Approaching Bose–Einstein condensation

Abstract

Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N0 by construction, leads to BEC in a straightforward way. For a density of states of the form G(ϵ)∝ϵγ, the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature TB, determined by an appropriate critical exponent. The condensate population N0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.