Canonical energy fluctuation identity

Statement

In the canonical ensemble with partition function Z(β) , the variance of the energy satisfies

In terms of temperature $T$ (with $\beta = 1/(k_B T)$), the heat capacity at constant volume satisfies

equivalently,

Key hypotheses

• The canonical ensemble is well-defined and $Z(\beta)$ is twice differentiable in $\beta$ (with differentiation under the integral justified).
• The energy variance is finite: $\langle H^2\rangle_\beta < \infty$.

Conclusion

• Energy fluctuations are controlled by curvature of $\log Z(\beta)$ in $\beta$.
• Positivity $\operatorname{Var}_\beta(H)\ge 0$ implies $C_V\ge 0$ under the usual identification (compare stability ).

Cross-links to definitions

• Variance in an ensemble
• Canonical partition function
• Canonical energy identity
• Heat capacity at constant volume

Proof idea / significance

Differentiate the identity ⟨H⟩ = -∂β log Z once more in $\beta$. The second derivative produces $\langle H^2\rangle_\beta - \langle H\rangle_\beta^2$, i.e. the variance. Translating derivatives from $\beta$ to $T$ yields the fluctuation–response relation linking equilibrium fluctuations to the measurable response coefficient $C_V$.