# Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma

## Abstract

We present a numerical solution of the quantum Lenard-Balescu equation using a spectral method, namely an expansion in Laguerre polynomials. This method exactly conserves both particles and kinetic energy and facilitates the integration over the dielectric function. To demonstrate the method, we solve the equilibration problem for a spatially homogeneous one-component plasma with various initial conditions. Unlike the more usual Landau/Fokker-Planck system, this method requires no input Coulomb logarithm; the logarithmic terms in the collision integral arise naturally from the equation along with the non-logarithmic order-unity terms. The spectral method can also be used to solve the Landau equation and a quantum version of the Landau equation in which the integration over the wavenumber requires only a lower cutoff. We solve these problems as well and compare them with the full Lenard-Balescu solution in the weak-coupling limit. Finally, we discuss the possible generalization of this method to include spatial inhomogeneity and velocity anisotropy.

- Authors:

- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Univ. of California, Los Angeles, CA (United States)
- Univ. Autonoma de Barcelona (Spain)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1399709

- Alternate Identifier(s):
- OSTI ID: 1327556

- Report Number(s):
- LLNL-JRNL-687277

Journal ID: ISSN 1070-664X; TRN: US1703090

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 23; Journal Issue: 9; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION

### Citation Formats

```
Scullard, Christian R., Belt, Andrew P., Fennell, Susan C., Janković, Marija R., Ng, Nathan, Serna, Susana, and Graziani, Frank R.
```*Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma*. United States: N. p., 2016.
Web. doi:10.1063/1.4963254.

```
Scullard, Christian R., Belt, Andrew P., Fennell, Susan C., Janković, Marija R., Ng, Nathan, Serna, Susana, & Graziani, Frank R.
```*Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma*. United States. doi:10.1063/1.4963254.

```
Scullard, Christian R., Belt, Andrew P., Fennell, Susan C., Janković, Marija R., Ng, Nathan, Serna, Susana, and Graziani, Frank R. Thu .
"Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma". United States.
doi:10.1063/1.4963254. https://www.osti.gov/servlets/purl/1399709.
```

```
@article{osti_1399709,
```

title = {Numerical solution of the quantum Lenard-Balescu equation for a non-degenerate one-component plasma},

author = {Scullard, Christian R. and Belt, Andrew P. and Fennell, Susan C. and Janković, Marija R. and Ng, Nathan and Serna, Susana and Graziani, Frank R.},

abstractNote = {We present a numerical solution of the quantum Lenard-Balescu equation using a spectral method, namely an expansion in Laguerre polynomials. This method exactly conserves both particles and kinetic energy and facilitates the integration over the dielectric function. To demonstrate the method, we solve the equilibration problem for a spatially homogeneous one-component plasma with various initial conditions. Unlike the more usual Landau/Fokker-Planck system, this method requires no input Coulomb logarithm; the logarithmic terms in the collision integral arise naturally from the equation along with the non-logarithmic order-unity terms. The spectral method can also be used to solve the Landau equation and a quantum version of the Landau equation in which the integration over the wavenumber requires only a lower cutoff. We solve these problems as well and compare them with the full Lenard-Balescu solution in the weak-coupling limit. Finally, we discuss the possible generalization of this method to include spatial inhomogeneity and velocity anisotropy.},

doi = {10.1063/1.4963254},

journal = {Physics of Plasmas},

number = 9,

volume = 23,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 2016},

month = {Thu Sep 01 00:00:00 EDT 2016}

}

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