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As can be inferred from present experiments in ultracold gases, thescattering length is a quantity that determines the thermodynamic state of thegas. As such, there exists a conjugate thermodynamic to it. Here, we show thatthe recently introduced "contact" is the conjugate of the inverse of thescattering length. We find that this identification allows for a derivation ofessentially all the known results regarding the contact. Using the mean-fieldtheory for the Bose-Einstein (BEC) to Bardeen-Cooper-Schriefer (BCS) crossover,we also find that the contact is proportional to the square of the gap. Weanalyze in detail both a homogenous balanced mixture of fermions and itsinhomogenous counterpart in a harmonic trap.

The thermodynamic origin of the Contact and its relation to the gap in the BEC-BCS crossover