High-frequency response of classical strongly coupled plasmas

The dynamic structure factor (DSF) of the Yukawa system is here obtained with highly converged molecular dynamics (MD) over the entire liquid phase. The data provide a rigorous test of theoretical models of ion-acoustic wave-dispersion relations, the intermediate scattering function, and the high-frequency response. We compare our MD results with seven diverse models, finding good agreement among those that enforce the three basic sum rules for dispersion properties, although one of the models has previously unreported spurious peaks. The MD simulations reveal that at intermediate frequencies $\omega$, the high-frequency response of the DSF follows a power law, going approximately as ${\omega }^{-p}$, where $p>0$, and $p$ shows nontrivial dependencies on the wave vector $q$ and the plasma parameters $\kappa$ and $\mathrm{\Gamma }$. In contrast, among the seven comparison models, the predicted high-frequency response is found to be independent of $\{q,\kappa ,\mathrm{\Gamma }\}$. This high-frequency power suggests a useful fitting form. In addition, these results expose limitations of several models and, moreover, suggest that some approaches are difficult or impossible to extend because of the lack of finite moments. We also find the double-plasmon resonance peak in our MD simulations that none of the theoretical models predicts.

Figure 1

The ($\kappa , \mathrm{\Gamma }$) pairs for which the DSF was computed using MD simulations. The solid blue and dashed green lines are the lower and upper limits of the liquid regime, respectively, and red dots indicate data points used in this study. Note that the points were chosen to span the strongly coupled liquid regime, and they roughly follow contours of constant effective coupling.

Figure 2

$S\left(q\right)$ and $I\left(q\right)$ calculated from the MD and HNCB results. $S\left(q\right)$ is shown in the left panels, and $I\left(q\right)$ is shown in the right panels. The MD results are shown with solid lines, the HNCB results are shown with dashed lines, and dotted lines indicate the HNCB results with Gaussian random noise added to mimic the MD simulations. Panels (a) and (c) show that the static structure factor from the MD and the HNCB results are in close agreement, and the addition of noise results in even better agreement. Panels (b) and (d) show that $I\left(q\right)$ obtained from the MD simulations is very different from $I\left(q\right)$ from the HNCB results at large $q$. Adding noise to the HNCB results yields an $I\left(q\right)$ similar to the MD results at large $q$, revealing the large sensitivity to statistical averaging; moreover, these results show that the addition of noise can either reduce or increase $I\left(q\right)$.

Figure 3

The DSF for three different numbers of particles: $N=2000,5000$, and $10000$. All simulations were performed with the plasma parameters ($\kappa =3, \mathrm{\Gamma }=1500$), and the wave number was set to either $q=1.86$ (left panel) or $q=3.40$ (right panel). The results of all three simulations are in good agreement, indicating convergence of the simulations at a modest particle number. The raw data are shown as individual points in both panels. The solid ($N=10000$), dashed ($N=5000$), and dotted ($N=2000$) lines are smoothed using a Savitzky-Golay filter.

Figure 4

The DSF for three different numbers of time steps. All simulations were performed with plasma parameters ($\kappa =3, \mathrm{\Gamma }=1500$), and the wave number was set to either $q=0.54$ (left panel) or $q=1.81$ (right panel). The results of all three sets of simulations exhibit very close agreement. These results reveal the importance of long runs for capturing the correct peak value of the DSF.

Figure 5

The DSF for three different choices of the time step. All simulations were performed with the plasma parameters ($\kappa =3, \mathrm{\Gamma }=1500$), and the wave number was set to either $q=0.54$ (left panel) or $q=1.81$ (right panel). The DSFs for all cases are in very close agreement.

Figure 6

Dispersion relations obtained from the theoretical models and the MD data. The plasma parameters are ($\kappa =0, \mathrm{\Gamma }=150$) in panel (a) and ($\kappa =2, \mathrm{\Gamma }=31$) in panel (b). The red band reflects damping and corresponds to the full-width-half-maximum of the MD results. The failure of the M model reflects the importance of including a $g\left(r\right)$ dependency; similarly, the lower frequencies in the MNS model reflect the lack of the high-frequency sum rule in that model. Results for the ion-acoustic wave in panel (b) show that all models except the M model predict the MD results well.

Figure 7

The DSF $S(q,\omega )$ obtained from the theoretical models and the MD data. The plasma parameters are ($\kappa =1, \mathrm{\Gamma }=217$), and the wave numbers are either $q=1.27$ (left panel) or $q=2.71$ (right panel). The HK and Mermin models fail to predict the MD results. In particular, the HK model predicts two peaks, which suggests that their third-order approximations may not converge fast enough.

Figure 8

The DSF obtained from the HK model. The plasma parameters are ($\kappa =0, \mathrm{\Gamma }=150$), and the wave numbers are $q=1.45,1.63,2.17,2.35$ in panels (a) through (d), respectively. The parameter ${\eta }_3$ was obtained from the MD results using the least squares fitting method. The HK model shows two peaks when the wave number is small; as the wave number increases, the second peak (on the right) weakens, while the left peak dominates the spectrum and yields a more reasonable result.

Figure 9

The DSF from the seven theoretical models and our MD results for different parameter values (upper panels) and the asymptotic power of the DSF from the MD results at the mid- to high-frequency limit (lower panels). Panels (a) and (b) show the DSF with ($\kappa =1, \mathrm{\Gamma }=72$) and ($\kappa =2, \mathrm{\Gamma }=158$), respectively, and panels (c) and (d) show the asymptotic power of the DSF from the MD results at high frequency for $\kappa =0$ and $\kappa =3$, respectively. We denote a portion of our MD results in square shapes to indicate a spurious result, ${\omega }^{-2}$, which arises from limitations when using numerical Fourier transforms. At intermediate frequencies, the power $p$ is obtainable from MD simulations and varies from $p=-\infty$ to $p\sim -6$. Table 1 shows that none of the theoretical models can predict the asymptotic power variation seen in the MD simulation results.

Figure 10

Log-log plot of the intermediate scattering function (ISF) $F(q,t)$ calculated using several theoretical models and from our MD data for different plasma parameter values. Panel (a) shows the ISF at the lower boundary of the liquid regime ($\kappa =0, \mathrm{\Gamma }=10$), and panel (b) shows the ISF near the upper boundary of the liquid regime ($\kappa =3, \mathrm{\Gamma }=1510$). The wave number is set to $q=5.6$ in both cases. Only the Mermin model shows fairly good agreement with the MD results (shown in red) at large $t$; all of the other models decay quickly.

Figure 11

A comparison of the DSF otained from the MD data and the fitting form for the DSF in (17). The MD DSF and the fitting form for the DSF are shown for moderate coupling regimes in panels (a) and (c) and for strong coupling regimes in panels (c) and (d); the wave number was set to $q=1.99$ in all cases.

Figure 12

The real and imaginary parts of $\chi (q,\omega )$ obtained from the MD and from the fitting form in (17). Panels (a) and (b) show moderate coupling regimes, and panels (c) and (d) show strong coupling regimes; the wave number was set to $q=1.99$ in all cases. These results show that the fitting form in (17) gives good agreement with the MD results.

Figure 13

The double-plasmon resonance peak in the DSF obtained from the MD results. The DSF is shown for several wave numbers for each of four different sets of plasma parameters. Panel (a) shows that the OCP exhibits the double-plasmon resonance peak at $\omega =2$. The peak amplitude of the double-plasmon resonance peak decreases as the wave number increases. Panels (b) through (d) suggest that the relative strength of the double-plasmon resonance peak decreases (larger width and smaller peak) with larger $\kappa$.

Figure 14

The DSF calculated from MD data obtained using different simulation-box sizes. The plasma parameters are $\kappa =0.1$ and $\mathrm{\Gamma }=10$. The properties of the second, double-plasmon resonance peak do not change with the size of the simulation box. The amplitude of the second peak decreases as the wave number increases, as expected.