Ideal gas in the microcanonical ensemble: I'm puzzled

  • #1
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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.

Homework Statement



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.

Homework 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.

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

F
 

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