Determination of the shear viscosity of the one-component plasma

The shear viscosity coefficient of the one-component plasma is calculated with unprecedented accuracy using equilibrium molecular dynamics simulations and the Green-Kubo relation. Numerical and statistical uncertainties and their mitigation for improving accuracy are analyzed. In the weakly coupled regime, our results agree with the Landau-Spitzer prediction. In the moderately and strongly coupled regimes, our results are found in good agreement with recent results obtained for the Yukawa one-component plasma using nonequilibrium molecular dynamics. A practical formula is provided for evaluating the viscosity coefficient across coupling regimes, from the weakly coupled regime to solidification threshold. The results are used to test theoretical predictions of the viscosity coefficients found in the literature.

Figure 1

Shear-viscosity coefficient of the Coulomb OCP $\left(\kappa =0\right)$ obtained by different authors [6, 7, 8, 9] using equilibrium molecular dynamics and the Green-Kubo relation. Results obtained using nonequilibrium molecular dynamics are compiled in Ref. [5].

Figure 2

Illustration of the calculation with equilibrium MD of the Green-Kubo expression (4) for the shear viscosity coefficient. Shown are results for the OCP with $\kappa =0$ and for three different values of the coupling parameter $\Gamma$: $\Gamma =0.5$ (upper row), $\Gamma =30$ (middle row), and $\Gamma =120$ (lower row). Results for the (dimensionless) shear-stress autocorrelation function ${\sum }_{x\ne y}^3\langle {\sigma }_{xy}\left(t\right){\sigma }_{xy}\left(0\right)\rangle$ obtained using Eq. (13) are shown in the left-hand column. Corresponding (dimensionless) cumulated sums $\eta \left(t\right)$ defined by Eq. (15) are shown in the right-hand column. In each figure, the results corresponding to five different simulation lengths are shown (in $1/{\omega }_p$ units); the other simulation parameters are listed in Table 1. In the left-hand side figures, the curves have been shifted vertically for clarity (the horizontal dashed lines corresponds to the shifted zero of the vertical axis).

Figure 3

Illustration of the convergence of the sum rule (16) with the length of the simulation $t_{\mathrm{run}}$ for $\Gamma =1$ (upper panel), $\Gamma =20$ (middle panel), and $\Gamma =175$ (lower panel). In all cases $\kappa =0$. Dashed lines represent Eq. (16) evaluated with the $g\left(r\right)$ obtained from the MD simulation, while the dots represents results of direct calculations of $J_{xy}\left(0\right)$ from the stress tensor.

Figure 4

Bulk modulus of the Coulomb OCP as a function of $\Gamma$. The vertical axis shows the dimensionless quantity ${\tilde{J}}_{xy}\left(0\right)=\frac{J_{xy}\left(0\right)}{{\left[m{\left(a{\omega }_p\right)}^2\right]}^2}=\frac{N}{{\left(3\Gamma \right)}^2}\frac{G}{n{k}_{B}T}$.

Figure 5

Illustration of the convergence of the detailed components of the sum rule (16) with the length of the simulation $t_{\mathrm{run}}$ for $\Gamma =2$ and $\kappa =0$. Dashed lines represent the exact value, and the dots represents results of direct calculations from the components of the stress tensor. Top panel: Kinetic-kinetic term (20). Middle panel: sr potential-potential term (23). bottom panel: Kinetic-potential term (25).

Figure 6

Shear viscosity coefficient across the fluid phase of the one-component plasma at $\kappa =0$ (blue open dots) and at $\kappa =2$ (black full dots) obtained with the equilibrium MD simulations described in the text. The lines between the dots are included to guide the eyes. At $\kappa =2$, the red dots show the nonequilibrium MD results of Donkó and Hartmann [5].

Figure 7

Comparison of the MD data (dots) with the fitting formula Eq. (34). The red line shows the full expression (34); the green dashed line shows the LS limit ${\eta }^{*}\left(\Gamma \right)=\frac{a}{{\Gamma }^{5/2}ln\left(\frac{b}{{\Gamma }^{3/2}}\right)}$; the blue line shows Eq. (31), i.e., ${\eta }^{*}\left(\Gamma \right)=\frac{a}{{\Gamma }^{5/2}ln(1+\frac{b}{{\Gamma }^{3/2}})}$; and the black dashed line shows the formula of Bastea [6].

Figure 8

Comparison of the MD results for the shear viscosity coefficient with the corrected Landau-Spitzer prediction discussed in Sec. 3, the theory of Vieillefosse-Hansen, the kinetic theories of Wallenborn-Baus and of Tanaka-Ichimaru, and the effective potential theory of Baalrud-Daligault. See the main text for a detailed comparison.

Figure 9

Comparison between the local field correction obtained from the hypernetted chain (HNC) approximation and Eq. (39) for weakly coupled OCP.

Figure 10

Contributions to the viscosity coefficient computed from MD for $\kappa =0$ (for finite $\kappa$ values, see Ref. [2]): kinetic (squares), potential (triangles), and total (circles). Also shown is the prediction of the effective potential theory (diamonds) and Landau-Spitzer theory (green line).