Bose-Einstein Condensation

Abstract

Bose-Einstein condensation (BEC) is a phenomenon that occurs in a macroscopic system of bosons (particles obeying ► Bose-Einstein statistics) at low temperatures: a nonzero fraction of all the particles in the system (thus a macroscopic number of particles) occupy a singleM one-particle state. This would, of course, happen for a system of distinguishable, noninteracting particles at zero temperature, but in this case the phenomenon disappears as soon as the temperature becomes comparable to the energy splitting between the single-particle groundstate and the first excited state — a quantity which tends to zero with the size of the system. By contrast, in BEC the macroscopic occupation occurs at all temperatures below a transition temperature, usually denoted T c , which while a function of intensive parameters such as density and interaction strength is constant in the thermodynamic limit. The fundamental reason for the occurrence of BEC lies in the requirement, which follows from considerations of quantum field theory, that the ► wave function of a system of identical bosons should be symmetric under the exchange of any two particles. This has the consequence that states that differ only by such an exchange must be counted as identical, i.e. counted only once. Thus, for example, while for a system of N distinguishable objects, which must be partitioned between two boxes, the number of ways of putting M of them into one box is given by the familiar binomial formula N!/(M!N — M!), for bosons there is exactly one way for each M. The effect is to remove the “entropic” factor, which for distinguishable objects militates against putting a large fraction of them in a single one-particle state.