Molecular dynamics study of the thermodynamics and transport coefficients of hard hyperspheres in six and seven dimensions

Molecular dynamics (MD) simulations are performed for six- and seven-dimensional hard-hypersphere fluids. The equation of state, velocity autocorrelation function, self-diffusion coefficient, shear viscosity, and thermal conductivity are determined as a function of density. The molecular dynamics results for the equation of state are found to be in excellent agreement with values obtained from theoretical approaches and previous MD simulations in seven dimensions. The short-time behavior of the velocity autocorrelation function is well described by the Enskog exponential approximation. The Enskog predictions for the self-diffusion coefficient and the viscosity agree fairly well with the simulation data at low densities, but underestimate these quantities at higher densities. Data for the thermal conductivity are in fine agreement with Enskog theory for all densities and dimensions studied.

Figure 1

Equation of state of hard-hypersphere fluids in six and seven dimensions. The symbols are the results of MD simulations. The circles are from the present work for $d=6$, and the triangles are from the present work for $d=7$. The squares are for MD simulation from Ref. 10 for $d=7$. The solid lines are the $Z_{[4,5]}$ Padé approximants, and dotted lines are the $Z_{[5,4]}$ Padé approximants.

Figure 2

The velocity autocorrelation function for $d=6$ as a function of time for $\rho =0.5$, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 (from right to left).

Figure 3

The short-time behavior of the velocity autocorrelation function for $d=6$: $\rho =0.6$ (circles), $\rho =1.0$ (squares), $\rho =1.4$ (triangles), and $\rho =1.8$ (diamonds). The lines are the Enskog predictions of Eq. (12).

Figure 4

The velocity autocorrelation function for $d=7$ as a function of time $\rho =0.6$, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, and 1.8 (from right to left). The dashed lines at $\rho =0.7$ and 0.9 are for $N=2187$, whereas all the other states are for $N=4000$.

Figure 5

The short-time behavior of the velocity autocorrelation function for $d=7$: $\rho =0.6$ (downward triangles), $\rho =0.9$ with $N=2187$ (diamonds), $\rho =1.2$ (upward triangles), $\rho =1.5$ (squares), and $\rho =1.8$ (circles). The lines are the predictions of Enskog theory [see Eq. (12)].

Figure 6

Variation of the self-diffusion coefficient for $d=3$ (circles, Ref. 11), $d=4$ (squares, Ref. 11), $d=5$ (upward triangles, Ref. 11), $d=6$ (diamonds, this work), and $d=7$ (downward triangles, this work) dimensions as a function of the Enskog time.

Figure 7

Log-log plot of the variation of the viscosity with $t_{\mathrm{avg}}$ for hard-hyperspherical fluids in dimensions 3–7. Symbols are from molecular dynamics simulations, and the curves are the predictions of Enskog theory [Eq. (16)]: $d=3$, circles and solid line; $d=4$, squares and long-dashed line; $d=5$, upward-triangles and dotted line; $d=6$, diamonds and dashed-dotted line; $d=7$, downward-triangles and short-dashed lines.

Figure 8

Log-log plot of the variation of the thermal conductivity with $t_{\mathrm{avg}}$ for hard-hyperspherical fluids in dimensions 3–7. Symbols are from molecular dynamics simulations and the curves are the predictions of Enskog theory [Eq. (17)]: $d=3$, circles and solid line; $d=4$, squares and long-dashed line; $d=5$, upward-triangles and dotted line; $d=6$, diamonds and dashed-dotted line; $d=7$, downward-triangles and short-dashed lines.