used (366a). Substituting (586) and (587) into (585) leads to (380).
Appendix E Uniqueness of the entropy-preserving distribution
Here, we prove that the Boltzmann–Gibbs distribution is the only distribution for which the entropy density is conserved. According to (C.3), is conserved when
| (588) |
(for all , , and , as well as all and ), where
| (589) |
Let us decompose the vector into components parallel and perpendicular to the vector :
| (590) |
where is a scalar function. (Because the velocities are functions of the momenta, one can as well consider as a function of and .) Due to the presence of the delta function in (588), the contribution of the first term to (588) is zero, so (588) can be written as
| (591) |
By considering this formula for parallel to (and thus perpendicular to ), one finds that . Combined with (589) and (590), this yields
| (592) |
Also, by swapping and , one finds that
| (593) |
Equation (592) yields, in particular, that464646The idea of this argument was brought to author’s attention by G. W. Hammett and is taken from (Landreman, 2017), where it is applied to single-species plasmas with a specific .
| (594a) | |||
| (594b) | |||
where we have assumed some coordinate axes in the momentum and velocity space labeled . Then,
| (595a) | |||
| (595b) | |||
where the derivative with respect to is taken at fixed and at fixed . Due to (592), is continuous for all and . (Here we consider only physical distributions, which are always differentiable.) Then, (595) leads to
| (596) |
By differentiating this with respect to , one obtains
| (597) |
whence (595a) yields
| (598) |
By repeating this argument for other axes and for instead of , one can also extend (598) to
| (599) |
Hence, is actually independent of the velocities; i.e. . Using this along with (593), one also finds that
| (600) |
Let us rewrite (592) as follows:
| (601) |
Here, the left-hand side is independent of and the right-hand side is independent of , so both must be equal to some vector
| (602) |
that is independent of both and . Because , this is equivalent to
| (603a) | |||
| (and similarly for ), where the integration constant is independent of both and . This is supposed to hold for any , so one can also write | |||
| (603b) | |||
where is any other species index. Subtracting equations (603) from each other gives
| (604) |
By differentiating this with respect to , one finds
| (605) |
By differentiating this further with respect to , one obtains . Then, (605) yields , and (604) yields . In other words, the functions , , and are independent of their second index and thus can as well be written as
| (606) |
But then, (600) and (602) also yield and . Therefore, (603) can be written as
| (607) |
which is the Boltzmann–Gibbs distribution (section 8.1). This proves that a plasma that conserves its entropy density necessarily has the Boltzmann–Gibbs distribution.
Appendix F Total momentum and energy
Here, we show that the total momentum and energy in the OC–wave representation equals the total momentum and energy in the particle–field representation.
F.1 Nonrelativistic electrostatic interactions
F.1.1 Momentum
F.1.2 Energy
F.2 Relativistic electromagnetic interactions
F.2.1 Momentum
Let us assume the notation and
| (613) |
Then, using (466) for , one can represent the OC momentum density as follows:
| (614) |
where we substituted (465) and used (314). Next, let us rewrite (468) as
| (615) |
where the last equality is due to Gauss’s law. This gives
| (616) |
Then, using (53) and also (354) for , one obtains
| (617) |
and thus (614) can be written as follows:
| (618) |
where we used (298) and (314) again. For an eikonal wave (471), which has (section 7.4.1), this gives
| (619) |
In case of a broadband spectrum, the same equality applies as well, because contributions of the individual eikonal waves to both left-hand side and the right-hand side are additive. (Alternatively, one can invoke (53) again.) This leads to (467).
F.2.2 Energy
Assuming the notation and using (466) for , one can represent the OC energy density as follows:
Using (610) and (613) for , one further obtains
Using (464) and proceeding as in section F.2.1, one can also cast this as follows:
| (620) |
Now, notice that
| (621) |
where is the oscillating-current density. From Ampere’s law,
| (622) |
Appendix G Selected notation
This paper uses the following notation (also see section 2 for the index convention):
| Symbol | Definition | Explanation |
| placeholder | ||
| complex conjugate | ||
| inverse | ||
| Hermitian adjoint | ||
| inverse Hermitian adjoint | ||
| transpose | ||
| section 4.2 | auxiliary notation | |
| section 6.5 | contribution from the microscopic part | |
| section 6.5 | contribution from the macroscopic part | |
| average part or, for eikonal waves, a quantity evaluated | ||
| on the local wavevector | ||
| oscillatory part | ||
| macroscopic part | ||
| microscopic part | ||
| operator | ||
| time derivative | ||
| (23) | Fourier image | |
| envelope of an eikonal (or monochromatic) wave | ||
| section 2.1.2 | anti-Hermitian part | |
| section 2.1.2 | Hermitian part | |
| partial derivative (but , , ) | ||
| partial derivative with respect to a lower-index quantity | ||
| (149) | auxiliary notation | |
| (116), (327) | convective time derivative | |
| (32) | Poisson bracket on | |
| (56) | Poisson bracket on | |
| commutator | ||
| (1), (59) | inner product on or on | |
| definition | ||
| section 2.1.3 | scalar product | |
| (30) | Moyal product on | |
| (72) | Moyal product on | |
| section 4.2 | times an infinitesimally small positive number | |
| principal-value integral | ||
| eigv | eigenvalue | |
| imaginary part | ||
| (146) | auxiliary notation | |
| real part | ||
| operator corresponding to a Weyl symbol on | ||
| operator corresponding to a Weyl symbol on | ||
| symb | same as or when the two are equal | |
| Weyl symbol of an operator on | ||
| Weyl symbol of an operator on | ||
| sign | ||
| trace | ||
| section 6.7 | part of a collision operator | |
| section 6.6 | particle’s total ponderomotive energy in on-shell waves | |
| (169) | auxiliary notation | |
| (182) | dressing function (since section 5.3) | |
| (202) | dressing function (a part of ) | |
| section 7.1 | dispersion function (one of ) | |
| section 7.1 | th eigenvalue of | |
| table 1 | OC momentum flux density of species | |
| table 1 | wave momentum flux density | |
| section 6.3 | dispersion matrix | |
| (220), (224) | dispersion operator | |
| section 6.3 | vacuum dispersion matrix | |
| (208) | vacuum dispersion operator | |
| (228) | Weyl symbol of | |
| section 6.1 | auxiliary operator | |
| (184), (203) | ponderomotive energy | |
| section 6.1 | generic interaction field | |
| (308) | complexified interaction field | |
| (139) | auxiliary notation | |
| section 6.1 | Weyl symbols of and | |
| section 6.1 | coupling operators | |
| (338), (377) | linear dissipation rate as a function of | |
| section 7.3 | local linear dissipation rate of an eikonal wave | |
| Kronecker symbol or delta function | ||
| section 3.1.1 | geometrical-optics parameter | |
| (456) | dielectric tensor | |
| (423) | parallel and transverse permittivity | |
| section 3.1.1 | small parameter proportional to the oscillation amplitude | |
| section 7.1 | polarization vector (one of ) | |
| section 7.1 | th eigenvector of | |
| eikonal phase | ||
| section 7.2.3 | auxiliary notation | |
| (86) | characteristic inverse scales in and , respectively | |
| charge density of species | ||
| (178) | auxiliary notation | |
| (229) | Weyl symbol of | |
| (229) | coupling operator | |
| (280) | entropy density | |
| (373) | sign of the action density | |
| electrostatic potential | ||
| any field | ||
| coordinate in the frequency space dual to | ||
| local frequency of an eikonal wave | ||
| frequency operator | ||
| (89) | Fourier image of | |
| (77), (82) | Fourier images of and | |
| section 6.8 | collision operator of species | |
| , | (44), (52) | Fourier images of and |
| section 5.1 | Weyl symbol of | |
| (177) | phase-space-diffusion coefficient | |
| (201), (284a) | momentum-diffusion coefficient (part of ) | |
| (168) | auxiliary notation | |
| (118) | diffusion operator on | |
| table 1 | OC energy density of species | |
| table 1 | wave energy density (also see (333) for eikonal waves) | |
| (187), (198) | OC distribution functions | |
| (267) | polarization drag for species | |
| section 4.2 | Weyl symbols of and | |
| (145) | approximation of to the zeroth order in | |
| (109) | effective Green’s operators on | |
| (237) | spectrum of the correlations between and | |
| section 3.2 | Green’s operator on | |
| (318) | action density of an eikonal wave | |
| section 7.4.3 | phase-space action density | |
| (57) | canonical Poisson structure | |
| (319) | action flux density of an eikonal wave | |
| particle Hamiltonian | ||
| (200) | OC Hamiltonian of species | |
| Hilbert space formed by functions on | ||
| Hilbert space formed by functions on | ||
| coordinate in the wavevector space dual to | ||
| wavevector operator on | ||
| (98) | extended Liouvillian (up to a factor ) | |
| (209) | coupling operator | |
| (73) | same as the Poisson bracket on | |
| (32) | same as the Poisson bracket on | |
| (213) | -dependent part of the plasma Lagrangian density | |
| (208) | Lagrangian density of in vacuum | |
| number of components of or of another vector field | ||
| dimension of the extended phase space | ||
| (199) | OC density | |
| big O (‘at most of the order of’) | ||
| table 1 | OC momentum density of species | |
| table 1 | wave momentum density (also see (333) for eikonal waves) | |
| table 1 | OC energy flux density of species | |
| table 1 | wave energy flux density | |
| section 6.8 | symmetrized coefficient in the collision operator | |
| (209) | coupling operator | |
| real axis | ||
| section 7.1 | action integral | |
| (223) | adiabatic action integral | |
| (399) | spectrum of the macroscopic oscillations | |
| (105) | shift operator (see also section 4.1) | |
| (285) | average Wigner function of the macroscopic field | |
| section 7.4.2 | average Wigner matrix of and () | |
| section 3.2 | unperturbed velocity in the space | |
| volume of -dimensional homogeneous plasma | ||
| (88) | Wigner function of | |
| , | (76), (81) | Weyl symbol of (Wigner function or matrix) |
| (87) | density operator on of | |
| , | (75), (80) | density operator on of a given field |
| section 5.6 | Wigner functions of and with as a parameter | |
| (43), (51) | Weyl symbol of (Wigner function or matrix) | |
| (42), (50) | density operator on of a given field | |
| section 6.4 | average Wigner matrix of the microscopic field | |
| coordinate in the extended phase space | ||
| operator of the position in the extended phase space | ||
| (243) | Weyl symbol of | |
| (242) | coupling operators on that enter | |
| differential | ||
| charge of species | ||
| distribution function | ||
| section 4.3 | initial conditions for and | |
| (355), (365) | rescaled phase-space action density | |
| section 7.4.2 | auxiliary notation () | |
| coordinate in the wavevector space dual to | ||
| local wavevector of an eikonal wave | ||
| wavevector operator | ||
| coordinate in the wavevector space dual to | ||
| local spacetime-wavevector of an eikonal wave | ||
| spacetime-wavevector operator | ||
| (16), (17) | eigenvector of corresponding to the eigenvalue | |
| section 4.1.1 | displacement in along unperturbed characteristics | |
| mass of species | ||
| number of spatial dimensions | ||
| number of spacetime dimensions | ||
| coordinate in the momentum space | ||
| position operator corresponding to the coordinate | ||
| coordinate in the wavevector space dual to | ||
| wavevector operator corresponding to the coordinate | ||
| coordinate in the wavevector space dual to | ||
| wavevector operator corresponding to the coordinate | ||
| species index | ||
| time | ||
| time operator | ||
| (92) | oscillating part of the phase-space velocity | |
| (114) | as an operator on | |
| (92) | average velocity in phase space or in physical space, | |
| (192) | or, since section 5.4, OC velocity | |
| (316) | group velocity as a function of | |
| (320) | local group velocity of an eikonal wave | |
| (313) | eikonal-wave frequency as a function of | |
| coordinate in space | ||
| ray coordinate in space | ||
| space-position operator | ||
| coordinate in spacetime | ||
| spacetime-position operator | ||
| (16), (17) | eigenvector of corresponding to the eigenvalue | |
| coordinate in phase space | ||
| phase-space position operator |
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