The objective in this section is to relate the Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac particle energy distributions of chapter 6 to the conclusions obtained in the previous section. The three distributions give the number of particles that have given single-particle energies.
In terms of the picture developed in the previous sections, they describe how many particles are on each energy shelf relative to the number of single-particle states on the shelf. The distributions also assume that the number of shelves is taken large enough that their energy can be assumed to vary continuously.
According to the conclusion of the previous section, for a system with given energy it is sufficient to find the most probable set of energy shelf occupation numbers, the set that has the highest number of system energy eigenfunctions. That gives the number of particles on each energy shelf that is the most probable. As the previous section demonstrated by example, the fraction of eigenfunctions that have significantly different shelf occupation numbers than the most probable ones is so small for a macroscopic system that it can be ignored.
Therefore, the basic approach to find the three distribution functions
is to first identify all sets of shelf occupation numbers
The final result is, of course, the particle energy distributions from
chapter 6:
It may be noted that
in the above distribution
laws is a temperature, but the derivation in the note did not
establish it is the same temperature scale that you would get with an
ideal-gas thermometer. That will be shown in section
11.14.4. For now note that
The derivation also did not show that
For permanent particles, increasingly large negative values of the
chemical potential
The symmetrization requirements for bosons and fermions can be ignored under conditions of very low particle densities.These are ideal gas conditions, section 11.14.4
Decreasing the temperature will primarily thin out the particle numbers at high energies. In this sense, yes, temperature reductions are indeed to some extent associated with (kinetic) energy reductions.