Molecular dynamics simulations and generalized Lenard-Balescu calculations of electron-ion temperature equilibration in plasmas

We study the problem of electron-ion temperature equilibration in plasmas. We consider pure H at various densities and temperatures and Ar-doped H at temperatures high enough so that the Ar is fully ionized. Two theoretical approaches are used: classical molecular dynamics (MD) with statistical two-body potentials and a generalized Lenard-Balescu (GLB) theory capable of treating multicomponent weakly coupled plasmas. The GLB is used in two modes: (1) with the quantum dielectric response in the random-phase approximation (RPA) together with the pure Coulomb interaction and (2) with the classical ($\hslash \to 0$) dielectric response (both with and without local-field corrections) together with the statistical potentials. We find that the MD results are described very well by classical GLB including the statistical potentials and without local-field corrections (RPA only); worse agreement is found when static local-field effects are included, in contradiction to the classical pure-Coulomb case with like charges. The results of the various approaches are all in excellent agreement with pure-Coulomb quantum GLB when the temperature is high enough. In addition, we show that classical calculations with statistical potentials derived from the exact quantum two-body density matrix produce results in far better agreement with pure-Coulomb quantum GLB than classical calculations performed with older existing statistical potentials.

Figure 1

Raw data (black dots) for the simulation of temperature equilibration in a hydrogen plasma with $n={10}^{25}$/cc and initial electron and proton temperatures of 1000 and 800 eV, respectively. Dunn-Broyles+Deutch SPs were used in the MD simulation. Solid blue and red curves show the fit to the data which is used to extract the value of ${\tau }_{ei}$.

Figure 2

Values of ${\tau }_{ei}$ extracted from MD runs for a hydrogen plasma ($n={10}^{25}$/cc, $T_e=1000$ eV, $T_p=800$ eV, Dunn-Broyles+Deutch potentials) as a function of the MD time step. The multiple points for each time step indicate statistically independent replicas.

Figure 3

Values of the initial ($t=0$) $d{T}_p/dt$ for H along an isochore at a density, $n={10}^{25}$/cc, for which $T_p(t=0)=0.8T_e(t=0)$. Results of GLB calculations of different types (see text) and MD results are shown.

Figure 4

$kF\left(k\right)$ (see text for definition) vs $k{r}_S$ for various ${\tau }_{ei}$ H simulations along the $n={10}^{25}$/cc isochore. These are the results of GLB calculations in the quantum-Coulomb (QC) mode. LFCs ($G_{ij}$) are set to zero.

Figure 5

$kF\left(k\right)$ vs $k{r}_S$ for $T_e=1$ keV and $T_p=0.8$ keV for H. Both QC and CS GLB results are shown. The statistical potential is of the Dunn-Broyles variety. LFCs ($G_{ij}$) are set to zero.

Figure 6

$kF\left(k\right)$ vs $k{r}_S$ for $T_e=1$ keV and $T_p=0.8$ keV for H. Both QC and CS GLB results are shown. Statistical potentials of both the Dunn-Broyles and modified-Kelbg varieties are considered. LFCs ($G_{ij}$) are set to zero.

Figure 7

$kF\left(k\right)$ vs $k{r}_S$ with and without LFCs for $T_e=1$ keV and $T_p=0.8$ keV for H. Classical-statistical potential GLB results are shown. The statistical potential is of the Dunn-Broyles variety.

Figure 8

$1-G_{ei}\left(k\right)$ vs $k{r}_S$ for $T_e=1$ keV and $T_p=0.8$ keV for H as calculated by HNC with the Dunn-Broyles potential.

Figure 9

Differences (including LFC vs not including LFCs) between absolute values of $d{T}_e/dt$ for H along the $T_e=1$ keV isotherm, where $T_p=0.8$ keV. The results are those of CS GLB with the Dunn-Broyles potential.

Figure 10

Radial distribution functions, $g_{\alpha \beta }\left(r\right)$, for H at $n={10}^{25}$/cc and $T_e=T_p=1$ keV. Both HNC (lines) and MD (points) results are shown, each using the Dunn-Broyles potential.

Figure 11

HNC results for $1-G_{ei}\left(k\right)$ for the classical positron-proton system at $n={10}^{25}$/cc, performed with the pure Coulomb interaction.

Figure 12

Effective $\ln {\lambda }_{ei}$ for the positron-proton system ($n={10}^{25}$/cc) vs the dimensionless plasma coupling, $g\equiv b_0/{\lambda }_{\mathrm{Debye}}$. Results are from classical GLB with the Coulomb interaction (black points) and the Dunn-Broyles statistical potential (red points). The green curve is a fit to the MD results reported in Ref. [10].

Figure 13

MD results (colored thick curves) and results of LS (black thin curves) for the scaled-mass ($\alpha =0.01$) Ar-doped H plasma. Species densities are $n_p={10}^{25}$/cc, $n_{\text{Ar}}={10}^{24}$/cc, and $n_e=2.8\times {10}^{25}$/cc. $Z_{\text{Ar}}$ is fixed at $+18$. MD was performed with the Dunn-Broyles potential, and the LS Coulomb logarithms were chosen according to the prescription outlined in Sec. 5.2 of Ref. [35].

Figure 14

LS results for the physical-mass ($\alpha =1$) Ar-doped H plasma with Coulomb logarithms all set to unity. Species densities are $n_p={10}^{25}$/cc, $n_{\text{Ar}}={10}^{24}$/cc, and $n_e=2.8\times {10}^{25}$/cc. $Z_{\text{Ar}}$ is fixed at $+18$.

Figure 15

LS results for the scaled-mass ($\alpha =0.01$) Ar-doped H plasma with Coulomb logarithms all set to unity. Species densities are: $n_p={10}^{25}$/cc, $n_{\text{Ar}}={10}^{24}$/cc, and $n_e=2.8\times {10}^{25}$/cc. $Z_{\text{Ar}}$ is fixed at $+18$.

Figure 16

The GLB $k$-integrand divided by the LS prefactor, $F\left(k\right)$, multiplied by $k$ for both QC and SP (Dunn-Broyles) cases for hydrogen at $n={10}^{25}$/cc, $T\equiv T_e=$ 100 000 eV, and $T_p=0.8T_e$. The vertical dotted lines are located at $k=a{k}_D$ and $b{k}_{\mathrm{th}}$, respectively, and define low-$k$, middle-$k$, and high-$k$ regions (see text). Between the dotted lines, $F\left(k\right)\sim 1/k$. As $T$ is increased further, the middle-$k$ region expands at the expense of the outer regions. QC and SP $F\left(k\right)$ only differ in the high-$k$ region.

Figure 17

The reciprocal of the fractional difference between QC and SP (Dunn-Broyles) rates, ${\nu }_{ie}\left(\mathrm{QC}\right)/[{\nu }_{ie}\left(\mathrm{QC}\right)-{\nu }_{ie}\left(\mathrm{SP}\right)]$, as a function of $T=T_e$ for the $n={10}^{25}$/cc isochore of hydrogen as computed by GLB ($\mathrm{LFCs}=0$). The semilog plot is linear for high $T$, indicating that the reciprocal of the fractional difference asymptotes to $\ln \left(T\right)$.