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Short-range correlations and entropy in ultracold atomic Fermi gases
The correlations of atomic Fermi gases at short distances
encapsulate essential thermodynamical information. We probe the finite
temperature correlation through its relation to the free energy and
entropy densities. In the low temperature limit the entropy is dominated
by phonon excitations and the correlations increase as $T^4$. In the
BEC limit, we calculate a boson model within the Bogoliubov
approximation to show explicitly how phonons enhance the fermion
correlations. In the high temperature limit, we show from the virial
expansion that the correlations decrease as $1/T$. The correlations
therefore reach a maximum at a finite
temperature. We infer the general structure of the isentropes of the
Fermi gas in the $T,-1/k_Fa$ plane, and the temperature dependence
of the correlations in the unitary, BEC, and BCS limits. Our
results compare well with measurements of the correlations via
photoassociation experiments at higher temperatures.