Fluctuations and equilibrium response

Write a classical Hamiltonian as $H_h=H_0-hM$, where the generalized field $h$ has the energy units needed to make $hM$ an energy. Assume that the observable $M$ itself has no explicit $h$ dependence. Differentiating its canonical expectation changes both the Boltzmann numerator and its normalizing partition function. The two terms combine into a variance multiplied by $\beta$. Thus the response is nonnegative at positive temperature for a stable canonical equilibrium with this conjugate-field convention.

The same simple formula holds for a quantum observable commuting with the Hamiltonian and entering through the same linear coupling. Noncommuting quantum observables require an imaginary-time correlation expression instead of blindly substituting an equal-time variance. Likewise, an observable with explicit field dependence contributes an extra derivative term. Keeping these restrictions visible prevents a compact classical identity from being mistaken for a universal formula for every optical or quantum response.

For $N$ independent variables $s_i=\pm1$, define $M=\sum_i s_i$ and energy $-hM$. A single moment has partition factor $2\cosh(\beta h)$, mean $\tanh(\beta h)$, and variance $\operatorname{sech}^2(\beta h)$. Independence makes means and variances additive. At zero field the mean magnetization vanishes, but the magnetization variance is $N$. A zero mean therefore does not imply that every microscopic configuration is unmagnetized.

At small field, the total mean is approximately $N\beta h$. This linear response requires $|\beta h|\ll1$; at larger field the exact hyperbolic tangent saturates instead of growing without bound. Distinguish the extensive total susceptibility from susceptibility per moment. A physical magnetic moment would introduce additional factors converting a laboratory magnetic field into the energy parameter $h$. Our dimensionless spins deliberately keep that unit conversion separate from the statistical derivation.

For interacting moments, the total variance contains a sum of connected correlations between different sites, not only individual-site variances. Long-range correlations near a critical point can strongly amplify susceptibility and disrupt ordinary inverse-square-root relative-noise scaling. In a finite system the correlations and variance remain finite. A fixed-magnetization ensemble would instead constrain the total variance, illustrating why one must match the ensemble and field contact before interpreting an observed fluctuation.

The static identity compares equilibria after a small parameter change; it gives no relaxation time by itself. Dynamical fluctuation-dissipation relations connect frequency-dependent response to time correlations under specified equilibrium and microscopic assumptions. A system driven far from equilibrium, aging over the measurement, or observed before it relaxes need not obey the equilibrium formula. Even in equilibrium, a short experimental trace can miss slow fluctuations and underestimate a susceptibility. The measurement window and equilibration protocol belong beside the algebra, rather than being treated as irrelevant details.