- #1
Pacopag
- 193
- 4
What is meant by "thermal average"?
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.
M = - \left({\partial F \over \partial H} \right)_T
F = -kT\ln T
Z = \sum_r e^{-\beta E_r}
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.
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.