Illustrative Example

Consider a very simple gas made up of two identical particles. Suppose that each particle can be in one of three possible quantum states, $s=1,2,3$ . Let us enumerate the possible states of the whole gas according to Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac statistics, respectively.

For the case of Maxwell-Boltzmann statistics, the two particles are considered to be distinguishable. Let us denote them $A$ and $B$ . Furthermore, any number of particles can occupy the same quantum state. The possible different states of the gas are shown in Table 8.1. There are clearly nine distinct states.

Table 8.1: Two particles distributed amongst three states according to Maxwell-Boltzmann statistics.

1	2	3
$AB$	$\cdots$	$\cdots$
$\cdots$	$AB$	$\cdots$
$\cdots$	$\cdots$	$AB$
$A$	$B$	$\cdots$
$B$	$A$	$\cdots$
$A$	$\cdots$	$B$
$B$	$\cdots$	$A$
$\cdots$	$A$	$B$
$\cdots$	$B$	$A$

For the case of Bose-Einstein statistics, the two particles are considered to be indistinguishable. Let us denote them both as $A$ . Furthermore, any number of particles can occupy the same quantum state. The possible different states of the gas are shown in Table 8.2. There are clearly six distinct states.

Table 8.2: Two particles distributed amongst three states according to Bose-Einstein statistics.

1	2	3
$AA$	$\cdots$	$\cdots$
$\cdots$	$AA$	$\cdots$
$\cdots$	$\cdots$	$AA$
$A$	$A$	$\cdots$
$A$	$\cdots$	$A$
$\cdots$	$A$	$A$

Finally, for the case of Fermi-Dirac statistics, the two particles are considered to be indistinguishable. Let us again denote them both as $A$ . Furthermore, no more than one particle can occupy a given quantum state. The possible different states of the gas are shown in Table 8.3. There are clearly only three distinct states.

Table 8.3: Two particles distributed amongst three states according to Fermi-Dirac statistics.

1	2	3
$A$	$A$	$\cdots$
$A$	$\cdots$	$A$
$\cdots$	$A$	$A$

It follows, from the previous example, that Fermi-Dirac (FD) statistics are more restrictive (i.e., there are less possible states of the system) than Bose-Einstein (BE) statistics, which are, in turn, more restrictive than Maxwell-Boltzmann (MB) statistics. Let

$$\xi = \frac{\mbox{ probability that the two particles are found i... ...te}}{\mbox{ probability that the two particles are found in different states}}.$$ (8.12)

For the case under investigation,

$$\xi_{\rm MB} = 1/2,$$ (8.13)

$$\xi_{\rm BE} =1,$$ (8.14)

$$\xi_{\rm FD} = 0.$$ (8.15)

We conclude that in Bose-Einstein statistics there is a greater relative tendency for particles to cluster in the same state than in classical (i.e., Maxwell-Boltzmann) statistics. On the other hand, in Fermi-Dirac statistics there is a lesser tendency for particles to cluster in the same state than in classical statistics.