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arXiv:cond-mat/0304286 · Revista Mexicana de Física

Bose-Einstein condensation in real space

J. J. Valencia, MANUEL DE LLANO, M. A. Solís

Published Apr 11, 2003• 0 citations
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Abstract

We illustrate how Bose-Einstein condensation occurs not only in momentum space but also in coordinate (or real) space. Analogies between the isotherms of a van der Waals gas of extended (or finite-diameter) identical atoms and the point (or zero-diameter) particles of an ideal Bose gas allow one to conclude that, in contrast to the van der Waals case, the volume per particle can go to zero in the pure Bose condensate phase precisely because the particle diameter is zero.

DOI
10.48550/arxiv.cond-mat/0304286

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We illustrate how Bose-Einstein condensation occurs not only in momentum space but also in coordinate (or real) space. Analogies between the isotherms of a van der Waals gas of extended (or finite-diameter) identical atoms and the point (or zero-diameter) particles of an ideal Bose gas allow one to conclude that, in contrast to the van der Waals case, the volume per particle can go to zero in the pure Bose condensate phase precisely because the particle diameter is zero.
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.
Schrödinger's paper (1926) on Bose–Einstein condensation was submitted for publication on 15 December, 1925, immediately preceding the first of his papers on wave mechanics. Historians of science (Hanle, 1977; Klein, 1964) have seen ‘an organic connection’ between the two. It came between the three papers of Einstein (1924, 1925 a, b ) on the boson gas and Uhlenbeck's doctoral thesis (Uhlenbeck, 1927), and did not receive the attention it deserved, although, with hindsight, it is seen to contain the gem of the idea which was later to resolve the conflict between Einstein and Uhlenbeck. Einstein based his prediction (Einstein, 1925 a ) of condensation in the free boson gas on a combination of Bose–Einstein statistics and the classical density of states in phase-space; this leads to an expression for the mean particle number density as a function of kinetic energy which saturates as the total density increases so that there is a maximum possible density of particles having nonzero kinetic energy. Einstein claimed that, if the particle density is increased above this maximum, the excess goes into a state with zero kinetic energy. This is the phenomenon of condensation. Uhlenbeck (1927) objected that the result holds only ‘when the quantization of translational motion is neglected’. The situation remained unclear until the Amsterdam meeting to celebrate the centenary (23 November 1937) of the birth of Johannes van der Waals; here Kramers pointed out the importance of the thermodynamic limit for the sharp manifestatioon of phase transitions.
This condition can be violated by increasing the density, or lowering the temperature. What then would be the value of z ? To answer this question, recall that in deriving Equation (18.1) in Section 14.12, we took the infinite-volume limit, and replaced the sum over states by an integration. This assigns zero weight to the state of zero momentum, for the volume element 4πp2dp vanishes at p = 0. In the formula (Equation [18.1]), n is therefore the density of particles with nonzero momentum, and this is the quantity bounded by Equation (18.4). When this bound is exceeded, the excess particles cannot disappear, as photons could, because their number is conserved. They are forced to go into the zero-momentum state, and this single quantum state becomes macroscopically occupied, that is, occupied by a finite fraction of the particles. This phenomenon is called Bose-Einstein condensation.
The properties of a trapped ideal Bose gas in n -dimensional space are studied. General analytic expressions of the critical temperature of Bose-Einstein condensation, the jump of heat capacity at and the fraction of condensation at temperatures below have been derived. How these physical quantities depend on the external potential, particle characteristics and space dimensionality are discussed. We find that when a proper external potential is taken, Bose-Einstein condensation may occur in any dimensional space.
The ideal Bose gas has two major shortcomings: at zero temperature, all the particles 'condense' at zero energy or momentum, thus violating the Heisenberg principle; the second is that the pressure below the critical point is independent of density resulting in zero incompressibility (or infinite isothermal compressibility) which is unphysical. We propose a modification of the ideal Bose gas to take into account the Heisenberg principle. This modification results in a finite (in)compressibility at all temperatures and densities. The main properties of the ideal Bose gas are preserved, i.e. the relation between the critical temperature and density, but the specific heat has a maximum at the critical temperature instead of a discontinuity. Of course interactions are crucial for both cases in order to describe actual physical systems.
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.