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Energy convexity and thermodynamic stability

Convexity of the energy fundamental relation as a stability condition, equivalent to entropy concavity and positivity of key response functions.

Energy convexity and thermodynamic stability

In the energy representation of equilibrium thermodynamics, the fundamental relation expresses the internal energy as a function of the extensive variables , typically $U = U(S,V,N,\dots)$ , where $S$ is thermodynamic entropy , $V$ is volume , and $N$ is particle number .

Definition (energy convexity). The equilibrium energy function $U(S,V,N,\dots)$ is convex if for any two admissible macrostates $X_1=(S_1,V_1,N_1,\dots)$ and $X_2=(S_2,V_2,N_2,\dots)$ and any $0\le \lambda\le 1$ ,

U\!\big(\lambda X_1+(1-\lambda)X_2\big)\ \le\ \lambda\,U(X_1)\ +\ (1-\lambda)\,U(X_2).

Here $\lambda X_1+(1-\lambda)X_2$ denotes the extensive variables obtained by taking weighted totals (the natural operation for composing macroscopic subsystems).

Physical interpretation. Energy convexity is a stability statement: at fixed total extensive variables, a system cannot lower its energy by splitting into two macroscopic regions (“phase separation”) with different local values of $(S,V,N,\dots)$ . If convexity fails, the system can reduce $U$ by separating into distinct macrostates, signaling instability of a homogeneous equilibrium. This criterion is one of the standard formulations of thermodynamic stability .

Equivalent stability formulation. Energy convexity is equivalent (under mild regularity assumptions) to concavity of the entropy representation $S=S(U,V,N,\dots)$ . These are the same stability content viewed in two Legendre-dual coordinate choices.

Local (differential) consequences. When $U$ is twice differentiable, convexity implies the Hessian of $U(S,V,N,\dots)$ is positive semidefinite. In particular, it forces positivity of several measurable response functions . For example, holding $V,N$ fixed one has

\frac{\partial^2 U}{\partial S^2}\Big|_{V,N}=\frac{T}{C_V}\ \ge\ 0,

so the heat capacity at constant volume satisfies $C_V\ge 0$ (assuming temperature $T>0$ ). Similarly, convexity in $V$ is tied to nonnegative adiabatic compressibility (and, in appropriate ensembles, the isothermal compressibility ).

Mathematical viewpoint. This is convexity in the usual sense of the epigraph definition of a convex function . It is also what ensures thermodynamic potentials related by Legendre transforms are well-defined and have the expected extremum principles.