# 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{sup 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 cos(2k{sub F}r)/r{sup 3}, which is a consequence of the divergence of the dielectric function at point k = 2k{sub F} in a quantum plasma, arises due to the finiteness of the Fermi-wavenumber and ismore »

- Authors:

- Faculty of Sciences, Department of Physics, Azarbaijan Shahid Madani University, 51745-406 Tabriz (Iran, Islamic Republic of)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 22408056

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Plasmas; Journal Volume: 22; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; BOHM CRITERION; DIELECTRIC MATERIALS; DIFFRACTION; DISPERSION RELATIONS; ELECTRON DENSITY; FLOW MODELS; HYDRODYNAMICS; LANDAU DAMPING; METALS; NEUTRON STARS; OSCILLATIONS; PERMITTIVITY; QUANTUM PLASMA; RANDOM PHASE APPROXIMATION; SEMICONDUCTOR MATERIALS; WAVELENGTHS; WHITE DWARF STARS

### Citation Formats

```
Akbari-Moghanjoughi, M., and International Centre for Advanced Studies in Physical Sciences and Institute for Theoretical Physics, Ruhr University Bochum, D-44780 Bochum.
```*Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited*. United States: N. p., 2015.
Web. doi:10.1063/1.4907167.

```
Akbari-Moghanjoughi, M., & International Centre for Advanced Studies in Physical Sciences and Institute for Theoretical Physics, Ruhr University Bochum, D-44780 Bochum.
```*Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited*. United States. doi:10.1063/1.4907167.

```
Akbari-Moghanjoughi, M., and International Centre for Advanced Studies in Physical Sciences and Institute for Theoretical Physics, Ruhr University Bochum, D-44780 Bochum. Sun .
"Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited". United States.
doi:10.1063/1.4907167.
```

```
@article{osti_22408056,
```

title = {Hydrodynamic limit of Wigner-Poisson kinetic theory: Revisited},

author = {Akbari-Moghanjoughi, M. and International Centre for Advanced Studies in Physical Sciences and Institute for Theoretical Physics, Ruhr University Bochum, D-44780 Bochum},

abstractNote = {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{sup 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 cos(2k{sub F}r)/r{sup 3}, which is a consequence of the divergence of the dielectric function at point k = 2k{sub 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 ω-k plane increases dramatically by increase of the electron number-density.},

doi = {10.1063/1.4907167},

journal = {Physics of Plasmas},

number = 2,

volume = 22,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}