Equation of State

Figure: Comparison of pressure vs. temperature for $r_s=2$ from the ideal Fermi gas, the Debye model, the Padé approximations in the chemical picture (PACH) by Ebeling and Richert (1985a), the EOS by Saumon and Chabrier (1992) and from PIMC simulations with VDM nodes, $\tau^{-1}=2\cdot10^6\,\rm K$, $n_{\rm A}=1$, and $n_{\rm E}=2$.

Figure: Internal energy per atom vs. temperature for $r_s=2$ from the methods in Fig.  and the activity expansion (Actex) by Rogers (1990).

Figure: Pressure vs. temperature for $r_s=2$ from the EOS by Saumon and Chabrier (1992), DFT-MD by Lenosky et al. (2000), linear mixing model by Ross (1998), wave packet MD by Knaup et al. (2000), from the fluid variational theory by Juranek and Redmer (2000) and from PIMC simulations with VDM nodes, $\tau^{-1}=2\cdot10^6\,\rm K$, $n_{\rm A}=1$, and $n_{\rm E}=2$.

Figure: Internal energy per atom vs. temperature for $r_s=2$ from the methods in Fig. 

Figure: Internal energy per atom vs. hydrogen density from PIMC simulation ($\circ$) with $\tau^{-1}=10^6\,\rm K$, $n_{\rm A}=1$, and $n_{\rm E}=2$ using free particle nodes are compared with the equation of state by Saumon and Chabrier (1992) (dashed lines).

Figure: Pressure vs. hydrogen density from PIMC simulations ($\circ$) with $\tau^{-1}=10^6\,\rm K$, $n_{\rm A}=1$, and $n_{\rm E}=2$ using free particle nodes are compared with the equation of state by Saumon and Chabrier (1992) (dash lines). $\Diamond$ show PIMC results with pressure correction $\Delta p= -\frac{n}{3}\;0.7\,$eV discussed section . The thin dashed line denotes the pressure of an ideal H$_2$ gas at $T=5000\,\rm K$.

The equation of state (EOS) is central interest in theoretical plasma physics since it is the basic thermodynamic quantity. It is also the key property to test the accuracy of different approaches to hot, dense hydrogen including analytical theories and numerical models. If the predicted EOS seems reasonable, one can have more confidence in all derived properties. The complete EOS data from our PIMC simulations can be found in tables in App. D.

First, we picked the density corresponding to $r_s=2$ and compared pressure and energy as a function of temperature. We separated the analysis in the high temperature behavior $k_BT \stackrel{\scriptstyle>}{\scriptscriptstyle\sim}\:1\,\rm Ry$ where thermal excitations dominate, and the low temperature regime $k_BT \stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:1\,\rm Ry$ where Coulomb effects and bound states are most relevant.

The pressure and energy in the high temperature regime are shown in Figs.  and . In the high temperature limit, kinetic effects are dominant and the hydrogen plasma behaves like a gas of non-interacting protons and electrons. The leading corrections are given by Debye screening effects (see App. C) that scale with the coupling parameter $\Gamma^{3/2}$. For small values of $\Gamma$, the (fully ionized) Debye model is a reliable approximation. One finds deviations of less than $20\%$ in pressure and energy for $\Gamma < 0.5$ at $r_s=2$. For higher values of $\Gamma$, quantum effects such as the formation of bound state at low density and degeneracy effects at high density limit the validity. Various extensions to the Debye model have been made, see (Ebeling et al., 1976). However, at sufficiently high $\Gamma$, the Debye model overestimates the screening drastically and leads to unphysically low, even negative pressures.

Figs.  and include a comparison with EOS model by Saumon and Chabrier (1992), the Padé approximations in the chemical picture (PACH) by Ebeling and Richert (1985a), and the activity expansion (Actex) by Rogers (1990). The observed deviations are a result of how interaction and degeneracy effects are treated in those models.

In Figs.  and , we compare pressure and energy in the low temperature regime. The EOS properties are determined by a strong coupling combined with a high degeneracy ( $T_F=145\,000\,\rm K$). The comparison includes EOS by Saumon and Chabrier (1992), the DFT-MD by Lenosky et al. (2000), the fluid variational theory by Juranek and Redmer (2000), the wave-packet MD by Knaup et al. (2000), the linear mixing model by Ross (1998), and PIMC simulations using free particle nodes and $\tau^{-1}=2\cdot10^6\,\rm K$. For this density, we found the best agreement of our results with the work by DFT-MD by Lenosky et al. (2000).

Finally, we present a comparison of internal energy and pressure as a function of density for different temperatures. Fig.  shows a reasonably good agreement in the energy between the EOS by Saumon and Chabrier (1992) and our simulation results over a broad range of densities. PIMC energies for low temperatures and densities are consistently lower by the order 1 or 2 eV per atom. For low density and high temperature, relatively large deviations were observed, which is surprising because in this regime, one expects both methods to work very well.

Studying the pressure as function of density in Fig. , one finds remarkably good agreement for $T=125\,000$ and $31\,250\,\rm K$. For 5000 K, the PIMC pressure is far too high because hydrogen under these conditions is a weakly interacting molecular gas with possibly a very small degree of dissociation caused by entropy effects. The PIMC pressure is about twice the expected value, which is a result of the inaccuracies in the pair density matrices discussed in section . After those corrections have been applied one finds better but not perfect agreement. New calculations with improved density matrices remain to be done.