Ensemble Covariance of Two Observables

For two observables $A$ and $B$ in a fixed ensemble (so ⟨·⟩ is defined), the covariance measures how their deviations from their means fluctuate together. It is the ensemble version of covariance .

The ensemble covariance is

where $\delta A$ and $\delta B$ are the fluctuations of $A$ and $B$.

Equivalently,

Special cases:

• $\mathrm{Cov}(A,A)=\mathrm{Var}(A)$, connecting covariance to variance .
• For local observables $A_x,B_y$, the quantity $\mathrm{Cov}(A_x,B_y)$ is precisely the two-point connected correlation .

Key properties

For real-valued observables:

• Symmetry: $\mathrm{Cov}(A,B)=\mathrm{Cov}(B,A)$.
• Bilinearity in each argument.
• Cauchy–Schwarz bound: $|\mathrm{Cov}(A,B)| \le \sqrt{\mathrm{Var}(A)\,\mathrm{Var}(B)}.$

These statements formalize the idea that covariance measures shared fluctuations.

Covariance and response (Gibbs ensembles)

In Gibbs-type ensembles built from a Hamiltonian and external fields, covariances arise as derivatives of expectations with respect to those fields (see fluctuation formulas from log Z ).

A standard schematic setting is a canonical weight proportional to $\exp[-\beta(H - hB)]$ (field $h$ coupled linearly to $B$). Then the sensitivity of $\langle A\rangle$ to the field is controlled by $\mathrm{Cov}(A,B)$:

This is a basic form of fluctuation–response, and it underlies definitions such as susceptibility .

Physical interpretation

• $\mathrm{Cov}(A,B)>0$ means that when $A$ fluctuates upward from its mean, $B$ tends to do the same.
• $\mathrm{Cov}(A,B)<0$ means upward fluctuations of $A$ tend to coincide with downward fluctuations of $B$.
• Spatially, $\mathrm{Cov}(A_x,B_y)$ diagnoses correlations between fluctuations at different locations and is the building block of the two-point correlation function and its connected part.