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The spherical model and Bose-Einstein condensation

John Edward Gough, Pulé, J.V.

Published Feb 15, 1993• 2 citations
Publisher

Abstract

In 1968 Gunton and Buckingham pointed out that there is a close relationship between the critical behaviour of the spherical model and that of the ideal Bose gas. In this paper we concentrate on this similarity between the two systems. We show that in the spherical model there is in fact Bose-Einstein condensation into the spin modes with low energy. As in the Bose gas we distinguish between generalised Bose-Einstein condensation and Bose-Einstein condensation. We find that the spherical model in certain cases has two critical temperatures: one temperature T c corresponding to the onset of generalised Bose-Einstein condensation (and of spontaneous magnetisation) and a lower temperature T m at which generalised condensation becomes condensation into the spin mode with the lowest energy

DOI
10.5169/seals-116564

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In 1968 Gunton and Buckingham pointed out that there is a close relationship between the critical behaviour of the spherical model and that of the ideal Bose gas. In this paper we concentrate on this similarity between the two systems. We show that in the spherical model there is in fact Bose-Einstein condensation into the spin modes with low energy. As in the Bose gas we distinguish between generalised Bose-Einstein condensation and Bose-Einstein condensation. We find that the spherical model in certain cases has two critical temperatures: one temperature T c corresponding to the onset of generalised Bose-Einstein condensation (and of spontaneous magnetisation) and a lower temperature T m at which generalised condensation becomes condensation into the spin mode with the lowest energy
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.
In this paper, it has been shown that the usual demonstrations of the phenomenon of Bose-Einstein condensation are invalid. Exact formal solutions have been derived for the partition functions and the mean occupation numbers in classical and Bose statistics, and the theory of fluctuations in the case of Bose statistics has been given. In a second paper, convenient approximations will be derived for the mean occupation numbers in Bose statistics, and it will be shown that for temperatures well above the critical temperature of the supposed Bose-Einstein condensation the well-known result for the mean occupation numbers is valid, whilst for temperatures rather lower than this critical value it will be shown that particles will condense into the lowest available state. The critical temperatureT0is given by where the summation is over all energy levels except the lowest.
Abstract This chapter describes the mechanism of Bose–Einstein condensation in the simplest ideal Bose gas case. An explicit expression for the critical temperature is derived in the analytically soluble case of a gas confined in a box. Various thermodynamic quantities are calculated above and below the critical temperature. Emphasis is given to the isothermal compressibility which diverges below the critical temperature. The behaviour of the off-diagonal one-body density matrix is discussed. Results for the fluctuations of the single-particle occupation numbers are derived in both the canonical and grand canonical ensembles.
The thermodynamic properties of an ideal gas of bosons moving freely within layered planes and having a finite amplitude for jumping between neighbouring planes, are derived. As long as this coupling is non-zero, the system becomes effectively two-dimensional at sufficiently high temperatures. At a correspondingly low temperature it undergoes a Bose-Einstein transition into a condensed phase as in three dimensions. Below the critical temperature the specific heat is a universal function of only one dimensionless system variable. The phase transition becomes weaker as the coupling between the planes decreases and eventually disappears when the planes decouple.