Question

Comparison of distribution functions for indistinguishable paricles

Answer 1

Explanation:

The distribution or probability density functions describe the probability with which one can expect particles to occupy the available energy levels in a given system. While the actual derivation belongs in a course on statistical thermodynamics it is of interest to understand the initial assumptions of such derivations and therefore also the applicability of the results.

The derivation starts from the basic notion that any possible distribution of particles over the available energy levels has the same probability as any other possible distribution, which can be distinguished from the first one.

In addition, one takes into account the fact that the total number of particles as well as the total energy of the system has a specific value.

Third, one must acknowledge the different behavior of different particles. Only one Fermion can occupy a given energy level (as described by a unique set of quantum numbers including spin). The number of bosons occupying the same energy levels is unlimited. Fermions and Bosons all "look alike" i.e. they are indistinguishable. Maxwellian particles can be distinguished from each other.

The derivation then yields the most probable distribution of particles by using the Lagrange method of indeterminate constants. One of the Lagrange constants, namely the one associated with the average energy per particle in the distribution, turns out to be a more meaningful physical variable than the total energy. This variable is called the Fermi energy, EF.

An essential assumption in the derivation is that one is dealing with a very large number of particles. This assumption enables to approximate the factorial terms using the Stirling approximation.

The resulting distributions do have some peculiar characteristics, which are hard to explain. First of all the fact that a probability of occupancy can be obtained independent of whether a particular energy level exists or not. It would seem more acceptable that the distribution function does depend on the density of available states, since it determines where particles can be in the first place.