  We describe the relation between the isothermal atomic compressibility and
density fluctuations in mixtures of two-component fermions with population or
mass imbalance. We derive a generalized version of the fluctuation-dissipation
theorem which is valid for both balanced and imbalanced Fermi-Fermi mixtures.
Furthermore, we show that the compressibility, its critical exponents, and
phase boundaries can be extracted via an analysis of the density fluctuations
as a function of population imbalance, interaction parameter or temperature.
Lastly, we demonstrate that in the presence of trapping potentials, the local
compressibility and local density-density correlations can be extracted via a
generalized fluctuation-dissipation theorem valid within the local density
approximation.
