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Homework Help: General thermo questions [Thermal average occupancy]

  1. Nov 5, 2009 #1

    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 [itex]N[/itex], which to me seems like it is the same as [itex]<N>[/itex] 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:


    [tex]\beta=\frac{1}{K_bT}[/tex], [tex]\lambda=e^{\beta\mu}[/tex]

    And the following definitions for [itex]<N>[/itex]:




    My question: What is the connection between the last two equations for [itex]<N>[/itex]? Thanks yall

  2. jcsd
  3. Nov 6, 2009 #2
    There are two ways of getting ensemble averages: you can either take an average weighted by the coefficients in the partition function

    Say, if

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

    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.
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