Classical representation of quantum systems at equilibrium

A quantum system at equilibrium is represented by an effective classical system, chosen to reproduce thermodynamic and structural properties. The motivation is to allow application of classical strong coupling theories and classical simulations like molecular dynamics and Monte Carlo to quantum systems at strong coupling. The correspondence is made at the level of the grand canonical ensembles for the two systems. The effective classical system is defined in terms of an effective temperature, local chemical potential, and pair potential. These are determined formally by requiring the equivalence of the grand potentials and their functional derivatives of the quantum and representative classical systems. The mapping is inverted using the classical density functional theory to solve for these three parameters. Practical forms of these formal solutions are obtained using the classical liquid state theories like hypernetted chain approximation (HNC). The mapping is applied to the ideal Fermi gas is demonstrated and the details of the thermodynamics of the effective system is derived explicitly. As the next application we consider the uniform electron gas and an explicit form for the effective interaction potential is obtained in the weak coupling limit. The pair correlation functions are calculated using the HNC equations and compared with path integral Monte Carlo data and other theoretical models like Perrot Dharma-wardana. Excellent agreement is obtained over a wide range of temperatures and densities. The last application is to the shell structure of harmonically bound charges. We show that in the mean field limit, the quantum effects of degeneracy and diffraction produce shells at very low temperatures.