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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?