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Energy fluctuations of canonical system

  1. Feb 9, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider a system of fixed volume in thermal contact with a reservoir. Show that the mean square fluctuation in the energy is

    < e-U >^2= t^2*(∂U/∂t) where U=<e>

    Hint: use the partition function to relate (∂U/∂t) to the mean square flucuation. Also, mulitply out the (...)^2 term. Note: the temperature t of a system is a quantity that by definition does not flucuate in value when the system is in thermal contact with a reservoir.

    2. Relevant equations

    U=Σε*exp(-ε/t)/Z=t^2*(∂logZ/∂t)
    Where the summation is over all states

    Z=Σexp(-ε/t)
    Where the summation is over all states

    3. The attempt at a solution
    1. Replace U in the mean square energy flucation term with the definition
    <ε-t^2*(∂logZ/∂t)>^2=t^2*(∂U/∂t)

    2. (∂logZ/∂t)=(1/z)*(∂Z/∂t)
    <ε-t^2*(1/z)*(∂Z/∂t)>^2=t^2*(∂U/∂t)

    3. (∂Z/∂t)=∑ε*t^-2*exp(-ε/t)
    <ε-t^2*(1/z)*∑ε*t^-2*exp(-ε/t)>^2=t^2*(∂U/∂t)

    Not sure where to go from here. My intution says to take the deriviative of U and substitue on the right hand side. But taking the deriviaitve of U (as its defined above) with respect to t introduces a second derivative of logZ and makes the problem a lot more difficult. I feel like there is an easier way?
     
  2. jcsd
  3. Feb 9, 2015 #2
    Hi. You already know that <ε> =(1/Z) Σε*exp(-ε/t)
    Now you need to find out an expression for <ε2>–<ε>2, so:
    How would you express <ε2> using the partition function?
    What function of the Z and derivatives of Z (with respect to time) would that correspond to?
     
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