QUANTUM STATISTICS AND DISCRETENESS - DIFFERENCES BETWEEN THE CANONICAL AND GRAND-CANONICAL ENSEMBLES FOR A FERMIONIC LATTICE-GAS

Studying the electronic properties of small metallic particles Kubo an d subsequently Denton, Muhlschlegel and Scalapino recognized the surpr ising effect that the mean occupation numbers of noninteracting fermio ns (electrons) are not given within the canonical ensemble by the usua l expression of Fermi-Dirac statistics. The fermions behave as if they are colder and observed deviations are most significant in the vicini ty 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 th e effect vanishes for very small and very large spacing between the en ergy levels in comparison to the thermal energy k(B)T. 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 a greement with our numerical results. The low-temperature aspects of th e effect are considered by applying a two-state model. The role of the finite-size effect is also discussed.