## Quantum Statistics

When we consider large numbers of particles, it becomes advantageous to talk of their properties in an average sense rather than describing each particle individually. The function that gives this description is the distribution function, which describes how many particles have energy between E and E+dE if there areN total particles. The most general form of the distribution might be f(x, y, z, vx, vy, vz, t), but if the system is time-independent and spatially uniform, it simplifies. Furthermore, if the distribution is isotropic, then we can express the distribution function as a function of speed, but for our purposes, the energy distribution is preferred. In formulating the distribution function, we will need to know how many kinds of particles there are and how they interact with one another. We shall deal first only with one kind of particle at a time and assume that they interact with one another only weakly, and we shall furthermore assume that the distribution is stationary in time. The fundamental technique in determining the distribution function is based

on counting and the calculus of variations in order to find the most probable distribution. We shall consider three kinds of particles: (a) Identical, distinguishable particles, (b) Identical, indistinguishable particles that obey the Pauli exclusion principle, (or particles with half-integral spin), and (c) Identical, indistinguishable particles that do not obey the Pauli exclusion principle, (or particles with integral spin). The first group is composed of classical particles (they have the same charge and mass, but we can imagine painting them different colors, or writing a number on them to tell them apart) and they obey Maxwell-Boltzmann statistics. The second group is composed of fermions and they obey Fermi-Dirac statistics. The third group is composed of bosons and they obey Bose-Einstein statistics.

We consider now an assembly of N particles, all of which belong to only one of the three groups defined above. We assume them to be noninteracting, and move under the influence of some potential V (r), and occupy specific energy states ψi with energy ²i, i = 0, 1, 2, . . . For simplicity, we shall assume all