  We show how the stability conditions for a system of interacting fermions
that conventionally involve variations of thermodynamic potentials can be
rewritten in terms of one- and two-particle correlators. We illustrate the
applicability of this alternative formulation in a multi-orbital model of
strongly correlated electrons at finite temperatures, inspecting the lowest
eigenvalues of the generalized local charge susceptibility in proximity of the
phase-separation region. Additionally to the conventional unstable branches, we
address unstable solutions possessing a positive, rather than negative
compressibility. Our stability conditions require no derivative of free energy
functions with conceptual and practical advantages for actual calculations and
offer a clear-cut criterion for analyzing the thermodynamics of correlated
complex systems.
