# Energy fluctuations of canonical system

1. Feb 9, 2015

### cdot

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. Feb 9, 2015

### Goddar

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?