1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Ideal gas in the microcanonical ensemble: I'm puzzled

  1. Nov 17, 2008 #1
    Hi all,

    this is about problem 8.2 in Huang's Statistical Mechanics.
    I think I've been able to solve it, but the solution raised a question about
    the Maxwell-Boltzmann distribution. So first I provide my solution to the
    problem, then discuss the apparently weird point.

    1. The problem statement, all variables and given/known data

    The problem requires to use the microcanonical formalism to derive the
    equations of state of the ideal quantum gas, i.e.

    [tex]N = \sum_j \frac{1}{z^{-1} e^{\beta \epsilon_p}\mp 1},\qquad
    \frac{PV}{kT} = \mp \sum_p \log(1\mp z e^{-\beta \epsilon_p})[/tex]

    where the upper and lower signs refer to Bose-Einstein and Fermi-Dirac
    statistics, respectively.

    2. Relevant equations

    I think one should use the constraints inherent in the microcanonical ensemble

    [tex] N = \sum_p n_p, \qquad E = \sum_p \ve_p n_p [/tex]

    along with the formula for the set of occupation numbers maximizing the entropy

    [tex]n_p = \frac{1}{z^{-1} e^{\beta \epsilon_p}\mp 1} \qquad (**) [/tex]

    and the the formula for the entropy thereby

    [tex]S = k \sum_p \left[\frac{\beta \epsilon_p - \log z }{z^{-1} e^{\beta \epsilon_p}\mp 1}\mp \log(1\mp z e^{-\beta \epsilon_p}) \right] [/tex]

    All these results are derived in section 8.5.

    3. The attempt at a solution

    The first equation of state is trivially obtained by plugging (**) in the constraint on [tex]N[/tex].
    The second equation of state was a bit harder, but at some point I recalled that E should be identified with the total internal energy, and [tex]z = e^{\beta \mu}[/tex]. Using this information and the constraints in the formula for the entropy one gets

    [tex] S = \frac{1}{T}(U-\mu N) \mp k \sum_p \log(1\mp z e^{-\beta \epsilon_p})[/tex]

    The second equation of state is obtained after recalling that the general form of the
    internal energy is [tex]U = TS - PV + \mu N[/tex].

    4. The weird point <===================

    So far, so good. However, it seems to me that adopting the same approach with the Maxwell-Boltzmann statistics produces a weird result. The MB entropy is

    [tex]S = k \sum_p z e^{-\beta \epsilon_p} (\beta \epsilon_p - \log z) [/tex]

    so that, if I identify the same quantities as above ( total energy and number) I get [tex]TS = E - \mu N[/tex]. But, assuming that [tex]U = TS - PV + \mu N[/tex] is true, wouldn't this mean [tex]PV=0[/tex] instead of the expected [tex]PV = NkT[/tex] ?

    Is there something I'm overlooking?

    Thanks a lot for any insight

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted