Molecular dynamics studies of electron-ion temperature equilibration in hydrogen plasmas within the coupled-mode regime

We use classical molecular dynamics (MD) to study electron-ion temperature equilibration in two-component plasmas in regimes for which the presence of coupled collective modes has been predicted to substantively reduce the equilibration rate. Guided by previous kinetic theory work, we examine hydrogen plasmas at a density of $n={10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and ${10}^7\mathrm{K}<T_e<{10}^9\mathrm{K}$. The nonequilibrium classical MD simulations are performed with interparticle interactions modeled by quantum statistical potentials (QSPs). Our MD results indicate (i) a large effect from time-varying potential energy, which we quantify by appealing to an adiabatic two-temperature equation of state, and (ii) a notable deviation in the energy equilibration rate when compared to calculations from classical Lenard-Balescu theory including the QSPs. In particular, it is shown that the energy equilibration rates from MD are more similar to those of the theory when coupled modes are neglected. We suggest possible reasons for this surprising result and propose directions of further research along these lines.

Figure 1

Effective proton temperature $T_i=\frac{2}{3}\langle E_i^{\text{kin}}\rangle$ versus time for a single $\mathit{NVE}$ run of $8.192\times {10}^5$ particles (protons plus electrons). Oscillations in $T_i$ correspond to underdamped transient relaxations in the ion-ion radial distribution function. Insets show the pair distribution function $g_{ii}(r,t)$, showing signs of crystalline order at early times. Long-range correlations are lost within 1.5 fs. The last inset shows the target conditions for this case, $T_p={10}^5\mathrm{K}$ and $T_e={10}^7\mathrm{K}$, which occur at $t=1.74\mathrm{fs}$.

Figure 2

Temperature histories are shown in light gray for three separate microcanonical MD simulations differing in total energy by a few eV per electron-ion pair. The temperature curves appear noisy because of intrinsic short-time fluctuations that are easily averaged over. Linear least-squares fits to portions of the time series are shown as black lines; $dT/dt$ is computed analytically from these fits. The fitted curves each reach the target ion temperature ($T_p={10}^5\mathrm{K}$) at times marked by $\times$. They reach the target electron temperature ($T_e={10}^7\mathrm{K}$) at times marked with $+$. By design, these times coincide for the middle microcanonical system.

Figure 3

The $k$ integrands $F\left(k\right)$ of various Lenard-Balescu expressions for the equilibration rate for the quantum hydrogen plasma with $n=1.0\times {10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and $T_e={10}^7\mathrm{K}$. The red curve shows $F\left(k\right)$ computed by Eq. (3), using the naive evaluation scheme outlined in Ref. [11]; the green curve shows $F\left(k\right)$ from Eq. (3), but computed with the pole-correction scheme outlined here; the blue curve shows $F\left(k\right)$ from the FGR expression of Eq. (10). The integrals under the two CM Lenard-Balescu $F\left(k\right)$ curves shown here (red and green) are equal to within $1\%$ for this case.

Figure 4

Instantaneous energy transfer rate per $e^{-}$ for hydrogen plasmas with $n={10}^{26}{\mathrm{cm}}^{-3}$ and $T_i={10}^5\mathrm{K}$, as a function of $T_e$. The solid red curve shows the quantum Coulomb prediction using the CM Lenard-Balescu approach; the solid blue curve shows the quantum Coulomb prediction with the FGR prescription; the dot-dashed red curve shows the classical QSP prediction using the CM Lenard-Balescu approach; the dot-dashed blue curve shows the classical QSP prediction with the FGR prescription.

Figure 5

Instantaneous $dT/dt$ calculated from the classical QSP Lenard-Balescu approach and extracted from MD using the identical QSPs for hydrogen plasmas with $n={10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and $T_e$ as shown on the $x$ axis. The uppermost, thick, dashed black curve is for LB FGR theory and the lowest, thick, solid black curve is for the LB CM theory. The disconnected black point corresponds to a local-field-corrected LB CM calculation using two-temperature HNC. Only the ion-ion local-field correction is included. Red symbols with error bars and thin solid lines correspond to MD for $d{T}_i/dt$ and blue symbols with error bars and thin dashed lines correspond to MD for $-d{T}_e/dt$. The green symbols with error bars and thin dotted lines show the MD results for the time rate of change of total energy, scaled by 2/3 from Table 1 to correspond to a temperature, i.e., $\frac{2}{3}d{E}_e/dt$ (see also Table 2). (the relation $d{E}_i/dt=-d{E}_e/dt$ holds exactly).

Figure 6

Plots of $g\left(r\right)$ from MD and the HNC approximation for the $T_e={10}^7\mathrm{K}$ case.

Figure 7

Close-up of the ion-ion pair distribution functions for $T_i={10}^5\mathrm{K}$ and a series of electron temperatures.

Figure 8

Instantaneous MD ion-ion pair distribution function $g_{ii}\left(r\right)$ for the hydrogen plasma with $n={10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and $T_e={10}^7\mathrm{K}$ (solid blue curve, essentially obscured by the green curve), compared with the ion OCP $g_{ii}\left(r\right)$ for the identical $n$, but with $T_i=6.777\times {10}^4\mathrm{K}$ (solid green curve). Also shown is the MD result for the same plasma, but with the mass of the protons artificially increased by a factor of 8 (dashed black curve).

Figure 9

The $k$ integrands $F\left(k\right)$ of Eq. (3) for the quantum Coulomb hydrogen plasma with $n={10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and $T_e={10}^7\mathrm{K}$. Here the Lindhard ${\chi }_i^0(k,\omega )$ is replaced by the corresponding function ${\tilde{\chi }}_i^0(k,\omega +i{\gamma }_i)$ of Refs. [66, 67]. Each curve represents a different choice of ${\gamma }_i$, expressed in atomic units (hartrees; see the legend). Note that the $F\left(k\right)$ for ${\gamma }_i={10}^{-12}$ hartree (solid black curve, not shown) and the $F\left(k\right)$ for ${\gamma }_i={10}^{-6}$ hartree (red curve) are essentially identical.

Figure 10

Effective Coulomb logarithms $\ln {\lambda }_{ei}$ for the quantum Coulomb hydrogen plasma with $n={10}^{26}{\mathrm{cm}}^{-3}, T_i={10}^5\mathrm{K}$, and $T_e={10}^7\mathrm{K}$. Here the Lindhard ${\chi }_i^0(k,\omega )$ in Eq. (3) is replaced by the corresponding function ${\tilde{\chi }}_i^0(k,\omega +i{\gamma }_i)$ of Refs. [66, 67]. Each point on the curve represents a different choice of ${\gamma }_i$. The CM LB RPA result is represented by the nearly flat line on the bottom left of the plot; the FGR result is at $\ln {\lambda }_{ei}\sim 0.95$.

Figure 11

Ion kinetic energy distributions from MD (purple), together with the closest-fit Maxwellian distribution.

Figure 12

The ${\mathcal{E}}_i$-dependent residual, together with a fit comprised (chiefly) of a single Laguerre polynomial.