What is meant by thermal average ?

[Pacopag]

What is meant by "thermal average"?

Homework Statement

I'm reading through Yeomans, Stat. Mech. of Phase transitions. I'm trying to verify equation 2.14
$$<M^2>-<M>^2 = k^2 T^2 {\partial^2 \over \partial H^2} \ln Z$$,
where k is Boltzmann constant, T is temperature, M is magnetization, H is magnetic field, and Z is partition function
I think that my main problem is that I don't know the precise definition of a thermal average (i.e. <...>). At least then I could start.

Homework Equations

$$M = - \left({\partial F \over \partial H} \right)_T$$
$$F = -kT\ln T$$
$$Z = \sum_r e^{-\beta E_r}$$

The Attempt at a Solution

Of course, I've gone so far as to plug everything in
$$<M^2>-<M>^2 = <k^2T^2\left({\partial \over \partial H}\ln Z \right)^2>-<-kT{\partial \over \partial H}\ln Z>^2$$

Then, I assume Maxwell-Boltzmann statistics, which I think means that
$$<x> = \sum_m x {e^{-\beta E_m}\over Z}$$
which leads to
$$<M^2>-<M>^2=\sum_m k^2T^2\left({\partial \over \partial H}\ln Z \right)^2 {e^{-\beta E_m}\over Z} - \sum_m \sum_n k^2 T^2 \left({\partial \over \partial H}\ln Z \right)^2 {e^{-\beta E_m}\over Z}{e^{-\beta E_n}\over Z}$$

This is where I'm stuck. I don't see how I'm going to get a second derivative from this. Thanks for any help.

 

[jbunten]

A good source for definition of thermal average can be found here in the Planck Distribution Function chapter on here:

http://en.wikibooks.org/wiki/Statistical_Mechanics/Thermal_Radiation


better late than never I suppose.