Grand responses in beta and chemical potential

For the grand preparation Grand canonical particle-exchange preparation, write let $\mathcal{G}$ and $m^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ be its disjoint-union state space and reference, define $H(N,z):=H_N(z)$ on sector ${\mathrm{\Gamma }}_N$, and define the particle-number statistic $\mathsf{N}(N,z):=N$. Write $m_N$ for $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in the sector sum, so $\mathrm{\Xi }(\beta ,\mu )={\sum }_N\int e^{-\beta H_N+\beta \mu N}d{m}_N\mathrm{.}$ Here $\beta >0$ and $\mu$ has units J per particle. Let $S_H:=\{(N,z)\in \mathcal{G}:H_N(z)<\mathrm{\infty }\}$, restrict $m^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ to this measurable set, and compute energy moments there; the grand weight is zero outside it and the restricted measure is sigma-finite. Assume the local envelopes of Dominated exponential normalization and covariance hold uniformly on this restricted space for its finite-valued sufficient statistics $H$ and $\mathsf{N}$; expectations written $⟨N⟩$ refer to $\mathsf{N}$. Then ${\mathrm{\partial }}_{\eta }\log \mathrm{\Xi }=⟨N⟩,{\mathrm{\partial }}_{\eta }^2\log \mathrm{\Xi }=\mathrm{Var}(N),\eta =\beta \mu ,$ and, at fixed $\mu$, $-{\mathrm{\partial }}_{\beta }\log \mathrm{\Xi }=⟨H-\mu N⟩,{\mathrm{\partial }}_{\mu }⟨N⟩=\beta \mathrm{Var}(N),{\mathrm{\partial }}_{\beta }⟨N⟩{\mathrm{\mid }}_{\mu }=-\mathrm{Cov}(N,H-\mu N)\mathrm{.}$ The covariance is computed in the full number-exchanging state. No fixed-$N$ derivative or thermodynamic-limit identification is implied.