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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