Introduction

Hydrogen is the most abundant element in the universe. Its properties are crucial for the evolution of stars and the characteristics of the Jovian planets (Jupiter, Saturn, Uranus, and Neptune). Despite the simple composition, its phase diagram is surprisingly complex and has been the topic of numerous experimental and theoretical approaches. Many of them were devoted to the high-temperature phase diagram, which is the subject of this work. At high temperature ($T>5000\,\rm K$), hydrogen is a dense, hot fluid that undergoes considerable structural changes. At low density ( $\rho \le 0.3 \, \rm gcm^{-3}$), one finds a plasma of free electrons and protons, an atomic and a molecular regime, while at high density hydrogen is expected to go into a metallic state. This transition has first been predicted by the pioneering work of Wigner and Huntington (1935) for $T=0$. Several attempts have been made to observe this transition experimentally. Diamond-anvil measurements (Silvera and Pravica, 1998; Mao and Hemley, 1994) have reached pressure up to 100 GPa at room temperature and shock wave experiments (Da Silva, 1997; Collins et al., 1998) achieved up to 390 GPa at much higher temperature ( $T \stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:10^5\,\rm K$). To date, no conclusive observation of metallic hydrogen has been made. However, in the gas gun shock wave experiments by Weir et al. (1996) reaching 140 GPa at $T < 5000\,\rm K$, a drop an increase in the conductivity over 4 orders of magnitude has been found, which is an indication that a metallic or nearly metallic state has been reached.

The simplicity of hydrogen also provides an uncluttered problem for theoretical consideration and computational methods. The main challenge lies in the complex interplay of different physical effects. Any theoretical approach must deal with strong coupling, degeneracy effects as well as with bound states. In this work, we apply path integral Monte Carlo simulations (PIMC) (Ceperley, 1995), a first principles simulation technique that describes all the mentioned effects. The main purpose of this work is to provide accurate numerical results for the equilibrium properties of hot, dense hydrogen. This topic has be studied extensively using various form of free energy models (see section 1.4). They inevitably require a number of uncontrolled approximations including fit parameters but have the advantage of low numerical requirements and that additional physical observables can be estimated that are not available in PIMC. With our PIMC simulation, we will provide an accurate equation of state table. One purpose would be that free energy model can be fit to it.

Furthermore, we use our equation of state to calculate the deuterium hugoniot, which can be directly compared to the above mentioned shock wave experiments. Since they provide this first direct measurements in this regime, the comparison is of particular significance to this work and will be discussed in detail in section .