Do hydrodynamic models based on time-independent density functional theory misestimate exchange effects? Comparison with kinetic theory for electrostatic waves

Abstract

We have extended previous quantum kinetic results to compute the exchange correction to the electrostatic electron susceptibility for arbitrary frequencies and wavenumbers in the low temperature limit. This has allowed us to make a general comparison with a much used hydrodynamic expression, based on density functional theory, for exchange effects. For low phase velocities, as for ion-acoustic waves, wave-particle interaction leads to a strong enhancement of the exchange correction and the hydrodynamic result is smaller by an order of magnitude. The hydrodynamic expression gives a useful approximation when the phase velocity is 2.5 times the Fermi velocity. If this condition is not fulfilled, the hydrodynamical theory gives misleading results. We discuss the implications of our results for model choice for quantum plasmas, especially regarding particle dispersive effects. arXiv:1809.05423v2 [physics.plasm-ph]

Quantum Kinetic Theory of Plasmas

“…The relative importance of exchange effects in plasmas is proportional to the parameter 12,13,[15][16][17][18], where the characteristic kinetic energy E k is given by E k = k B T F for a degenerate plasma (with T F > T ) and by E k = k B T for the non-degenerate case (with T > T F ). The above suggests that exchange effects are as important as the more basic particle dispersive effects, described by the Wigner-Moyal equation of the previous section.…”

“…Another basic, and important, phenomenon is the degeneracy effect [1-5, 10, 11], entering when the density becomes high enough to make the Fermi temperature significant in relation to the thermodynamic temperature. A related phenomena, also dependent on the antisymmetry of the manybody wavefunction of electrons, but considerably more complicated to model mathematically, is the exchange interaction [12][13][14][15][16][17][18], whose importance probably has been undervalued in the recent plasma research literature. Upgrading from the Schrödinger Hamiltonian to the Pauli-Hamiltonian, the spin dynamics enter the picture [19][20][21][22][23][24][25], with physics such as the magnetic dipole force, spin precession, and (spin) magnetization currents.…”

Exchange fluid model derived from quantum kinetic theory for plasmas

“…For instance, for high-frequency waves, the DFT-based hydrodynamical model predicts the wrong sign of the exchange contribution, for short wavelengths. [18] For these reasons, it is an essential task to derive exchange macroscopic models starting right from the kinetic theory and evaluating the moments dynamics. The price for the choice of the moments method focused on the time-evolution of average quantities like number density, fluid velocity field, pressure dyad, etc.…”

Blow-up solitons at the nonlinear stage of the two-stream instability in quantum plasmas

“…[19]. Recently, it was also shown [20] that the hydrodynamic model gives a decent approximation of the exchange effects for sufficiently long wavelengths.…”

“…Singularities in the denominators in eqs. (19) and (20) are avoided, as usual, using Landau's rule by replacing ω → ω + i0 and then (ω − kv)…”