Fluctuation of an Observable

In statistical mechanics, an observable $A$ assigns a numerical value to each microstate (for a classical system, a point in phase space ). Once an ensemble measure is chosen (e.g. canonical ensemble or microcanonical measure ), $A$ becomes a random variable in the probabilistic sense (compare random variable ), and its ensemble mean is ⟨A⟩ .

The fluctuation (or centered observable) associated with $A$ is

This is an observable with zero ensemble mean:

If $A_x$ is a local observable (e.g. a spin or density at site/position $x$), then its fluctuation is $\delta A_x = A_x - \langle A_x\rangle$. Fluctuations are the basic objects behind variances , covariances , and connected correlation functions .

Key formulas

• Variance (typical size of fluctuations):

  $$\mathrm{Var}(A) \;=\; \langle (\delta A)^2\rangle.$$

  This is the content of ensemble variance .

• Covariance (joint fluctuations):

  $$\mathrm{Cov}(A,B) \;=\; \langle \delta A\,\delta B\rangle,$$

  as in ensemble covariance .

• Connected two-point correlations (spatial correlations of fluctuations): for local observables $A_x,B_y$,

  $$\langle A_x B_y\rangle_c \;=\; \langle \delta A_x\,\delta B_y\rangle,$$

  which is the two-point case of connected correlations and relates to the two-point correlation function .

Physical interpretation

$\delta A$ measures how much $A$ typically deviates from its thermodynamic “typical value” $\langle A\rangle$ under the chosen ensemble. In a finite system these deviations are unavoidable (thermal noise). In many systems and for many self-averaging observables, relative fluctuations shrink as system size grows toward the thermodynamic limit , which is why macroscopic macrostates can be stable even though underlying microstates fluctuate.