5.1 Ideal quantum gas: Method of the most probable distribution

Now we batch together all states having energies in an interval . The number of states in such a ``cell'' is named . The values of the 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 particles should best be distributed over the 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 by . The reason for using is that ``'' is reserved for the quantum numbers.

A specific distribution
of the particles to the cells is more probable if its
**multiplicity** is larger, meaning that we can allot the
particles in more different ways to the
states in each cell - always keeping in mind the
Fermi or Bose rules:

(5.1) |

The distribution having the largest multiplicity
may again be determined by Lagrange variation with the conditions
and
:

(5.2) |

(5.3) |

It is easy to interpret the Lagrange parameters and . As in Section 2.2 one compares the consequences of the population densities given above to empirical/thermodynamical facts, finding that is related to temperature as , and that 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 -plane is spanned by integers ; each quantum state is represented by a point.
A specific state of a system of fermions is represented by a set of inhabited points on that plane.

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

- assign particles randomly to the states on -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] |

2005-01-25