Commuting quantum grand responses

At each parameter pair, the postulate gives $p_j>0$ and ${\sum }_j{p}_j=1$; the open compact-envelope hypothesis makes $e^{-\beta (H-\mu N)}$ trace class locally, so [F2] supplies the corresponding normal density interpretation. Its spectral expectation of each moment-controlled diagonal observable is the displayed absolutely convergent weighted sum.

To differentiate the countable partition function in either natural parameter, apply the scalar mean-value theorem to each exponential difference quotient. Its absolute value is bounded by the corresponding $\mathrm{\mid }{E}_j\mathrm{\mid }$ or $\mathrm{\mid }{n}_j\mathrm{\mid }$ factor times the exponential at an intermediate parameter; the assumed envelope dominates these terms and all second derivative terms. Since that envelope is summable, dominated convergence for series permits termwise first and second differentiation. The quotient rule then gives ${\mathrm{\nabla }}_{({\theta }_E,{\theta }_N)}\log \mathrm{\Xi }=(⟨H⟩,⟨N⟩)$ and Hessian entries $⟨X_a{X}_b⟩-⟨X_a⟩⟨X_b⟩$, for $X=(H,N)$, exactly the canonical response interface [L1].

Since ${\theta }_E=-\beta$ and ${\theta }_N=\beta \mu$, at fixed $\beta$, ${\mathrm{\partial }}_{\mu }=\beta {\mathrm{\partial }}_{{\theta }_N}$, so ${\mathrm{\partial }}_{\mu }\log \mathrm{\Xi }=\beta ⟨N⟩$ and ${\mathrm{\partial }}_{\mu }⟨N⟩=\beta \mathrm{Var}(N)$. At fixed $\mu$, ${\mathrm{\partial }}_{\beta }\log \mathrm{\Xi }=-⟨H⟩+\mu ⟨N⟩$, giving the stated grand-energy identity. A covariance matrix is positive semidefinite because $c^T\mathrm{Cov}(X)c=⟨(c\cdot (X-⟨X⟩){)}^2⟩\ge 0$. These are predictions conditional on the adopted preparation and the stated common-basis and moment-envelope hypotheses.