Quantum statistics and discreteness. Differences between the canonical and grand canonical ensembles for a fermionic lattice gas

Abstract

AbstractStudying the electronic properties of small metallic particles Kubo and subsequently Denton, Mühlschlegel and Scalapino recognized the surprising effect that the mean occupation numbers of noninteracting fermions (electrons) are not given within the canonical ensemble by the usual expression of Fermi‐Dirac statistics. The fermions behave as if they are colder and observed deviations are most significant in the vicinity of the Fermi level. The results of our Monte Carlo simulations, as well as our complementary numerical and analytical results are in very good agreement with the predictions of Denton et al. We found that the effect vanishes for very small and very large spacing between the energy levels in comparison to the thermal energy kBT. Associated with this effect are the particle‐hole correlations. Extending the method used by Denton et al. we derive an analytical formula for the particle‐hole correlation functions whose predictions are again in very good agreement with our numerical results. The low‐temperature aspects of the effect are considered by applying a two‐state model. The role of the finite‐size effect is also discussed.