General thermo questions [Thermal average occupancy]

[IHateMayonnaise]

Howdy,

Just studying for a test, need to clear something up and I can't find it in any of my books.

My question is in regards to $N$, which to me seems like it is the same as $<N>$ also known as the thermal average occupancy. This quantity represents the thermal average number of the orbitals in the system while in thermal and diffusive contact with a reservoir. In such a domain, we want to use the grand partition function:

$$z=\sum_{ASN}e^{-\beta(N\mu-\varepsilon_s)}=\sum_{ASN}\lambda^Ne^{(-\beta\varepsilon_s)}$$

where
$$\beta=\frac{1}{K_bT}$$, $$\lambda=e^{\beta\mu}$$

And the following definitions for $<N>$:

$$<N>=\frac{1}{z}\sum_{ASN}Ne^{-\beta(N\mu-\varepsilon_s)}$$

and

$$<N>=\lambda\sum_{S}e^{-\beta\varepsilon_s}$$

My question: What is the connection between the last two equations for $<N>$? Thanks yall

IHateMayonnaise

 

[clamtrox]

There are two ways of getting ensemble averages: you can either take an average weighted by the coefficients in the partition function


Say, if

$$Z = \sum_{\mathrm{states}} \rho, \hspace{0.5cm} \mathrm{then} \hspace{0.5cm} \langle N \rangle = \frac{1}{Z} \sum_{\mathrm{states}} N \rho.$$

The second way that I know of to calculate averages is to derivate the thermodynamic potential twice. Your second formula looks like it might be something of the sort, but I'm really not very sure.