Energy Fluctuations in Canonical Ensemble

[chiaki]

Homework Statement

In deriving <E2>-<E>2

starting from <E>=U=sum(Eiexp(-beta Ei))/sum(exp(-beta Ei). the taking derivative of U with respect to beta, the book always notes E (thus Volume) is held constant. what i am trying to do is taking the derivative of U with respect to beta or T (temp) and V (volume). but i get stuck

Homework Equations

<E>=U= sum(Ei*exp(-beta*Ei)/sum(exp(-beta*Ei)
dU=dU/dT+dU/dV

The Attempt at a Solution

I applied the above equation dU to U as listed above. i performed the quotient and product rule obtaining terms partial derivative with respect to V and T. I tried to look for a way to combine terms and cancel terms. but I cannot. anyone help thank you.

 

[nickjer]

Why are you taking the derivative of U with respect to V? It only asked you to take the derivative with respect to beta.

 

[chiaki]

in the notes in the book its hold Ei constant, i want to perform the derivative more generally allowing Ei to vary

 

[nickjer]

But I am pretty sure, they take a partial derivative with respect to beta. So you don't worry about N or V.