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Canonical energy variance equals ${\partial }_{\beta }^2\log Z=k_B{T}^2{C}_V$.

For an equilibrium entropy $S(E,V,N)$ that is concave in energy at fixed volume and particle number, the thermodynamic relation $(\partial S/\partial E{)}_{V,N}=1/T$ gives, wherever the derivatives are finite and invertible,Strict concavity therefore implies positive heat capacity at constant volume. A straight coexistence segment can instead have constant temperature over an energy interval; finite strictly positive heat-capacity conclusions do not apply there.

There is also a direct canonical ensemble statement for a temperature-independent Hamiltonian operator:The numerator is the canonical energy variance. Positivity follows from a variance, without assuming an ideal gas or noninteracting particles.

The concavity hypothesis is essential. Isolated systems with long-range gravitational interactions can have negative heat capacity in a microcanonical ensemble; the familiar virialized-gas example does not belong to a stable canonical branch in contact with an infinite heat bath. A claim that every stable equilibrium has strictly positive heat capacity would erase this ensemble distinction.

For independent particles with interconverting state families $i$, the equilibrium partition function is a sum $q={\sum }_i{q}_i$ and the family probabilities are $x_i=q_i/q$. Let $u_i$ be the conditional mean one-particle energy in family $i$ and $c_i=d{u}_i/dT$. The total canonical energy variance separates into within-family and between-family contributions, givingThe final positive term is absent if family fractions are held fixed. A frozen ortho-para mixture and an equilibrating mixture can therefore have different heat capacities at the same instantaneous composition.

For a fixed number of non-interacting molecules, $Z_N=q^N/N!$ in the dilute ideal gas limit. The temperature-independent factorial drops out of the internal energy derivative:Differentiate once more at fixed volume to obtainPrimes denote fixed-volume derivatives. If $q$ factorizes into translational, rotational and other independent contributions, their heat capacities add. In particular, the three-dimensional translational factor contributes $3R/2$, while a constant spin degeneracy factor contributes zero. The formula also follows from the positive canonical energy variance, $C_V=\mathrm{Var}(E)/(k_{\mathrm{B}}{T}^2)$.