How to get mean occupation numbers by Grand partition function?

[hokhani]

How to calculate <n_i ^2> for an ideal gas by the grand partition function (<n_i> is the occupation number)? In other words, I like to know how do we get to the formula <n_i>=-1/\beta (\frac{\partial q}{\partial\epsilon}) and <n_i ^2>=1/Z_G [-(1/\beta \frac{\partial }{\partial\epsilon})^2 Z_G]?

Z_G is grand partition function , q=ln Z_G and \epsilon is the energy of the corresponding level.

 

[vanhees71]

Define
Z(\beta,\alpha)=\mathrm{Tr} \exp(-\beta \hat{H}+\alpha \hat{N}O.
Here, I assume we have a non-relativistic system with conserved particle number \hat{N}.
Then you get
\frac{\partial}{\partial \alpha} Z=Z \langle N \rangle, \quad \frac{\partial^2}{\partial \alpha^2} Z=Z \langle N^2 \rangle.
Now you have
\frac{\partial}{\partial \alpha} \ln Z=\frac{1}{Z} \frac{\partial Z}{\partial \alpha}=\langle N \rangle
and then
\frac{\partial^2}{\partial \alpha^2} \ln Z=\frac{1}{Z^2} \left (\frac{\partial Z}{\partial\alpha} \right )^2+\frac{1}{Z} \frac{\partial^2 Z}{\partial \alpha^2}=\langle N^2 \rangle -\langle N \rangle^2=\sigma_N^2.
So the 2nd derivative of the GK potential wrt. to \alpha is the standard deviation \sigma_N^2 of the particle number, not the expectation value of the particle number squared!

More conventional is to write \alpha = \beta \mu, where \mu is the chemical potential, but then it's a bit inconvenient for evaluating expectation values of the particle number and its powers (or equivalently cumulants of the particle number).

 

[hokhani]

Define
Z(\beta,\alpha)=\mathrm{Tr} \exp(-\beta \hat{H}+\alpha \hat{N}O.
Here, I assume we have a non-relativistic system with conserved particle number \hat{N}.
Then you get
\frac{\partial}{\partial \alpha} Z=Z \langle N \rangle, \quad \frac{\partial^2}{\partial \alpha^2} Z=Z \langle N^2 \rangle.
Now you have
\frac{\partial}{\partial \alpha} \ln Z=\frac{1}{Z} \frac{\partial Z}{\partial \alpha}=\langle N \rangle
and then
\frac{\partial^2}{\partial \alpha^2} \ln Z=\frac{1}{Z^2} \left (\frac{\partial Z}{\partial\alpha} \right )^2+\frac{1}{Z} \frac{\partial^2 Z}{\partial \alpha^2}=\langle N^2 \rangle -\langle N \rangle^2=\sigma_N^2.
So the 2nd derivative of the GK potential wrt. to \alpha is the standard deviation \sigma_N^2 of the particle number, not the expectation value of the particle number squared!

More conventional is to write \alpha = \beta \mu, where \mu is the chemical potential, but then it's a bit inconvenient for evaluating expectation values of the particle number and its powers (or equivalently cumulants of the particle number).

Thank you very much for your good response but what I meant by n_i is the number of particles in the ith single particle state with energy \epsilon_i and not the total number of particles, N.
I study the book "statistical mechanics by Pathria". Reading this book is somewhat difficult. Also I think this book (although is a very good book) hasn't well set apart the borders between quantum and classical treatment. Could you please tell me, if there is any, another good reference in that level instead?