Chapter 2 - Various statistics

Abstract

This chapter discusses the Fermi–Dirac and Bose–Einstein Distribution Functions. The Fermi–Dirac and Bose–Einstein distribution functions may be derived using statistical mechanics. The quantity μ in the Fermi distribution is the chemical potential. It is a function of temperature and is chosen in such a way that the total number of particles in the system is N. The Fermi distribution function f(ɛ), at absolute zero changes discontinuously from value 1 (filled) to value 0 (empty) at ɛ= ɛF= μ, where ɛF is the Fermi energy. At all temperatures, f(ɛ) is equal to 1/2 when ɛ= μ, as then, the denominator of the equation has the value of 2. To derive a very simple formula of the Boltzmann factor, one has to consider a small system with two states, one at energy 0 and one at energy ɛ, which are placed in thermal contact with a large system, called the reservoir. The total energy of the combined system is U. When the small system is in energy 0, the reservoir has energy U and it has g(U) states accessible to it. When the small system is in energy ɛ, the reservoir has energy U-ɛ and it has g(U-ɛ) states accessible to it.