{'fraction_non_alphanumeric': 0.04239897370109044, 'fraction_numerical': 0.019820397690827454, 'mean_word_length': 4.407908428720083, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Exchange effects in a cold plasma

We have studied the exchange corrections to linear electrostatic wave propagation in a plasma using a quantum kinetic formalism. Specifically we have considered the zero temperature limit. In order to simplify the calculations we have focused on the long wavelength limit, i.e. wavelengths much longer than the de Broglie wavelength. For the case of ion-acoustic waves we have calculated the exchange correction both to the damping rate and the real part of the frequency. For Langmuir waves the frequency shift due to exchange effects is found. Our results are compared with the frequency shifts deduced from commonly used exchange potentials which are computed from density functional theory.


I. INTRODUCTION
Recently much work has been devoted to quantum plasmas, see e.g. the books and review articles [1][2][3][4]. The research is motivated by an interest in e.g. quantum wells [5], spintronics [6], plasmonics [7], laser plasma interaction [8], astrophysical applications [9] or general theory development [10]. The theoretical descriptions range from hydrodynamical equations (e.g. Refs. [1,4,11]) to quantum kinetic models (e.g. Refs. [3,10,12,13]] and field theoretical approaches (e.g. Ref. [14]). Most models include the physical effects of particle dispersion and Fermi pressure, and in some cases the magnetic dipole force and magnetization due to the electron spin [10,12,15,16]. An important effect that is sometimes accounted for (e.g. Refs. [11,[17][18][19][20][21][22][23]) but often overlooked is the exchange effects resulting from the total antisymmetry of the electron wavefunction. A popular approach to include the effects of exchange interaction has been to apply density dependent potentials deduced from density functional theory (DFT) [11,[19][20][21][22]. An advantage with this is that the resulting fluid models becomes comparatively simple once the exchange potentials are established. As a consequence problems involving both high-frequency dynamics [11,19] as well as low-frequency (ion-acoustic) dynamics [19,20] can be addressed in a straightforward way, also for nonlinear problems [19]. A drawback is that the calculation of DFT potentials typically involve approximations (e.g. the local density approximation (LDA)) whose accuracy can be hard to estimate beforehand. Thus there is a general need to validate results derived from DFT by independent methods.
In the present paper we calculate the exchange contribution to the ion-acoustic dispersion relation using quantum kinetic theory derived from first principles. Previous works along this line [17,18] have assumed the ordering T ≫ T F (where T is the temperature and T F is the Fermi temperature), which has prevented a direct comparison with results based on DFT potentials that have considered the opposite ordering. In this paper we focus on the low-temperature limit T ≪ T F and evaluate the exchange contribution to the ion-acoustic dispersion relation in the Hartree-Fock approximation to first order in perturbation theory. We deduce that the effects of the exchange term is to increase the phase-velocity of the ion-acoustic mode and to increase the linear damping rate (which is due to wave-particle interaction). Moreover, as a confirmation of the correctness of the quantum kinetic formalism, we compare results from our quantum kinetic formalism with previous results for the exchange contribution of high-frequency Langmuir waves. In this latter case we recover the results of Refs. [24][25][26] exactly within the outlined approximation scheme.
Finally we compare our findings with results based on commonly used DFT potentials [11,19,20,22]. As the DFT potentials are incorporated in a fluid formalism, no comparison can be made for the damping due to wave-particle interaction. In general we find a qualitative agreement. In particular the frequency shift in the different formalisms has the same scaling with the parameters (density and wavenumber) both for ion-acoustic and Langmuir waves. However, there is a discrepancy concerning a numerical factor. This is discussed in more detail in the final section of the manuscript.

II. THE EXCHANGE CORRECTION AT T = 0 K
In a previous paper, Ref. [17], the exchange contribution to the evolution equation of the Wigner function was derived (see Eq. (7) of Ref. [17]). The correction was obtained by writing down the first equation in the BBGKY-hierarchy and writing the two-particle density matrix as a anti-symmetric product of one-particle density matrices. The treatment was here limited to electrostatic fields. For a generalization allowing for electromagnetic fields, see Ref. [18]. Eq. (7) of Ref. [17] was further simplified by considering a plasma without spin polarization and summing over all spin states, and also by taking the long scale limit (where the macroscopic scale length is assumed to be much longer than the de Broglie wavelength). The long scale assumption implies that the Wigner function reduces to the Vlasov limit, in which case the Wigner function becomes similar to a classical distribution function. This means that the evolution equation reduces to the Vlasov equation with a correction term due to the exchange effects. The resulting expression (Eq. (12) of Ref. [17]) with the exchange correction written in the right hand side reads Here f denotes the electron distribution function, E is the electrical field, −e and m e is the electron charge and mass respectively, h = 2πh is Planck's constant and we use x and r for position vectors and p and q for momentum vectors. Furthermore, ∂ i x ≡ ∂ /∂ x i and analogously for ∂ i p and ∂ i r . An arrow above an operator indicates in which direction it acts. We have also used the summation convention so that a sum over in-dices occurring twice in a term is understood. Finally V (r) = e 2 /4πε 0 |r| is the Coulomb potential. Here we will use Eq.
(1) to consider linear wave propagation treating the exchange term on the right hand side perturbatively. Thus we linearize Eq. (1) and make a plane wave ansatz To first order in a long wavelength expansion, the linearized version of Eq. (1) then reduces to Previously the correction to the dispersion relations for ion acoustic waves (see Ref. [17]) and for Langmuir waves (see Ref. [18]) has been found considering a Maxwellian background, by making the approximation f 1 ≈f 1 in the exchange term.
the solution when exchange effects are neglected.
Here we will study the same problems but for the case when T = 0 K. As we will demonstrate the problem is analytically tractable owing to the simple form of the background distribution function f 0 which is now given by where p F =h(3π 2 n 0 ) 1/3 is the Fermi momentum (we will also use the notation v F = p F /m e for the electron Fermi velocity below), and n 0 is the equilibrium electron number density.
In particular, taking f 1 =f 1 in the exchange term, combining Eqs. (2) and (3) with Poisson's equation and computing the ion charge density classically we obtain where is the combined ion and electron susceptibility in the absence of exchange effects. Here we have neglected a contribution from a small but finite ion Fermi velocity (i.e. we have used that v Fi ≪ ω/k). In the expression above ω e and ω i denote the electron and ion plasma frequency respectively. The exchange correction is given by By changing integration variables to u 1 = p + q, u 2 = p − q, with the Jacobian equal to 1/8, we can use the properties of the Dirac delta functions to reduce the problem to an integral over two spheres. Introducing spherical coordinates u i = u i (cos ϕ i sin θ i , sin ϕ i sin θ i , cos θ i ) we explicitly get χ 1 = m 2 e q 4 I ′ /8π 6 ε 2 0 k 2h4 with I ′ given by where ψ is the angle between u 1 and u 2 and α = ω/(kv F ) is the dimensionless phase velocity. By the spherical law of cosines cos ψ = cos θ 1 cos θ 2 + sin θ 1 sin θ 2 cos(ϕ 1 − ϕ 2 ). By changing variables to ϕ 1 ,φ = ϕ 1 −ϕ 2 , the ϕ 1 integral is trivial. Theφ integral can then be performed, giving a factor π/(a 2 − b 2 ) 1/2 where a = 1 − cosθ 1 cosθ 2 and b = sin θ 1 sin θ 2 . Since a 2 − b 2 = (cos θ 1 − cosθ 2 ) 2 , we have that I ′ = −2 −1 π 2 I where I is the double integral At this point, no approximations have been made beyond those leading to the evolution equation (2), taking f 1 =f 1 , and T = 0 K. For these approximations the exchange correction is therefore proportional to (9) for all frequency regimes. Thus Poisson's equation, with exchange corrections, is where χ 0 is given by Eq. (5). From now on we must treat the low-frequency ion-acoustic case separately from the highfrequency Langmuir case in order to simplify the expression for I. a. Ion-acoustic waves The integral I can be evaluated analytically in terms of α with computer algebra software. We will consider the quasi-neutral limit (ω ≪ ω i ) where the left hand side of Eq. (10) is negligible. For ion-acoustic waves we can then make the approximation that α = m e /(3m i ) when solving the integral (9). It is also possible to plug in the value of α and evaluate the integral numerically. The dispersion relation for ion-acoustic waves with exchange corrections is then To the best of our knowledge this result has not been derived before. b. Langmuir waves In the high-frequency regime ω/k ≫ v F , an expansion of I in powers of v F k/ω can be made. With the lowest order non-vanishing correction, the dispersion relation in this regime is This is in exact agreement with previous calculations using several different methods, see Refs. [24][25][26].

III. DFT COMPARISON
Eq. (9) together with the specific results (11) and (12) are the main results of the present paper. The exact agreement of (12) with previous results also confirms the validity of (9). A strength of the quantum kinetic formalism is that it follows from first principles, and that it can address wave particle interaction, such as the enhanced Landau damping rate found in Eq. (11). However, a drawback is that the formalism is difficult to use for more complicated problems. Thus there is a need to develop theories that are easier to apply in a more general context. One such possibility is offered by DFT. A commonly used exchange potential (see e.g. Refs. [11,19,20,22]) computed from DFT is that has been used for low-frequency ion-acoustic phenomena (e.g. Refs. [19,20] ) as well as high-frequency Langmuir waves (e.g. Refs. [11,19]). Including the contribution from Eq. (13) in the electron momentum equation (see e.g. Ref. [11]) and treating the term as a small perturbation when calculating the ion-acoustic dispersion relation, we get a qualitative agreement with Eq. (11). In particular the ion-acoustic frequency is decreased and this change scale ash 2 ω 2 e /m 2 e v 4 F , in accordance with (11). The numerical value of the frequency shift deviates rather significantly from our result, however. We note that if we make the adjustment 0.985 → 6.52 of the numerical pre-factor in (13) we would get agreement with the real part of the frequency in Eq. (11). Here we stress that the wave damping cannot be compared with the DFT formalism, as this requires a quantum kinetic framework.
The same comparison can be made in the high-frequency regime. Thus we again include the contribution from (13) in the electron momentum equation, calculating the frequency shift of the Langmuir dispersion relation. Also here we have qualitative agreement, i.e. the Langmuir frequency increases proportional to k 2 , and the scaling with density is in accordance with the factorh 2 ω 2 e /m 2 e v 4 F . This time the numerical accuracy is better, although not perfect, and we need to make a less significant substitution of the numerical factor in Eq. (13), 0.985 → 1.23 in order to get agreement with our result (12). Thus we conclude that the DFT exchange potential Eq. (13) are in qualitative agreement with the results obtained here, but that the numerical accuracy is better for Langmuir waves than for ion-acoustic waves.

IV. SUMMARY AND CONCLUSION
In the present paper we have computed the exchange contribution to the ion-acoustic dispersion relation in a plasma, using a quantum kinetic formalism. The validity of our approach have been confirmed by comparison with similar results for high-frequency Langmuir waves [24,25], in which case we get exact agreement for the exchange correction. This is to be expected, as our formalism as well as that of Refs. [24,25] is based on first principles. While the quantum kinetic formalism is of fundamental importance, it has the drawback of producing complicated formulas (cf. Eq. (1)) that can only be solved perturbatively. In fact even a perturbative treatment is far from straightforward. Thus there is a need for a formalism that can be used in more complex situations. Such a possibility is offered by DFT, where the resulting exchange potentials can be used in fluid theories, and straightforwardly applied to a number of linear and nonlinear problems [11,19,20,22]. However, as the computation of DFT potentials typically involves approximations (e.g. the local density approximation (LDA)) whose accuracy is unknown, there is a need to evaluate DFT potentials against independent methods. A key motivation in the present paper has been to evaluate the DFT potentials used in Refs. [11,19,20,22] against the quantum kinetic results computed in our formalism. This comparison reveals that there is a reasonable qualitative agreement. In both cases the relative magnitude of the exchange term scales ash 2 ω 2 e /m 2 e v 4 F , and both the Langmuir and ionacoustic wave frequencies decreases due to the exchange interaction, in agreement with Eqs. (11) and (12). The numerical value of this frequency shift differ somewhat, however. To some extent this can be fixed by replacing the numerical pre-factor in the exchange potential. If this approach is chosen, we note that different substitutions must be used for lowfrequency and high-frequency phenomena (i.e. 0.985 → 6.52 in the former case, 0.985 → 1.23 in the latter case). It is an open question to what extent this result is robust (i.e. whether the same numerical coefficient is a good approximation for different problems at a given frequency scale), or if more advanced expressions for the exchange potentials are needed to cover a broad spectrum of problems.
The fact that there is not a perfect agreement between the DFT exchange potential and the quantum kinetic theory should not be overly surprising. The DFT potentials used in Refs. [11,19,20,22] have not necessarily been optimized for the situation we have been studying. In our case the fields are dynamically varying (such that time-dependent density functional theory (TDDFT) applies), and the system is weakly collisional such that the collision-free Vlasov equation holds to leading order. The systems of relevance for the present study generally include plasmas with a high density and a modest temperature T ≪ T F . In a laboratory context this applies to e.g. solid state plasmas and inertial confinement fusion plasmas before the heating stage, and in an astrophysical context this include e.g. white dwarf stars. In that case the perturbed distribution function will be comparatively far from thermodynamic equilibrium, and approaches which do not include the full quantum kinetic details run the risk of losing information. Nevertheless an approach based on density functional theory is a valuable option, in particular for problems where the approach used here becomes too complicated.