5.1 Ideal quantum gas: Method of the most probable distribution

We consider a system of $N$ independent particles in a cubic box with ideally elastic walls. In the spririt of the kinetic theory of dilute gases we explore the state space for the individual particles, which Boltzmann dubbed $\mu$-space. In the quantum case this space is spanned by the quantum numbers $n_{x,y,z} = 1, 2 \dots$ pertaining to the momentum eigenstates having eigenvalues $\vec{p} \equiv (h/2L) \vec{n}$ and energies $E_{\vec{n}} = p^{2}/2m = (h^{2}/8mL^{2}) \vert \vec{n} \vert^{2}$.

Now we batch together all states having energies $E_{\vec{n}}$ in an interval $\left[ E_{j}, \Delta E \right]$. The number of states in such a ``cell'' is named $g_{j}, \;\; j=1,2, \dots m$. The values of the $g_{j}$ are not important; they should only be large enough to allow the application of Stirling's formula.

As before we try to answer the question how the $N$ particles should best be distributed over the $m$ cells. To do so we change the notation from the one used in Section 2.2, in that we denote the number of particles in cell $j$ by $f_{j}$. The reason for using $f_{j}$ is that ``$n$'' is reserved for the quantum numbers.

A specific distribution $\vec{f} \equiv \left\{ f_{j}; j=1, \dots m \right\}$ of the $N$ particles to the $m$ cells is more probable if its multiplicity $W$ is larger, meaning that we can allot the $\left\{ f_{j}\right\}$ particles in more different ways to the $\left\{ g_{j}\right\}$ states in each cell - always keeping in mind the Fermi or Bose rules:

$${\rm Fermi-Dirac:}\; \;W= \prod_{j=1}^{m} {g_{j} \choose f_{... ...stein:} \; \; W= \prod_{j=1}^{m} {g_{j}+f_{j}-1 \choose f_{j}}$$ (5.1)

To compare: the multiplicity given in Section 2.2, pertaining to a classical distribution (see equ. 2.20) would in our present notation read $W= \prod_{j=1}^{m} g_{j}^{f_{j}}/f_{j}!$.

The distribution $f_{j}^{*}$ having the largest multiplicity may again be determined by Lagrange variation with the conditions $\sum_{j}f_{j}E_{j}=E$ and $\sum_{j}f_{j}=N$:

$$\delta \ln W - \beta \delta \left( \sum_{j}f_{j}E_{j}-E \right) + \gamma \delta \left( \sum_{j}f_{j} -N \right) = 0$$ (5.2)

Writing $z \equiv e^{\gamma}$ we find for

$${\rm\bf Fermi-Dirac:}\; \; f_{j}^{*} = \frac{g_{j}}{z^{-1}e^... ...nstein:}\; \; f_{j}^{*} =\frac{g_{j}}{z^{-1}e^{\beta E_{j}}-1}$$ (5.3)

This is the most probable distribution of the particles upon the cells. Since $g_{j}$ denotes the number of states in cell $j$, we have for the average population number of each state

$$\langle f_{\vec{n}} \rangle \equiv \frac{f_{j}^{*}}{g_{j}} =... ... \;\;\; (+ \dots {\rm Fermi;}\; - \dots {\rm Bose})$$ (5.4)

It is easy to interpret the Lagrange parameters $\beta$ and $z$. As in Section 2.2 one compares the consequences of the population densities given above to empirical/thermodynamical facts, finding that $\beta$ is related to temperature as $\beta = 1/kT$, and that $z = e^{\gamma}=e^{\mu/kT}$ is identical to the fugacity.

For a better understanding of this derivation, let us interpret its premises as a set of rules in a game of fortune, as we have done in Chapter 2. By running the applet EFRoulette we may indeed play that game - for Fermi particles at least - and compare its outcome with the result just given.

These are the rules:

For non-interacting particles in a square box the $\mu$-plane is spanned by integers $n_{x},n_{y}$; each quantum state is represented by a point. A specific state of a system of $N$ fermions is represented by a set of $N$ inhabited points on that plane.

To find the average (and also most probable!) distribution of particles on states,

- assign $N$ particles randomly to the states on $\mu$-plane

- make sure that the sum of the particle energies equals the given system energy,
AND
- discard all trials in which a state is inhabited by more than one particle

- determine the mean number of particles in each state; sort the result according to the state energies

Applet LBRoulette: Start

Playing the Fermi-Dirac game. [Code: EFRoulette]