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Grand partition function Z of a system

  1. Jun 2, 2012 #1
    The grand partition function Z of a system is given by formula:

    Z = Ʃ exp ((-Ei/KbT) + (μni/KbT))

    where , 1, 2... i E i= are permitted energy levels, μ is the chemical
    potential, , 1,2... i n i= are number of particles of different types.
    Taking into account that averaged internal energy

    U = Ʃ Pi(Ei-μni) show that

    U = Kb(T^2)(d(lnZ)/dT)

    any help?
     
  2. jcsd
  3. Jun 4, 2012 #2
    The grand partition function is as follows:

    [itex]\Large{Z = \sum_{i}{e^{(\frac{-\epsilon_i}{kT}) + (\frac{{\mu}n_i}{kT})}}}[/itex]

    Remembering that the expectation value of the energy is the following:

    [itex]\large{<U> = \sum_{i}{P(i)({\epsilon}_i-{\mu}n_i)}}[/itex]

    (where P(i) is the probability of finding the system in the ith state..)
    Show that:

    [itex]\large{<U> = kT^2\frac{d(ln(Z))}{dT}}[/itex]

    Just dressing up your equations in Latex for the practice. This proof should in a standard Thermal Physics text, but unfortunately I am without mine at the moment :(
     
    Last edited: Jun 4, 2012
  4. Jun 4, 2012 #3
    Note:
    [tex]
    \frac{d}{dT}\ln(Z)=\frac{1}{Z}\frac{dZ}{dT}
    [/tex]
    also, you should look up the definition of the canonical probability distribution [itex]P(\sigma_i)[/itex]. With those definitions, you should be on your way. Also remember that the temperature derivative can pass through the sum in the first equation.
     
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