Virial expansions for quantum plasmas: Diagrammatic resummations.

Abstract

We are studying the equilibrium properties of quantum Coulomb fluids in the low-density limit. In the present paper, we only consider Maxwell-Boltzmann statistics. Use of the Feynman-Kac path-integral representation leads to the introduction of an equivalent classical system made of filaments interacting via two-body forces. All the corresponding Mayer-like graphs diverge because of the long-range Coulombic nature of the filament-filament potential. Inspired by the work of Meeron [J. Chem. Phys. 28, 630 (1958); Plasma Physics (McGraw-Hill, New York, 1961)] for purely classical systems, we show that these long-range divergencies can be resummed in a systematic way. We then obtain a formal diagrammatic representation for the particle correlations of the genuine quantum system. The prototype graphs in these series are made of root and internal filaments, connected by two-body resummed bonds according to well-defined topological rules. The resummed bonds depend on the particle densities and decay faster than the bare Coulomb potential because of screening. Some bonds decay algebraically as 1/${\mathit{r}}^{3}$ in accord with the absence of exponential clustering, while the other ones are short ranged. This ensures the integrability of all the above prototype graphs. Moreover, we show that the filament densities, which are the statistical weights of the filaments in these graphs, can themselves be calculated in terms of the particle densities via a well-behaved diagrammatic series. This provides a useful algorithm for expanding the Maxwell-Boltzmann thermodynamic functions in powers of the particle densities, as to be described in a second future paper. The exchange effects due to Fermi or Bose statistics will be considered in a third paper.