PX284 - G2 - large systems

Z_{N} = (Z_{1})^{N}

⟨ E ⟩ = - N \frac{d \ln Z}{d β}

$⟨ E ⟩$ is extensive
extensive: scales with the size of the system. eg: $⟨ E ⟩ = U$ , hence $C_{V}$
intensive: local properties. eg: temperature
so, $σ_{E} \propto \sqrt{C_{V}} \propto \sqrt{N}$
and, the relative size of fluctuation:

\frac{σ_{E}}{⟨ E ⟩} \propto \frac{\sqrt{N}}{N} \propto \frac{1}{\sqrt{N}}

so, if $N \to \infty$ , $\frac{σ_{e}}{⟨ E ⟩} \to 0$
conclusions:
- fluctuations are important in small systems, but are negligible for macroscopic systems
- in general, deviations from any quantity about the average (eg. pressure, magnetization) become insignificant in the large $N$ limit
note: fluctuations are important near critical points
eg: for the phases of water
- near the critical point, $C_{V}$ can diverge
- fluctuations become significant
- fluctuations in density can be observed - critical opalescence