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.

 

[paranormal]

It seems like you are on the right track in terms of using the quotient and product rule to take the derivative of U with respect to both temperature and volume. However, it is important to note that in the canonical ensemble, the volume is typically held constant. Therefore, when taking the derivative with respect to T and V, the term for dU/dV should be zero since the volume is not changing.

To find <E2>-<E>2, you can use the formula <E2>=sum(Ei2*exp(-beta*Ei)/sum(exp(-beta*Ei). Then, using the formula for variance, you can subtract <E>2 from <E2> to get the desired result.

It is also important to note that in the canonical ensemble, energy fluctuations are dependent on temperature and the specific heat capacity. Therefore, when taking the derivative with respect to T, you may need to consider the specific heat capacity term.

Overall, it is important to carefully consider the conditions and assumptions of the canonical ensemble when deriving equations and taking derivatives. I would suggest reviewing the relevant equations and their derivations to ensure accuracy in your calculations.