Vanishing of relative fluctuations in the thermodynamic limit

Under extensivity, canonical energy fluctuations are O(N) and relative fluctuations are O(N^{-1/2}), hence vanish as system size grows.

Vanishing of relative fluctuations in the thermodynamic limit

Statement (self-averaging of extensive observables)

Consider a sequence of systems of size $N$ (or volume $|\Lambda|$ ) in the canonical ensemble at fixed temperature $T$ , with Hamiltonian $H_N$ .

Assume the mean energy and constant-volume heat capacity are extensive:

$\langle H_N\rangle = \Theta(N)$ ,
$C_V(N) = \Theta(N)$ , where $C_V(N)=\left(\frac{\partial \langle H_N\rangle}{\partial T}\right)_{V,N}$ .

Then

\frac{\sqrt{\operatorname{Var}(H_N)}}{\langle H_N\rangle} \;=\; O(N^{-1/2}) \quad \text{as } N\to\infty,

so the relative energy fluctuations vanish:

\frac{\sqrt{\operatorname{Var}(H_N)}}{\langle H_N\rangle}\;\longrightarrow\;0.

Key hypotheses

Canonical equilibrium is well-defined for each $N$ (finite partition function).
Extensivity: $\langle H_N\rangle$ and $C_V(N)$ scale linearly in $N$ (away from anomalous regimes such as critical points where scaling may change).
Smoothness in $T$ sufficient to define $C_V(N)$ .

Conclusions

Energy is self-averaging: typical fluctuations are negligible compared to the mean at large $N$ .
Quantitative scaling: $\operatorname{Var}(H_N)=k_B T^2 C_V(N)=\Theta(N),\qquad \sqrt{\operatorname{Var}(H_N)}=\Theta(\sqrt{N}).$
This provides a basic mechanism behind equivalence of ensembles for macroscopic observables.

Cross-links (definitions and supporting results)

Proof idea / significance

Combine $\operatorname{Var}(H_N)=k_B T^2 C_V(N)$ (from energy–C_V fluctuations ) with extensivity: $C_V(N)=\Theta(N)$ and $\langle H_N\rangle=\Theta(N)$ imply $\sqrt{\operatorname{Var}(H_N)}=\Theta(\sqrt{N})$ while the mean is $\Theta(N)$ , giving a $1/\sqrt{N}$ relative scale. This is one of the simplest “thermodynamic limit = law of large numbers” statements in equilibrium statistical mechanics.