Energy fluctuations and heat capacity in the canonical ensemble

Statement (canonical energy fluctuations)

Let a system at fixed volume $V$ and particle number $N$ be described in the canonical ensemble at temperature $T$ (inverse temperature $\beta = 1/(k_B T)$), with Hamiltonian $H$ and finite canonical partition function $Z(\beta)$.

Then the energy variance satisfies

where $C_V = \left(\frac{\partial \langle H\rangle}{\partial T}\right)_{V,N}$ is the heat capacity at constant volume and $\operatorname{Var}_\beta(H)$ is the variance computed with respect to the canonical state.

Key hypotheses

• The canonical measure/state exists: $Z(\beta) < \infty$ in a neighborhood of the $\beta$ of interest.
• The map $\beta \mapsto \log Z(\beta)$ is twice differentiable (enough to justify differentiating under the normalization).
• $V$ and $N$ are held fixed when defining $C_V$.

Conclusions

• Energy fluctuations are controlled by the thermodynamic response $C_V$:
  • $C_V \ge 0$ implies $\operatorname{Var}_\beta(H)\ge 0$ (consistency check).
  • Large $C_V$ corresponds to large energy fluctuations (e.g. near criticality).
• Equivalent differential form: $\operatorname{Var}_\beta(H)=\frac{\partial^2}{\partial \beta^2}\log Z(\beta).$

Cross-links (definitions and upstream identities)

• ensemble average , variance of an observable
• canonical energy fluctuation identity
• definition of C_V

Proof idea / significance

Differentiate $\log Z(\beta)$:

• First derivative gives the mean energy, $\langle H\rangle_\beta = -\partial_\beta \log Z(\beta)$.
• Second derivative yields $\partial_\beta^2 \log Z(\beta)=\operatorname{Var}_\beta(H)$. Convert $\partial/\partial\beta$ to $\partial/\partial T$ using $\beta=1/(k_B T)$ to obtain $\operatorname{Var}_\beta(H)=k_B T^2 C_V$. This is the basic “fluctuation–response” relation for energy.