Connected correlations as cumulants

Connected correlation functions are joint cumulants; they are generated by derivatives of log Z with respect to sources.

Connected correlations as cumulants

Connected correlation functions (also called connected correlators or truncated correlations) isolate the genuinely collective part of fluctuations. Mathematically, they are joint cumulants; physically, they are the pieces of correlation that cannot be explained by lower-order factorization.

This construction is the many-body analogue of the cumulant generating function and sits behind the derivative identities in fluctuation formulas from log Z .

Two-point case: covariance

For observables $A$ and $B$ in an equilibrium ensemble (see ensemble average ), the connected two-point function is

\langle AB\rangle_c \;=\; \langle AB\rangle \;-\; \langle A\rangle\,\langle B\rangle.

This is exactly the covariance (see covariance of observables ), and it vanishes when $A$ and $B$ are independent under the ensemble.

When $A$ and $B$ are local observables at points $x,y$ , this quantity is the connected two-point correlation function , closely related to the ordinary two-point correlation function .

Higher order: subtracting disconnected pieces

For three observables $A,B,C$ , the connected three-point function (third cumulant) is

\langle ABC\rangle_c ={} \langle ABC\rangle -\langle AB\rangle\langle C\rangle -\langle AC\rangle\langle B\rangle -\langle BC\rangle\langle A\rangle +2\langle A\rangle\langle B\rangle\langle C\rangle.

This is the part of $\langle ABC\rangle$ that remains after subtracting all lower-order products.

In general, the $n$ -point connected correlator is the joint cumulant $\kappa(A_1,\dots,A_n)$ , obtained by a combinatorial inclusion–exclusion over partitions of $\{1,\dots,n\}$ .

Generating functional viewpoint: sources and log Z

Introduce sources (fields) coupling to observables. In a classical many-body setting, one writes schematically

Z[J] \;=\; \int \exp\!\left(-\beta H(x) + \int J(r)\,A(r;x)\,\mathrm{d}r\right)\,\mathrm{d}\mu(x),

where $J$ is a space-dependent source and $A(r;x)$ is the microscopic observable density at position $r$ .

Then the functional

W[J] \;=\; \log Z[J]

generates connected correlations:

The first functional derivative of $W$ gives the mean field $\langle A(r)\rangle$ .
The second derivative gives the connected two-point function $\langle A(r)A(r')\rangle_c$ .
Higher derivatives give higher connected correlators.

This is the many-body/statistical-mechanics realization of the probabilistic fact that the log of a moment-generating object generates cumulants.

In finite-dimensional “source vectors” $\lambda$ , the same statement is precisely the content of deriving observables from log Z .

Physical meaning

Clustering and independence: If distant regions become independent (in a suitable limit), connected correlators decay; this is quantified by the correlation length .
Response theory: Connected two-point functions control linear response. For a field $h$ conjugate to $A$ , one typically has $\frac{\partial}{\partial h}\langle A\rangle \propto \langle A^2\rangle_c,$ connecting equilibrium fluctuations to susceptibilities (see susceptibility ).
Diagrammatics: In perturbation expansions, “connected diagrams” contribute to $W=\log Z$ , while $Z$ itself sums both connected and disconnected contributions. This is the combinatorial reason $\log Z$ isolates cumulants.

Thus, “connected correlation = cumulant” is not only a definition; it is the organizing principle that turns $\log Z$ into the natural generator of physically meaningful correlations.