Ensemble average

Definition.
An observable is a function $A$ of the microstate (classically, a function on phase space ; on a discrete state space, a function on the set of states). Given an ensemble described by a probability measure $P$ on microstates, the ensemble average of $A$ is the expectation

If $P$ has a density $\rho(x)$ with respect to the phase-space volume element $d\Gamma$, then

For a discrete ensemble with probabilities $\{p_i\}$, this is $\langle A\rangle=\sum_i A_i p_i$.

Canonical and microcanonical cases.

• In the canonical ensemble , $\rho_\beta(x)=e^{-\beta H(x)}/Z$ where $H$ is the Hamiltonian and $Z$ is the partition function . Then $\langle H\rangle$ is the internal energy predicted at temperature $T$.
• In the microcanonical ensemble , $P$ is (approximately) uniform on the energy shell , so $\langle A\rangle$ is the phase-space average of $A$ over that constraint surface.

Physical interpretation.
$\langle A\rangle$ is the equilibrium prediction for repeated sampling of the system under the macroscopic constraints defining the ensemble. In many-body systems, large-$N$ behavior often makes $\langle A\rangle$ representative of typical outcomes (self-averaging), and different ensembles can give the same limit for suitable observables (equivalence of ensembles).

Fluctuations around the mean.
Once $\langle A\rangle$ is defined, fluctuations are quantified by the variance

and for two observables $A,B$ by the covariance

These are organized systematically by fluctuation formulas and by cumulant identities derived from $\ln Z$ (or $\ln \Xi$) in fluctuations from log Z .