11.4 Particle-Energy Distribution Functions

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 $\vec{I}$ that have the given energy, and then among these pick out the set that has the most system eigen­functions $Q_{\vec{I}}$. There are some technical issues with that, {N.24}, but they can be worked out, as in derivation {D.58}.

The final result is, of course, the particle energy distributions from chapter 6:

$$\iota^{\rm {b}}=\frac{1}{e^{({\vphantom' E}^{\rm p}- \mu)/... ...\frac{1}{e^{({\vphantom' E}^{\rm p}- \mu)/{k_{\rm B}}T} + 1}.$$

Here $\iota$ indicates the number of particles per single-particle state, more precisely, $\iota$ $\vphantom0\raisebox{1.5pt}{$=$}$ $I_s$$\raisebox{.5pt}{$/$}$$N_s$. This ratio is independent of the precise details of how the shelves are selected, as long as their energies are closely spaced. However, for identical bosons it does assume that the number of single-particle states on a shelf is large. If that assumption is problematic, the more accurate formulae in derivation {D.58} should be consulted. The main case for which there is a real problem is for the ground state in Bose-Einstein condens­ation.

It may be noted that “$T$” 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 $T$ will normally have to be positive. Otherwise the derived energy distributions would have the number of particles become infinity at infinite shelf energies. For some weird system for which there is an upper limit to the possible single-particle energies, this argument does not apply, and negative temperatures cannot be excluded. But for particles in a box, arbitrarily large energy levels do exist, see chapter 6.2, and the temperature must be positive.

The derivation also did not show that $\mu$ in the above distributions is the chemical potential as is defined in general thermo­dynamics. That will eventually be shown in derivation {D.62}. Note that for particles like photons that can be readily created or annihilated, there is no chemical potential; $\mu$ entered into the derivation {D.58} through the constraint that the number of particles of the system is a given. A look at the note shows that the formulae still apply for such transient particles if you simply put $\mu$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0.

For permanent particles, increasingly large negative values of the chemical potential $\mu$ decrease the number of particles at all energies. Therefore large negative $\mu$ corre­sponds to systems of very low particle densities. If $\mu$ is sufficiently negative that $e^{({\vphantom' E}^{\rm p}-\mu)/{k_{\rm B}}T}$ is large even for the single-particle ground state, the $\pm1$ that charac­terize the Fermi-Dirac and Bose-Einstein distributions can be ignored compared to the exponential, and the three distributions become equal:

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.