Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited

Abstract

In this paper, we revisit the hydrodynamic limit of the Langmuir wave dispersion relation based on the Wigner-Poisson model in connection with that obtained directly from the original Lindhard dielectric function based on the random-phase-approximation. It is observed that the (fourth-order) expansion of the exact Lindhard dielectric constant correctly reduces to the hydrodynamic dispersion relation with an additional term of fourth-order, beside that caused by the quantum diffraction effect. It is also revealed that the generalized Lindhard dielectric theory accounts for the recently discovered Shukla-Eliasson attractive potential (SEAP). However, the expansion of the exact Lindhard static dielectric function leads to a k 4 term of different magnitude than that obtained from the linearized quantum hydrodynamics model. It is shown that a correction factor of 1/9 should be included in the term arising from the quantum Bohm potential of the momentum balance equation in fluid model in order for a correct plasma dielectric response treatment. Finally, it is observed that the long-range oscillatory screening potential (Friedel oscillations) of type cosð2k F rÞ=r 3 , which is a consequence of the divergence of the dielectric function at point k ¼ 2k F in a quantum plasma, arises due to the finiteness of the Fermi-wavenumber and is smeared out in the limit of very high electron number-densities, typical of white dwarfs and neutron stars. In the very low electron number-density regime, typical of semiconductors and metals, where the Friedel oscillation wavelength becomes much larger compared to the interparticle distances, the SEAP appears with a much deeper potential valley. It is remarked that the fourth-order approximate Lindhard dielectric constant approaches that of the linearized quantum hydrodynamic in the limit if very high electron number-density. By evaluation of the imaginary part of the Lindhard dielectric function, it is shown that the Landau-damping region in x-k plane increases dramatically by increase of the electron number-density. V C 2015 AIP Publishing LLC. [http://dx.

Statically screened ion potential and Bohm potential in a quantum plasma

“…The comparison with the two other potentials, the ones of Stanton and Murillo (SM) [19] and of Akbari-Moghanjoughi (AM) [18] which do not exhibit a comparable minimum reveals the origin of this discrepancy (see below). The SM (and AM) potential exhibits very good agreement with the RPA potential at T = 0 indicating that it correctly captures the long-wavelength properties of the RPA, in contrast to the SE potential.…”

“…For a comprehensive comparison we consider a broad density range, covering 6 orders of magnitude with the three values of the Brueckner parameter, r S = 25.27, 2.3, 0.25 (similar values were studied in Ref. [18]). The first observation, cf.…”

“…Quantum hydrodynamics, as used in the plasma physics community, is a zero-temperature theory, so the results of SE [13] for the effective ion potential and the recent correction by Akbari-Moghanjoughi [18] are restricted to the ground state. At the same time, for relevant applications to dense plasmas, warm dense matter or laser plasmas, finite temperature effects are usually non-negligible.…”

Theoretical foundations of quantum hydrodynamics for plasmas

“…Manfredi and Haas [31] derived the Fermi pressure and a quantum correction in the form of the Bohm potential using a semi-classical Hartree ansatz for the N -electron wave functions with identical amplitude for all singleelectron orbitals [38]. However, in order to reach agreement with the results of the more fundamental kinetic theory in its simplest form -the random phase approximation (RPA) -both, the Fermi pressure and the Bohm potential have to be "corrected" by constant pre-factors [35,[43][44][45]. Similarly, in plasmonics, the QHD theory is used with one or, sometimes, two fitting parameters corresponding to the prefactors of the Fermi pressure and the Bohm potential, but with an additional exchange correlation potential contribution, which is valid only in the static case.…”

Nonlocal orbital-free kinetic pressure tensors for the Fermi gas

“…The previous example hides a very serious pathology: the kinetic/microscopic dynamic behavior of fermionic systems is not captured by the functional 1. From this point of view, the limitations of QHD are surprisingly rare brought into attention 17,[29][30][31] . The general recipe is to use the approximation [1] with λ = 1/9 for equilibrium configurations while, in the linear regime, the functional is modified to:…”