The well-known cumulant-summation formula is applied to the low-temperature calculation of the quantum-mechanical partition functions for many particle systems. The motivation for this method is to be able to deal with many-particle states directly and avoid the usual starting point of independent single-particle states. The test case chosen for this method was the calculation of the canonical partition function and the grand-canonical partition function for the degenerate free-electron gas. The result is that the expected equivalence of the canonical and grand-canonical ensembles is not found at low temperatures. The role of few-particle fluctuations in the grand-canonical ensemble is shown to be extremely important to the thermal properties of the degenerate free-electron gas and in the difference between the ensembles.

• Received 28 October 1991

DOI:https://doi.org/10.1103/PhysRevB.46.1338