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arXiv:cond-mat/0305614 · Physical Review E

Quantum gas-liquid condensation in an attractive Bose gas

Shun-ichiro Koh

Published Jul 1, 2005• 4 citations
arXivHTMLPDF

Abstract

Gas-liquid condensation (GLC) in an attractive Bose gas is studied on the basis of statistical mechanics. Using some results in combinatorial mathematics, the following are derived. (1) With decreasing temperature, the Bose-statistical coherence grows in the many-body wave function, which gives rise to the divergence of the grand partition function prior to Bose-Einstein condensation. It is a quantum-mechanical analogue to the GLC in a classical gas (quantum GLC). (2) This GLC is triggered by the bosons with zero momentum. Compared with the classical GLC, an incomparably weaker attractive force creates it. For the system showing the quantum GLC, we discuss a cold helium 4 gas at sufficiently low pressure.

DOI
10.1103/physreve.72.016104

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Gas-liquid condensation (GLC) in an attractive Bose gas is studied on the basis of statistical mechanics. Using some results in combinatorial mathematics, the following are derived. (1) With decreasing temperature, the Bose-statistical coherence grows in the many-body wave function, which gives rise to the divergence of the grand partition function prior to Bose-Einstein condensation. It is a quantum-mechanical analogue to the GLC in a classical gas (quantum GLC). (2) This GLC is triggered by the bosons with zero momentum. Compared with the classical GLC, an incomparably weaker attractive force creates it. For the system showing the quantum GLC, we discuss a cold helium 4 gas at sufficiently low pressure.
Gas-liquid condensation (GLC) in the attractive uniform Bose gas is studied by applying the idea of the Yang-Lee zeros to its grand partition function. The followings are proved: (i) When the temperature decreases and the density increases, the GLC occurs prior to the Bose-Einstein condensation (BEC), (ii) an explosive growth of the Bose-statistical coherence to a macroscopic scale occurs simultaneously with the GLC, which are triggered by bosons with a zero momentum, and (iii) the GLC in the Bose gas is not only a condensation in coordinate space, but also in momentum space. On the basis of these results, a comparative study of the BEC and the GLC is developed. Further, we discuss its implication to the trapped atomic gas.
From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.
Bose–Einstein condensation is a phenomenon, first proved by Einstein, in which a gas of bosons condenses into the lowest quantum state below a certain critical temperature. It occurs due to the symmetry, under particle exchange, of the boson wave function. The chapter discusses temperature, the Kelvin scale, the concept of absolute zero, and how laser cooling can reach ultralow temperatures. The property of bosons in which they often prefer to be in the same state is seen to apply to lasers, superfluid liquid helium, and to dilute nearly ideal gases. The latter are trapped in magnetic or optical potential wells where the condensation can be observed directly. These large condensate matter waves show quantum interference in a variety of possible experiments.
Bose-Einstein condensation, observed in either strongly interacting liquid helium or weakly interacting atomic Bose gases, is widely known to be a second-order phase transition. Here we predict a first-order Bose-Einstein condensation in a cloud of harmonically trapped bosons interacting with both attractive two-body interaction and repulsive three-body interaction, characterized respectively by an sss-wave scattering length a<0a<0a<0 and a three-body scattering hypervolume D>0D>0D>0. It happens when the harmonic trapping potential is weak, so with increasing temperature the system changes from a low-temperature liquidlike quantum droplet to a normal gas and therefore experiences a first-order liquid-to-gas transition. At large trapping potential, however, the quantum droplet can first turn into a superfluid gas, rendering the condensation transition occurring later from a superfluid gas to a normal gas smooth. We determine a rich phase diagram and show the existence of a tricritical point, where the three phases, i.e., quantum droplet, superfluid gas, and normal gas, meet. We argue that an ensemble of spin-polarized tritium atoms could be a promising candidate to observe the predicted first-order Bose-Einstein condensation, across which the condensate fraction or central condensate density jumps to zero and the surface-mode frequencies diverge.
It has been shown that Mayer's theory of imperfect gases can be handled approximately as a problem in association in which the actual gas is replaced by a mixture of ideal gases consisting of clusters of all sizes and between which dissociative equilibrium exists. Kahn and Uhlenbeck have shown how Mayer's theory can be generalized to apply to quantum degenerate gases. The purpose of the present paper is to examine the clustering approximation to such a theory of the quantum degenerate Bose-Einstein gas. The symmetrized wave functions for the clustering assembly are set up and the partition function evaluated for all ways of clustering. Equations of state are deduced that include condensation into the liquid phase as a clustering avalanche at the saturation pressure. It is found that clustering has the effect of lowering the lambda-temperature due to statistical degeneracy right down to the saturated vapor line, so that the lambda-transition is entirely inhibited from the gas phase.
The condition which has to be fulfilled in order that the Bose-Einstein condensation be accompanied by a discontinuity of the heat capacity at the condensation temperature is discussed. The nature of the transition depends on the distribution of the energy levels of the individual particles forming the gas. The condensation is studied for several types of individual energy spectra. It is shown that all those Bose-Einstein (B.E.) assemblies which admit a non-ambiguous definition of the statistical distribution function present, beside interesting analogies, a qualitatively different behavior from that observed in liquid helium. It appears that as long as the transition in momentum space of non-ideal B.E. systems is not understood, the possibility of connection between B.E. condensation and the transition observed in liquid helium remains an open problem.