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Canonical energy identity

In the canonical ensemble, the mean energy equals minus the derivative of log partition function with respect to β.
Canonical energy identity

Statement

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

⟨H⟩β  =  −∂∂βlog⁡Z(β). \langle H\rangle_\beta \;=\; -\frac{\partial}{\partial \beta}\log Z(\beta).

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

⟨H⟩β  =  ∂∂β(βF(β)). \langle H\rangle_\beta \;=\; \frac{\partial}{\partial \beta}\big(\beta F(\beta)\big).

Key hypotheses

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

Conclusion

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