Sound speed in Yukawa one-component plasmas across coupling regimes

A many-body system of charged particles interacting via a pairwise Yukawa potential, the so-called Yukawa one-component plasma (YOCP), is a good approximation for a variety of physical systems. Such systems are completely characterized by two parameters: the screening parameter, $\kappa$, and the nominal coupling strength, $\mathrm{\Gamma }$. It is well known that the collective spectrum of the YOCP is governed by a longitudinal acoustic mode, both in the weakly and strongly coupled regimes. In the long-wavelength limit, the linear term in the dispersion (i.e., $\omega =sk)$ defines the sound speed $s$. We study the evolution of this latter quantity from the weak- through the strong-coupling regimes by analyzing the dynamic structure function $S(\mathbf{k},\omega )$ in the low-frequency domain. Depending on the values of $\mathrm{\Gamma }$ and $\kappa$ and $w=s/v_{\mathrm{th}}$ (i.e., the ratio between the phase velocity of the wave and the thermal speed of the particles), we identify five domains in the $(\kappa ,\mathrm{\Gamma })$ parameter space in which the physical behavior of the YOCP exhibits different features. The competing physical processes are the collective Coulomb-like versus binary-collision-dominated behavior and the individual particle motion versus quasilocalization. Our principal tool of investigation is molecular dynamics (MD) computer simulation from which we obtain $S(\mathbf{k},\omega )$. Recent improvements in the simulation technique have allowed us to obtain a large body of high-quality data in the range $\mathrm{\Gamma }=\{0.1-10000\}$ and $\kappa =\{0.5-5\}$. The theoretical results based on various models are compared in order to see which one provides the most cogent physical description and the best agreement with MD data in the different domains.

Figure 1

Left: $\mathrm{\Gamma }\text{−}\kappa$ parameter space. Right: ${\mathrm{\Gamma }}_{\mathrm{eff}}\text{−}\kappa$ parameter space. The boundaries are defined in the text. The black dots indicate all the simulations performed.

Figure 2

Plots of $S(\mathbf{k},\omega )$ obtained from MD simulation at the lowest $ka$ value. Left: $\mathrm{\Gamma }\text{−}\kappa$ values pertaining to Domain 1. Right: Evolution of the minimum in $S(\mathbf{k},\omega )$ as the system enters Domain 3.

Figure 3

(a)–(c) Color maps of $S(\mathbf{k},\omega )$, calculated using Eqs. (15) and (21), showing the dispersion (or lack thereof) of the longitudinal acoustic mode; the dashed lines indicate the acoustic mode $\omega \left(k\right)=s_{\mathrm{RPA}}k$, where $s_{\mathrm{RPA}}$ is given by Eq. (19). (d)–(f) Line plots of $S(\mathbf{k},\omega )$ at three $ka$ values showing the presence (or lack thereof) of the acoustic peak. The $\mathrm{\Gamma },\kappa$ values are indicated in the insets. Panels (a) and (d) belong to Domain 1, (b) and (e) are on the boundary between Domain 1 and Domain 2, and (c) and (f) belong to Domain 2.

Figure 4

Comparison between RPA sound (19) and fluid sound speeds (22) for an adiabatic $\gamma =1$, isothermal $\gamma =5/3$, process. $\gamma =3$ corresponds to the sound speed from Eq. (20). The KT line is the sound speed calculated in Ref. [16]. The dashed vertical lines represent the boundaries between the Domains. $\kappa$ values are indicated in the top left corner of each plot.

Figure 5

Comparison between quartic fit (solid lines) and MD simulations (diamonds) for two $\kappa$ values. The values of $\kappa$ are indicated in the top left corner of each plot. For clarity, only the first few $\mathrm{\Gamma }$ values are indicated as the lines become densely packed at higher $\mathrm{\Gamma }$. The missing labels are $\mathrm{\Gamma }=\{10,20,40,60,80,100,120,150,200\}$

Figure 6

Plots comparing $S(\mathbf{k},\omega )$ obtained from MD simulations (solid lines) and from RPA calculations Eq. (21) (dashed lines), at the three lowest $ka$ values: $ka=\left\{0.1289,0.2578,0.3867\right\}$. The arrow indicates the direction of increasing $ka$. $\mathrm{\Gamma }$ and $\kappa$ values are indicated in the top left corner of each plot.

Figure 7

Comparison between theoretical models and MD simulations (black squares) with 5% error bars. The blue solid line (RPA) corresponds to Eq. (19), the green dashed line (Hydro) to (22), the red dotted line (KT) to Ref. [16], cyan circles (QLCA) to Eq. (34), and magenta diamonds (Feynman ansatz) to Eq. (27). $\kappa$ values are indicated in the insets.

Figure 8

Same as Fig. 7, but for $\kappa =2.0,3.0$. In this plot, error bars indicate a 10% error.

Figure 9

Polar plot comparing the lattice sound speeds and the QLCA speeds. $\theta$ represents the polar and $\varphi$ the azimuthal angles. Inner lines refer to an fcc and outer lines to a bcc crystalline structure.

Figure 10

Comparison between the relative (right) HWHM obtained from the RPA $S(\mathbf{k},\omega )$ and the ratio of the imaginary over the real part of the complex solution of $\varepsilon (k,\omega )=0$. $\kappa =0.5$.

Figure 11

Damping coefficient extracted from fits to MD simulations at the two lowest $ka$ values. The top left panel shows also the exponential decay of the (Landau) damping obtained from the complex solution of $\varepsilon (k,\omega )=0$. $\kappa$ values are indicated in the top left corner of each plot.

Figure 12

$\sigma$ parameter extracted from fits to MD simulations at the lowest $ka$ value.