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(The question is by a friend of mine, who is a physicist.)

Background: There have been many research papers about electrostatic waves in a degenerate electron gas in a neutralizing background, and there are two widely known theoretical approaches to study them:

  1. The simplest approach is to use the Vlasov equation and assume the unperturbed distribution function to be the Fermi-Dirac distribution, thereby ignoring all quantum mechanical effects except for the shape of the distribution. The dispersion relation is thus obtained by linearizing the Vlasov equation with respect to perturbations.

  2. A more sophisticated approach is to use the quantum mechanical analogue of the Vlasov equation, namely the Wigner equation. The dielectric response function derived by that approach is called the Lindhard dielectric function, and the dispersion relation is obtained by equating it to zero. Physically, this approach is an upgrade to include quantum tunneling.

Both approaches are collisionless, i.e., mean-field, and are thus strictly valid only in the limit of infinitely small quantum coupling parameter Γ, which is a measure of the non-ideality of a quantum plasma and is, with the accuracy of a numerical factor of the order of unity, the square of the ratio of the plasma frequency multiplied by h-bar to the Fermi energy. In this limit (Γ→0), quantum tunneling effects vanish, so in this limit the two approaches coincide and give identical dispersion relations. The difference arises only at finite, non-zero Γ, but at finite Γ there must also be some contribution of collisions, which are ignored by the nature of the above approaches.

Question: How does it make sense to include quantum tunneling, but ignore collisions? In other words, why do scientists believe or presume that at small but finite Γ, the collisional correction to the dispersion relation of electrostatic waves in a degenerate electron gas is less important than the one due to quantum tunneling?

To illustrate my question, here is an example of using the second approach. The link is to an arXiv preprint, but the work was published in Phys. Rev. B and cited almost 500 times. Have a look at Eq. 38. That's the dispersion relation derived from the Wigner equation and explicitly expanded in a power series of Γ. My question is why the effects ignored by the mean-field approximation are believed, expected, or presumed not to substantially affect the first-order and higher-order terms of that expansion.

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  • $\begingroup$ Your question explained in the paper you cite before and after equation (9). $\endgroup$ – Alex Trounev Jun 30 at 23:05
  • $\begingroup$ @AlexTrounev : >> Your question explained in the paper you cite before and after equation (9) << That explanation is clearly wrong. In particular, it basically says that at T=0 the Wigner equation is an exact model for a degenerate electron gas regardless of the value of the quantum coupling parameter, as if there were no effects beyond the mean-field approximation at T=0. That's nonsensical, and you can look in Phys. Rev. B 4, 4312 (1971) to see a dielectric function of a degenerate electron gas derived by including effects that are beyond the Wigner equation model. $\endgroup$ – Mitsuko Jul 1 at 0:11
  • $\begingroup$ Even so this is just a model with a limiting application. They tried to develop mixture from hydrodynamics and quantum mechanics. If you like this approach, then you can add viscous term to the hydrodynamic part to take in to account collisions. $\endgroup$ – Alex Trounev Jul 1 at 14:58
  • $\begingroup$ @AlexTrounev : Just adding a "viscous term" would be merely a phenomenological approach, and, besides, what collision rate do I use in that term? How can I relate it with the parameters of the plasma? How can I estimate it? That's what I'm looking for. $\endgroup$ – Mitsuko Jul 2 at 6:40
  • $\begingroup$ According to this meta post, this question was posted to be “intentionally boring” and “uninteresting”. $\endgroup$ – G. Smith Jul 4 at 17:50

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