Entropy is concave in energy (at fixed V,N)

Statement

Let $S(U,V,N)$ denote the thermodynamic entropy of an equilibrium thermodynamic system as a function of internal energy $U$, with $V,N$ held fixed. Under the usual stability/additivity assumptions of equilibrium thermodynamics, $S(\,\cdot\,,V,N)$ is concave: for any $U_1,U_2$ and any $\lambda\in[0,1]$,

If $S$ is twice differentiable in $U$, this is equivalent to

Key hypotheses

• Equilibrium description is valid (see thermodynamic equilibrium ).
• Additivity/weak coupling for composite systems and entropy maximization consistent with the second law .
• Fixed $(V,N)$ throughout.

Conclusions

• $S(U,V,N)$ is concave in $U$ at fixed $(V,N)$.
• If differentiable, $1/T = \partial S/\partial U$ with temperature $T$; concavity implies $T$ is nondecreasing in $U$.
• In particular, the heat capacity at constant volume satisfies $C_V=(\partial U/\partial T)_{V,N}\ge 0$ (a standard facet of thermodynamic stability ).

Proof idea / significance

Consider two weakly interacting copies of the same system with total energy $U_{\mathrm{tot}}$ and fixed $(V,N)$ per copy. The second law asserts the equilibrium energy split maximizes total entropy $S(U_1)+S(U_{\mathrm{tot}}-U_1)$. This midpoint-maximization yields midpoint concavity, and hence full concavity by standard interpolation arguments. Concavity encodes stability: small energy exchanges do not decrease total entropy and response coefficients (like $C_V$) have the expected sign.