Composition fluctuations and the geometry of phase diagrams in multicomponent mixtures

The connection between the composition fluctuations and the geometry of the phase diagram, discussed previously for ternary mixtures, is extended to mixtures of arbitrary many components. It is shown that for two coexisting phases the phase diagram determines all but n + 1 of the n(n + 1) independent fluctuations in the two phases. For three coexisting phases all the composition fluctuations in the three phases are determined by the phase diagram up to a scale factor. New geometric constraints on phase diagrams are derived. For four-component mixtures the constraints are determined explicitly.