# Ideal gas in the microcanonical ensemble: I'm puzzled

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.

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

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

$$N = \sum_p n_p, \qquad E = \sum_p \ve_p n_p$$

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

$$n_p = \frac{1}{z^{-1} e^{\beta \epsilon_p}\mp 1} \qquad (**)$$

and the the formula for the entropy thereby

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

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 $$N$$.
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 $$z = e^{\beta \mu}$$. Using this information and the constraints in the formula for the entropy one gets

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

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

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

$$S = k \sum_p z e^{-\beta \epsilon_p} (\beta \epsilon_p - \log z)$$

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

Is there something I'm overlooking?

Thanks a lot for any insight

F