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

so the relative energy fluctuations vanish:

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)

• energy fluctuations vs. C_V
• canonical energy fluctuation identity
• heat capacity at constant volume
• large-deviation characterization of equilibrium
• large deviation principle