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Path Integral Monte Carlo Simulations of Hot Dense Hydrogen

Burkhard Militzer

Thesis for the degree of Doctor of Philosophy in Physics in the Graduate College
of the University of Illinois at Urbana-Champaign, 2000.

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Path integral Monte Carlo (PIMC) simulations are a powerful computational method to study interacting quantum systems at finite temperature. In this work, PIMC has been applied to study the equilibrium properties of hot, dense hydrogen in the temperature and density range of $5000 \leq T \leq 10^6 \,\rm K$ and $10^{-3} \leq \rho \leq
2.7 \,\rm gcm^{-3}$. We determine the equation of state (EOS) and the high temperature phase diagram. Under these conditions, hydrogen is a dense fluid that exhibits a molecular, an atomic and a plasma regime at low density. A high density, it is predicted to go into a metallic state. The determination of these properties has direct application to the understanding of brown dwarfs and Jovian planets.

The restricted PIMC method relies on a nodal surface, taken from a trial density matrix, in order to deal with Fermi statistics. The PIMC method has been applied extensively using free particle nodes. In this work, we develop a variational technique that allows us to obtain a variational many-body density matrix (VDM). In a first application to hydrogen, we derive a VDM that describes the principle physical effects in high temperature hydrogen such as ionization and dissociation.

In the PIMC simulation, we employ this more realistic density matrix in order to replace the free particle nodes and study the effect on the derived thermodynamic properties. The modifications are particularly significant at low temperature and high density where PIMC using free particle nodes have suggested a first order plasma phase transition. We critically review these findings and show improved results from simulations with VDM nodes.

The recent laser shock wave experiments are of particular relevance to this research because they represent the first direct EOS measurements in the megabar regime. We estimate the shock Hugoniot from the calculated EOS and compare with the experimental findings. We study finite size effects and the dependence on the time step of the path integral and on the type of nodes.

Furthermore, we extend the restricted PIMC method to open paths in order to determine off-diagonal density matrix elements and apply this method to the momentum distribution of the electron gas and to the natural orbitals in hydrogen.

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