Thermodynamic convexity and stability

An isolated additive macroscopic system has maximum entropy at stable equilibrium for fixed total energy, volume, and amount. Imagine redistributing a little energy between two equal subsystems while preserving the total. The first entropy variation vanishes when their temperatures agree. The second must be nonpositive, so a further spontaneous redistribution cannot increase the already maximal entropy. This gives concavity of entropy in its extensive coordinates on the ordinary stable equilibrium domain.

The equivalent energy representation is convex in entropy, volume, and amount. Extensivity creates a zero-curvature scaling direction when all extensive coordinates scale together, so the general Hessian condition is semidefinite rather than automatically strictly positive. Restricting amount and considering independent fluctuations in a smooth stable single phase can give strict positivity in the remaining relevant directions. Constraints can remove some candidate variations, and nonadditive systems such as self-gravitating matter lie outside the simple bulk argument used here.

At fixed bath temperature, volume, and amount, Helmholtz free energy is minimized over allowed internal configurations. Temperature itself is imposed by the bath, so this does not require F to have a minimum as a function of its temperature argument. In fact $F_{TT}=-C_V/T$ is negative for a stable positive-capacity phase. By contrast, its volume curvature at fixed temperature is $-(\partial p/\partial V)_T=1/(V\kappa_T)$ and is positive for positive compressibility.

Thus F has different curvature signs in its different natural variables. The Gibbs potential is similarly concave in its intensive temperature and pressure arguments on stable branches. These signs reflect a Legendre transform, not an instability. Diagnose stability using the appropriate constrained variations and responses instead of asserting that “free energy must be convex in everything.” When a homogeneous equation of state predicts pressure rising with volume at fixed temperature, however, the resulting negative compressibility does identify a mechanical instability in this ordinary setting.

For two internal coordinates, positive individual diagonal curvatures are necessary but insufficient. A sufficiently large cross coupling can create a downhill combined direction even when varying either coordinate alone raises the potential. The Hessian eigenvalues or the two-by-two determinant reveal this possibility. A negative eigenvalue means local instability. A zero eigenvalue requires examination of higher orders or a symmetry or coexistence constraint; it is not by itself proof of either a strict minimum or an instability.

Local stability also differs from global equilibrium. A metastable free-energy minimum can persist even though a distant configuration has lower free energy, because a barrier separates them. In bulk phase separation, the stable free energy forms a convex envelope rather than retaining an unstable homogeneous segment. A spinodal marks loss of local stability, while coexistence is set by equality of phase potentials. These distinctions connect curvature measurements to observable responses without pretending that curvature alone determines nucleation barriers or transition rates.