Connected correlations as derivatives of expectations

In Gibbs ensembles, derivatives with respect to conjugate parameters yield covariances (connected two-point functions) and higher cumulants.

Connected correlations as derivatives of expectations

Let $\mu_\lambda$ be a Gibbs/ensemble measure (e.g. canonical ensemble or finite-volume Gibbs measure ) whose Hamiltonian depends on a real parameter $\lambda$ via a linear coupling to an observable $A$ :

H_\lambda = H_0 - \lambda A.

Let $\langle \cdot\rangle_\lambda$ denote ensemble averages with respect to $\mu_\lambda$ .

Statement

For any integrable observable $B$ ,

\frac{\partial}{\partial\lambda}\,\langle B\rangle_\lambda = \beta\Big(\langle A\,B\rangle_\lambda-\langle A\rangle_\lambda\,\langle B\rangle_\lambda\Big).

In other words,

\frac{\partial}{\partial\lambda}\,\langle B\rangle_\lambda = \beta\,\mathrm{Cov}_\lambda(A,B),

where $\mathrm{Cov}_\lambda(A,B)$ is the (two-point) connected correlation, i.e. the connected two-point function at the level of observables.

In particular, for the partition function $Z(\lambda)$ (canonical or lattice ),

\frac{\partial^2}{\partial\lambda^2}\log Z(\lambda) = \beta^2\,\mathrm{Var}_\lambda(A),

with variance as in variance in an ensemble .

More generally, if one introduces sources $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_k)$ coupled to observables $(A_1,\dots,A_k)$ , then mixed partial derivatives of $\log Z(\boldsymbol{\lambda})$ generate higher connected correlations (cumulants).

Key hypotheses

The parameter enters linearly: $H_\lambda=H_0-\lambda A$ (or, more generally, differentiably with $\partial_\lambda H_\lambda=-A$ ).
The relevant derivatives can be passed under the integral/sum defining $\langle\cdot\rangle_\lambda$ (automatic in finite volume with bounded observables).
$\beta>0$ is fixed (inverse temperature).

Conclusions

Response of $B$ to the field conjugate to $A$ is controlled by their covariance.
Taking $B=A$ recovers a fluctuation identity: $\partial_\lambda\langle A\rangle_\lambda=\beta\,\mathrm{Var}_\lambda(A)\ge 0.$
This packages many “susceptibility = fluctuation” statements, including susceptibility–variance for magnetization as a special case.

Proof idea / significance

Write $\langle B\rangle_\lambda=\frac{1}{Z(\lambda)}\sum B\,e^{-\beta(H_0-\lambda A)}$ (or the analogous integral). Differentiate using the product/quotient rule: one term differentiates the numerator (producing $\beta\langle AB\rangle_\lambda$ ) and one term differentiates $\log Z(\lambda)$ (subtracting $\beta\langle A\rangle_\lambda\langle B\rangle_\lambda$ ). This is the basic mechanism behind fluctuation–response and generating-function methods.