Mixed energy–number fluctuation identity (grand canonical)

Statement

In the grand canonical ensemble with density proportional to $\exp(-\beta(H-\mu N))$, the following differentiation identities hold for any observable $A$ (when justified):

Specializing gives a mixed fluctuation identity linking energy–number covariance to parameter derivatives:

Key hypotheses

• The grand canonical state exists and is differentiable in $(\beta,\mu)$.
• Differentiation can be interchanged with the trace/integral defining expectations.

Conclusions

• $\mu$-response produces covariance with $N$; $\beta$-response produces covariance with $H-\mu N$.
• Energy–number correlations are controlled by mixed thermodynamic responses: $\partial_\mu\langle H\rangle = \beta\,\mathrm{Cov}(H,N)$ and $\partial_\beta\langle N\rangle = -\mathrm{Cov}(N,H)+\mu\,\mathrm{Var}(N)$.
• Together with number fluctuation identity , these yield a closed set of fluctuation/response relations for $(H,N)$.

Proof idea / significance

Write $\langle A\rangle = \Xi^{-1}\mathrm{Tr}\!\left(Ae^{-\beta(H-\mu N)}\right)$. Differentiate in $\mu$ or $\beta$ and apply the quotient rule; the difference between differentiating the numerator and correcting for the derivative of $\Xi$ produces a covariance term. The final displayed identity follows by choosing $A=H$ and $A=N$ and eliminating $\mathrm{Cov}(N,H)$ in favor of $\partial_\beta\langle N\rangle$ and $\mathrm{Var}(N)$. This is the mixed (energy–number) analog of fluctuation–dissipation relations.