  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 critical exponents for the compressibility can be
extracted via an analysis of the density fluctuations when phase transitions
occur as a function of population imbalance or interaction parameter. In
addition, we demonstrate that in the presence of trapping potentials, the local
compressibility, local spin-susceptibility and local density-density
correlations can be extracted from experimental data via a generalized local
fluctuation-dissipation theorem which is valid beyond the local density
approximation. Lastly, we use the local density approximation to calculate
theoretically the local compressibility, local spin-susceptibility and local
density-density fluctuations to compare with experimental results as they
become available.
