Ensemble Variance of an Observable

Given an observable $A$ and an ensemble (so that ⟨·⟩ is defined), the variance is the basic scalar measure of the size of fluctuations. It is the statistical-mechanics specialization of variance .

The ensemble variance of $A$ is

where $\delta A$ is the fluctuation of the observable .

Equivalently,

Basic facts (for real-valued $A$):

• $\mathrm{Var}(A)\ge 0$.
• $\mathrm{Var}(A)=0$ iff $A$ is almost surely constant under the ensemble measure (no thermal fluctuations of $A$ in that ensemble).

Variance from the partition function

In a Gibbs-type ensemble, variances appear as second derivatives of $\log Z$ with respect to parameters coupled to observables; see fluctuation formulas from log Z and observables from log Z .

A canonical example is the energy $H$ in the canonical ensemble with partition function $Z(\beta)$. Then

where $\beta$ is the inverse temperature .

Using $\beta = 1/(k_B T)$ (with Boltzmann constant $k_B$ and thermodynamic temperature $T$), one obtains the standard fluctuation relation

where $C_V$ is the heat capacity at constant volume ; this is emphasized in specific heat from fluctuations .

Physical interpretation

• $\mathrm{Var}(A)$ sets the typical fluctuation scale of $A$: roughly, $A$ wanders around $\langle A\rangle$ by an amount of order $\sqrt{\mathrm{Var}(A)}$.
• For extensive observables (often scaling like system size), $\mathrm{Var}(A)$ often scales extensively as well, so the relative size $\sqrt{\mathrm{Var}(A)}/|\langle A\rangle|$ can shrink with system size, supporting macroscopic reproducibility in the thermodynamic limit .