The shear viscosity has been shown to be equal to the product of pressure andrelaxation time in normal scale-invariant fluids, but the presence ofsuperfluidity at low temperatures can alter the relation. By using themean-field BCS-Leggett theory with a gauge-invariant linear response theory forunitary Fermi superfluids, we present an explicit relation betweenthermodynamic quantities, including the pressure and chemical potential, andtransport coefficients, including the shear viscosity, superfluid density, andanomalous shear viscosity from momentum transfer via Cooper pairs. The relationis modified when pairing fluctuations associated with noncondensed Cooper pairsare considered. Within a pairing fluctuation theory consistent with theBCS-Leggett ground state, we found an approximate relation for unitary Fermisuperfluids. The exact mean-field relation and the approximate one with pairingflucutaions advance our understanding of relations between equilibrium andtransport quantities in superfluids, and they help determine or constrainquantities which can be otherwise difficult to measure.
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