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Homework Statement
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.
Homework Equations
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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
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?