Standard deviation of the energy of a system in a heat bath

[Narcol2000]

Given the mean energy of a system in a heat bath is

$$\bar{E} = - \frac{\partial ln(Z)}{\partial \beta}$$

Where Z is the partition function and $$\beta = k_BT$$

Why is the standard deviation of E defined by:

$$(\Delta E)^2 = \frac{\partial^2 ln(Z)}{\partial \beta ^2}$$

I can't seem to find any proof of how the second derivative is related to the standard deviation.

 

[atyy]

Eric Poisson has a derivation on p40 of his Statisical Physics II notes: http://www.physics.uoguelph.ca/~poisson/research/notes.html.

 

[Narcol2000]

Thanks a lot those notes are also pretty useful in other respects as well.