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American Journal of Physics

Bose-Einstein Condensation of an Ideal Gas

A. Casher, M. Revzen

Published Dec 1, 1967• 10 citations
Publisher

Abstract

The well-known free Bose-gas condensation is described as a condensation into a coherent state. The analogy of the phase transition that occurs here with more transparent symmetry-breaking transition is demonstrated.
DOI
10.1119/1.1973806

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The peculiar condensation in phase space of ideal Bose-Einstein gases is discussed when the internal energy states of the atoms or molecules is taken into account and when the gas of atoms without internal structure is placed in an external conservative field of force. The inclusion of the internal energy states decreases while the application of the external field increases the condensation temperature from its value in absence of either of these causes. The transition phenomenon, apart from the preceding change in the condensation temperature, proceeds in the same way as in the case of the free gas atoms without internal structure; in particular, the order of the transition remains invariable, as the continuity of the heat capacity at constant volume is maintained at the transition point.
Bose-Einstein condensation is a unique phase transition in that it is not driven by inter-particle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question \emph{`How is this ideal Bose gas condensation modified in the presence of interactions between the particles?' } This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers.
We present a short review of how the effective action formalism, well known in relativistic quantum field theory, can be used to discuss Bose-Einstein condensation of non-relativistic gases.This method lends itself very naturally to an interpretation of Bose-Einstein condensation in terms of symmetry breaking.It also allows for the definition of a very elegant regularization technique involving generalized ␨ -functions.We show how this method can be used to recover the well known results for the free boson gas, as well as the charged boson gas in a constant magnetic field.A general criterion for interpreting Bose-Einstein condensation in terms of a phase transition with symmetry breaking is given.Finally we present an analysis of Bose-Einstein condensation in a harmonic oscillator confining potential trap, and show how the results of this simple model are in excellent agreement with experiment.