  We apply variational principles from statistical physics and the Landau
theory of phase transitions to multicomponent alloys using the
multiple-scattering theory of Korringa-Kohn-Rostoker (KKR) and the coherent
potential approximation (CPA). This theory is a multicomponent generalization
of the $S^{(2)}$ theory of binary alloys developed by G. M. Stocks, J. B.
Staunton, D. D. Johnson and others. It is highly relevant to the chemical phase
stability of high-entropy alloys as it predicts the kind and size of
finite-temperature chemical fluctuations. In doing so it includes effects of
rearranging charge and other electronics due to changing site occupancies. When
chemical fluctuations grow without bound an absolute instability occurs and a
second-order order-disorder phase transition may be inferred. The S$^{(2)}$
theory is predicated on the fluctuation-dissipation theorem; thus we derive the
linear response of the CPA medium to perturbations in site-dependent chemical
potentials in great detail. The theory lends itself to a natural interpretation
in terms of competing effects: entropy driving disorder and favorable pair
interactions driving atomic ordering. To further clarify interpretation we
present results for representative ternary alloys CuAgAu, NiPdPt, RhPdAg, and
CoNiCu within a frozen charge (or band-only) approximation. These results
include the so-called Onsager mean field correction that extends the
temperature range for which the theory is valid.
