1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: What is meant by thermal average ?

  1. Mar 30, 2009 #1
    What is meant by "thermal average"?

    1. The problem statement, all variables and given/known data
    I'm reading through Yeomans, Stat. Mech. of Phase transitions. I'm trying to verify equation 2.14
    [tex]<M^2>-<M>^2 = k^2 T^2 {\partial^2 \over \partial H^2} \ln Z[/tex],
    where k is Boltzman 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.

    2. Relevant equations
    [tex]M = - \left({\partial F \over \partial H} \right)_T[/tex]
    [tex]F = -kT\ln T[/tex]
    [tex]Z = \sum_r e^{-\beta E_r}[/tex]

    3. The attempt at a solution
    Of course, I've gone so far as to plug everything in
    [tex]<M^2>-<M>^2 = <k^2T^2\left({\partial \over \partial H}\ln Z \right)^2>-<-kT{\partial \over \partial H}\ln Z>^2[/tex]

    Then, I assume Maxwell-Boltzmann statistics, which I think means that
    [tex]<x> = \sum_m x {e^{-\beta E_m}\over Z}[/tex]
    which leads to
    [tex]<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}[/tex]

    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.
  2. jcsd
  3. Jan 12, 2010 #2
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook