Fluctuations and phase diagrams in ternary mixtures

A binary or liquid ternary mixture has two independent densities, say x1 and x2. There are three independent fluctuations in these densities, namely 〈(δx1)2〉, 〈(δx2)2〉, and 〈δx1 δx2〉. It is demonstrated that there exists a homogeneous linear relation between these fluctuations, if the mixture coexists with a second phase. The coefficients of this relation are simple geometric quantities of the phase diagram. If the mixture coexists with two other phases, there exist two such linear relations. In this case the fluctuations can be determined up to a common factor from the phase diagram alone.