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We study the change in condensation energy from a single pair of fermionicatoms to a large number of pairs interacting via the reduced BCS potential. Wefind that the energy-saving due to correlations decreases when the pair numberincreases because the number of empty states available for pairing gets smaller("moth-eaten effect"). However, this decrease dominates the 3D kinetic energyincrease of the same amount of noninteracting atoms only when the pair numberis a sizeable fraction of the number of states available for pairing. As aresult, in BEC-BCS crossover of 3D systems, the condensation energy per pairfirst increases and then decreases with pair number while in 2D, it always iscontrolled by the "moth-eaten effect" and thus simply decreases.

Competition between BCS-pairing and "moth-eaten effect" in BEC-BCS crossover