Effect of correlation on viscosity and diffusion in molecular-dynamics simulations

In the warm dense matter (WDM) regime, material properties like diffusion and viscosity can be obtained from lengthy quantum molecular dynamics simulations, where the quantum behavior of the electrons is represented using either Kohn-Sham or orbital-free density functional theory. To reduce the simulation duration, we fit the time dependence of the autocorrelation functions (ACFs) and then use the fit to find values of the diffusion and viscosity. This fitting procedure avoids noise in the long time behavior of the ACFs. We present a detailed analysis of the functional form used to fit the ACFs, which is always a more efficient means to obtain mass transport properties. We use the fits to estimate the statistical error of the transport properties. We apply this methodology to a dense correlated plasma of copper and a mixture of carbon and hydrogen. Both systems show structure in their ACFs and exhibit multiple time scales. The mixture contains different structural forms of the ACFs for each component in the mixture.

Figure 1

Computer-generated data from an OFMD run on Cu at $T=100$ eV and $\rho =67.4$ g/${\mathrm{cm}}^3$. (a) The green “x” are the numerical data, while the light-red solid line is the fit to Eq. (10). (b) The numerically integrated data represented by green “x” and the fit to the analytic integral of Eq. (10) is the light-red solid line. In both panels, the solid black line is the example of a single exponential fit, which is given by $a_i\equiv 0$ $\forall$ $i>0$.

Figure 2

(a) Pair correlation function for Cu at $T=100$ eV at various densities. The strong peak around $r=2.2{\mathrm{a}}_0$ is indicative of transient, collective motion. It is this correlation that gives rise to structure in the VACF. (b) The VACF for various densities of 1 (solid green), 10 (dashed purple), 30 (dotted light-red), and 67.4 (dash-dot violet) g/${\mathrm{cm}}^3$, and all are at $T=10$ eV. There is a clear emergence of structure as the density goes up.

Figure 3

(a) VACF for C in the CH mixture using Eq. (12) with $i=2$, and (b) VACF for H in the CH mixture using Eq. (10). The green “x” marks are the simulation data, and solid light-red line is the fits using the respective equations. The solid black line is the single exponential fit, $a_i\equiv 0$ $\forall$ $i>0$ in Eq. (12).

Figure 4

VACF for H (a) and C (b) for densities 1 (solid green), 4 (dash purple), 8 (short-dash light-red), 12 (dotted light-blue), and 16 (dash-dot violet) g/${\mathrm{cm}}^3$, all at $T=10$ eV.

Figure 5

(a) VACF for H in CH at 1 and 16 $\text{g}/{\mathrm{cm}}^3$. ${\tau }_0^{1g/cc}=33.6$ fs and ${\tau }_0^{16g/cc}=12.6$ fs. In (b) and (c) we plot the integrated value and fit of the 1 and 16 $\text{g}/{\mathrm{cm}}^3$ simulation, respectively. The 16 g/${\mathrm{cm}}^3$ fit is to Eq. (12) with $i=1$ while 1 g/${\mathrm{cm}}^3$ is a fit to Eq. (12) with $a_i\equiv 0$ $\forall$ $i>0$.

Figure 6

The STACF for Cu at $T=100$ eV and $\rho =67.4$ $\text{g}/{\mathrm{cm}}^3$. (a) The STACF simulation data (denoted by a green “x”) and the solid light-red line is the fit using Eq. (17). (b) The numerically integrated simulation data with the fit (solid light-red line) the fit to the analytic integral of Eq. (17). The two values obtained are statistically the same.

Figure 7

(a) STACF simulation data and (b) numerical integral (given by green “x”) for the CH system at $T=10$ eV and $\rho =16$ $\text{g}/{\mathrm{cm}}^3$. The fits are to Eq. (17) and its analytic integral. The numbers are statistically the same.