Fluctuation–response equivalence (canonical covariance identities)

Statement

Let $\langle \cdot \rangle$ denote ensemble averages in the canonical ensemble at inverse temperature $\beta$. Suppose the Hamiltonian is perturbed by a coupling $h$ to an observable $A$:

Then, whenever differentiation under the integral/trace is justified,

where $\mathrm{Var}_h(A)$ is the variance computed in the perturbed ensemble.

More generally, for two observables $A,B$ with a coupling $g$ to $B$ via $H_g=H_0-gB$,

with $\mathrm{Cov}_g(A,B)=\langle AB\rangle_g-\langle A\rangle_g\langle B\rangle_g$.

Key hypotheses

• Equilibrium canonical state exists and is differentiable in the coupling (see canonical ensemble and canonical partition function ).
• Observable(s) and perturbation are such that exchanging differentiation and expectation is valid.

Conclusions

• Response (a derivative of a mean) equals a fluctuation (variance/covariance).
• Susceptibilities are variances/covariances; e.g. a linear susceptibility is a special case of susceptibility .

Proof idea / significance

Write $\langle A\rangle_h = Z(h)^{-1}\,\mathrm{Tr}\!\left(Ae^{-\beta H_h}\right)$ (or the corresponding phase-space integral), with $Z(h)$ the partition function. Differentiate numerator and denominator in $h$ and use the product rule. The cancellation produces a covariance. This is the basic bridge between measurable responses and equilibrium fluctuations.