Concavity of Helmholtz free energy in temperature

Statement

Let $F(T,V,N)$ be the Helmholtz free energy of a stable equilibrium system. At fixed $V,N$ and for $T>0$,

where $C_V$ is the constant-volume heat capacity .

Equivalently, $T\mapsto F(T,V,N)$ is concave (and $T\mapsto -F(T,V,N)$ is convex).

Key hypotheses

• Equilibrium thermodynamics applies (see thermodynamic equilibrium ).
• $F$ is twice differentiable in $T$ at fixed $V,N$.
• Thermodynamic stability, in particular $C_V\ge 0$ (see thermodynamic stability and $C_V$ ).
• Temperature $T>0$ (see temperature ).

Key conclusions

• Curvature identity:

  $$\left(\frac{\partial^2 F}{\partial T^2}\right)_{V,N} \le 0.$$

• Since $S = -(\partial F/\partial T)_{V,N}$, it follows that

  $$\left(\frac{\partial S}{\partial T}\right)_{V,N} = \frac{C_V}{T} \ge 0,$$

  so entropy is nondecreasing with temperature at fixed $V,N$.

• The concavity implies a tangent-line bound: for any $T_0$,

  $$F(T,V,N) \le F(T_0,V,N) + \left(\frac{\partial F}{\partial T}\right)_{V,N}\Big|_{T_0}\,(T-T_0),$$

  i.e. $F$ lies below its tangent lines as a function of $T$ (at fixed $V,N$).

Cross-links to definitions

• Helmholtz free energy $F$
• heat capacity at constant volume $C_V$
• temperature , entropy
• thermodynamic stability

Proof idea / significance

Start from the fundamental differential of Helmholtz free energy,

At fixed $V,N$, this gives $S = -(\partial F/\partial T)_{V,N}$. Differentiate once more in $T$:

Using $C_V = T(\partial S/\partial T)_{V,N}$ yields the identity

Stability implies $C_V\ge 0$, hence concavity in $T$.

Significance: the temperature-curvature of $F$ is controlled by $C_V$; this is a compact way to encode thermal stability and governs how quickly the free energy changes with temperature.