Virial expansions for quantum plasmas: Maxwell-Boltzmann statistics

This paper is devoted to the calculation of the density expansions (at fixed temperature) of the Maxwell-Boltzmann thermodynamic functions for a quantum plasma. We start from a standard identity that relates the free energy to the particle correlations. These correlations are represented by diagrammatic series, which have been introduced in a previous paper. In the corresponding graphs, the ordinary points are replaced by extended objects, the filaments, which are linked by resummed bonds depending on the particle densities ρ. A scaling analysis of the spatial integrals involved in the graphs shows that the free energy can be represented by a double integer series in ρ^1/2 and lnρ. Furthermore, we derive simple rules that give the leading order in ρ of the contribution from every previous graph. The exact density expansion of the free energy is explicitly calculated up to order ρ^5/2. In the corresponding expression, the contributions of various physical effects, such as screening, diffraction, or recombination, are clearly identified. At the order ρ², we recover the expansion obtained via the effective-potential method. Our present terms of order ρ^5/2 correctly reproduce results that are known in some particular limits.