Canonical energy identity

Statement

Let a classical system be distributed according to the canonical ensemble with inverse temperature $\beta$ and partition function $Z(\beta)$. Then the canonical mean energy (internal energy) satisfies

Equivalently, for the canonical free energy free energy $F(\beta)=-(1/\beta)\log Z(\beta)$,

Key hypotheses

• A well-defined canonical ensemble with normalizing constant $Z(\beta)<\infty$.
• Differentiation under the integral defining Z(β) is justified (e.g., dominated convergence).

Conclusion

• The mean energy is obtained by differentiating $\log Z(\beta)$ with respect to $\beta$.
• This provides an efficient route from partition function to thermodynamic energy.

Cross-links to definitions

• Canonical ensemble
• Canonical partition function
• Ensemble average
• Statistical free energy
• Internal energy (thermodynamics)

Proof idea / significance

Differentiate $\log Z(\beta)$ using $Z(\beta)=\int e^{-\beta H}\,d\lambda$ (with the appropriate reference measure). The derivative brings down a factor of $-H$, and division by $Z(\beta)$ converts the result into a canonical expectation. This identity is the starting point for fluctuation formulas such as canonical energy fluctuation identity .