Mean energy and preassure inolving partition function

[rayman123]

Homework Statement

We had a lecture about partition function, canonical ensemble etc.
Can someone explain to me how this work out this formula

Homework Equations

we are supposed to find the mean energy and preasure of a gas with given partition function

The Attempt at a Solution

mean energy is given $$\overline{-}U=\sum_{r}E_{r}p_{r}$$
I know also that Boltzmann's probability distribution is described by
$$p_{r}= \frac{e^{-\beta E_{r}}}{\sum_{r}e^{-\beta E_{r}}}$$

because the partition function is definied as $$z=\sum_{r}e^{-\beta E_{r}^}$$


so rewriting now the Boltzmann's probablility distribution I get
$$p_{r}= \frac{e^{-\beta E_{r}}}{z}$$

Homework Statement

now going back to the mean energy I can write
$$\overline{-}U=\frac{1}{z}\sum_{r}E_{r}e^{-\beta E_{r}$$

These are operations I do not understand. Could someone explain them step by step ?

$$\sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z$$

and the final one

$$U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}$$

 

[zhermes]

These are operations I do not understand. Could someone explain them step by step ?
$$\sum_{r}E_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}\sum_{r}e^{-\beta E_{r}}= -\frac{\partial}{\partial \beta}z$$

The first part of the equation is how you find the (ensemble) average energy---you find the weighted average over all ensemble states. The second part is showing that you can rewrite this as the partial derivative with respect to beta; if you take the derivative of the Boltzmann factor WRT beta, the energy E gets pulled down
$$\frac{\partial}{\partial \beta}e^{-\beta E_{r}}= E_{r} e^{-\beta E_{r}}$$
In the expression using the partial derivative wrt beta, the sum over Boltzmann factors is the definition of the partition function. So you end up with the partial derivative of the partition function = the average energy.

and the final one
$$U= -\frac{1}{z}\frac{\partial z}{\partial \beta}=-\frac{\partial lnz}{\partial \beta}$$

Again, this is just rewriting things with derivatives. Using the chain rule, if you take the derivative of lnz, you get a 1/z term out front.

'z' is some function of beta $$z = z(\beta)$$, so using the chain rule: $$\frac{\partial ln z(\beta)}{\partial \beta} = \frac{1}{z} \frac{\partial z(\beta)}{\partial \beta}$$

Does that make sense?

 

[rayman123]

yes it does!:) thank you