Stability implies concavity/convexity of thermodynamic potentials

Thermodynamic stability forces entropy to be concave in extensive variables (equivalently, energy is convex), yielding positivity of response functions.

Stability implies concavity/convexity of thermodynamic potentials

Statement

Consider an equilibrium thermodynamic system with a differentiable entropy representation $S=S(U,V,N)$ for the entropy . Assume standard thermodynamic stability in the entropy-maximum sense: for fixed additive constraints, the equilibrium state maximizes total entropy under allowed exchanges (see thermodynamic stability ).

Then $S(U,V,N)$ is a concave function of the extensive variables: for $0\le \lambda\le 1$ and any two admissible states,

S(\lambda x + (1-\lambda) y)\ \ge\ \lambda S(x) + (1-\lambda) S(y), \qquad x=(U,V,N),\ y=(U',V',N').

Equivalently, in the energy representation $U=U(S,V,N)$ , the internal energy is convex in $(S,V,N)$ .

Key hypotheses

Existence of equilibrium states and differentiable fundamental relations for $S(U,V,N)$ or $U(S,V,N)$ .
Additivity/ability to form composite systems and exchange conserved quantities.
Stability as an entropy maximum (or, dually, minimum of a suitable free energy at fixed intensive parameters).

Conclusions

Concavity of $S$ implies the Hessian of $S$ (where it exists) is negative semidefinite; convexity of $U$ implies the Hessian of $U$ is positive semidefinite.
Standard response-function positivities follow (under appropriate regularity and choice of independent variables), e.g.
- Heat capacity at constant volume satisfies $C_V\ge 0$ in stable regions (see also C_V positivity from stability ).
- Isothermal compressibility satisfies $\kappa_T\ge 0$ (see compressibility positivity ).
Stability restricts curvature of thermodynamic potentials obtained by Legendre transforms (e.g., convexity/concavity properties of Helmholtz free energy and Gibbs free energy ).

Proof idea / significance

The core idea is the “mixing” argument: take two macrostates, form a composite system that can redistribute the conserved quantities, and use the entropy-maximum principle (a formulation of the second law ) to show that the entropy of the combined equilibrium state dominates the weighted average of entropies. That inequality is precisely concavity.
Concavity/convexity is the mathematical backbone of stability, ensuring uniqueness of equilibrium in single-phase regions and governing fluctuations/response via curvature.