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We consider equilibrium level occupation numbers in a Fermi gas with a fixednumber of particles, n, and finite level spacing. Using the method ofgenerating functions and the cumulant expansion we derive a recurrence relationfor canonical partition function and an explicit formula for occupation numbersin terms of single-particle partition function at n different temperatures. Weapply this result to a model with equidistant non-degenerate spectrum andobtain close-form expressions in terms of q-polynomials and Rogers-Ramanujanpartial theta function. Deviations from the standard Fermi-Dirac distributioncan be interpreted in terms of a gap in the chemical potential between theparticle and the hole excitations with additional correlations at temperaturescomparable to the level spacing.

Exact canonical occupation numbers in a Fermi gas with finite level spacing and a q-analog of Fermi-Dirac distribution