PX284 - G1 - energy deviation

in a canonical ensemble, $T$ is fixed, but $U$ (and hence $F, S, p$ , etc) is calculated from an average energy, $⟨ E ⟩$ , so, in principle, could fluctuate

⟨ E ⟩ = \frac{1}{Z} \sum_{i} E_{i} \exp (- β E_{i}) = - \frac{1}{Z} \frac{d Z}{d β}

similarly,

⟨ E^{2} ⟩ = \frac{1}{Z} \sum_{i} E_{i}^{2} \exp (- β E_{i}) = - \frac{1}{Z} \frac{d^{2} Z}{d β^{2}}

note:

\frac{d ⟨ E ⟩}{d β} = - \frac{1}{Z} \frac{d^{2} Z}{d β^{2}} + \frac{1}{Z^{2}} {(\frac{d Z}{d β})}^{2} = - (⟨ E^{2} ⟩ - ⟨ E ⟩^{2}) = - σ_{E}^{2}

which is the variance of energy

furthermore:

\frac{d ⟨ E ⟩}{d β} = \frac{d T}{d β} \frac{d ⟨ E ⟩}{d T} = - k_{B} T^{2} C_{V}

thus, the standard deviation of energy:

σ_{E} = T \sqrt{k_{B} C_{V}}

conclusions

there is a finite probability for fluctuations in energy (and hence in $F, S, p,$ etc)
the size of the fluctuations increase with increasing heat capacity