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How to get mean occupation numbers by Grand partition function?

  1. Jan 4, 2014 #1
    How to calculate [itex]<n_i ^2>[/itex] for an ideal gas by the grand partition function ([itex]<n_i>[/itex] is the occupation number)? In other words, I like to know how do we get to the formula [itex]<n_i>=-1/\beta (\frac{\partial q}{\partial\epsilon})[/itex] and [itex]<n_i ^2>=1/Z_G [-(1/\beta \frac{\partial }{\partial\epsilon})^2 Z_G][/itex]?

    [itex]Z_G[/itex] is grand partition function , q=[itex]ln Z_G[/itex] and [itex]\epsilon[/itex] is the energy of the corresponding level.
     
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  3. Jan 4, 2014 #2

    vanhees71

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

    More conventional is to write [itex]\alpha = \beta \mu[/itex], where [itex]\mu[/itex] 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).
     
  4. Jan 5, 2014 #3
    Thank you very much for your good response but what I meant by [itex]n_i[/itex] is the number of particles in the ith single particle state with energy [itex]\epsilon_i[/itex] 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?
     
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