Fermi–Dirac and Bose–Einstein distributions: an alternative approach

In this paper, we present an approach to learning about statistical distributions of quantum particles suitable for Advanced Placement high school courses and early-year undergraduate physics and chemistry courses. The procedure developed here uses selected tools known from phenomenological thermodynamics, particularly the concepts of chemical potential and chemical equilibrium, which allow us to circumvent the extensive mathematical apparatus traditionally employed for obtaining equilibrium conditions. To this end, we (a) introduce the notion of site , i.e. an abstract ‘place’ that is able to accept particles; (b) assume that sites having different occupation numbers can be treated as different elementary substances ; (c) consider the change of occupation number as a reaction between elementary substances; and (d) derive quantum distributions by assuming that at equilibrium the driving force, i.e. the difference of chemical potentials of ‘educts’ and ‘products’ vanishes. Assuming that the available sites can be occupied either by at most one particle or by any number of them, we obtain the Fermi–Dirac and the Bose–Einstein distributions, respectively. We illustrate a number of examples, including the Planck distribution for the blackbody radiation and the pressure of the degenerate electron gas in white dwarfs. The paper ends with a comparison of quantum and classical Boltzmann distributions and with some remarks from an educational perspective.