Mode Expansion in Equilibrium Statistical Mechanics. I. General Theory and Application to the Classical Electron Gas

Abstract

A new expansion for the Helmholtz free energy of a classical system is presented. The potential energy of the system is assumed to be composed of two parts: a “reference system” potential energy and a perturbation potential, which is the sum of two‐particle potentials. The two‐particle perturbation potential energy is assumed to have a Fourier transform. Collective variables, which are the Fourier transforms of the single‐particle density, are introduced, and the canonical ensemble partition function is expressed as an infinite series. The first term in the result for the Helmholtz free energy is the reference system free energy. The second is a mean field term, and the third is the random phase approximation. Subsequent terms involve correlations among the collective variables in the reference system. The general results are applied to the special case in which the reference system is the ideal gas. A density and temperature dependent renormalized potential arises from the analysis in a straightforward way. When the results are specialized to the case of the Coulomb potential, agreement with the ionic cluster theory is obtained, and the renormalized potential is the usual Debye–Hückel potential. Further applications of the technique are mentioned.