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Bose-Einstein condensation in arbitrarily shaped cavities

Klaus Kirsten, David J. Toms

Published Jan 1, 1999• 0 citations
Publisher

Abstract

We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density 3of states factor obtained from spectral analysis. The application to some simple cavities is discussed. ©1999 The American Physical Society.

DOI
10.25405/ncl.34594413

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Bose-Einstein condensation in arbitrarily shaped cavities

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We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed.

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Bose-Einstein condensation in arbitrarily shaped cavities

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We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed. [S1063-651X(99)03101-3].

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We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density 3of states factor obtained from spectral analysis. The application to some simple cavities is discussed. ©1999 The American Physical Society.
We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed.
We discuss the phenomenon of Bose-Einstein condensation of an ideal nonrelativistic Bose gas in an arbitrarily shaped cavity. The influence of the finite extension of the cavity on all thermodynamical quantities, especially on the critical temperature of the system, is considered. We use two main methods that are shown to be equivalent. The first deals with the partition function as a sum over energy levels and uses a Mellin-Barnes integral representation to extract an asymptotic formula. The second method converts the sum over the energy levels to an integral with a suitable density of states factor obtained from spectral analysis. The application to some simple cavities is discussed. [S1063-651X(99)03101-3].
The topic of Bose–Einstein condensation in a uniform, non-interacting gas of bosons is treated in most textbooks on statistical mechanics. In the present chapter we discuss the properties of a non-interacting Bose gas in a trap. We shall calculate equilibrium properties of systems in a semi-classical approximation, in which the energy spectrum is treated as a continuum. For this approach to be valid the temperature must be large compared with Δ∊/ k , where Δ∊ denotes the separation between neighbouring energy levels. As is well known, at temperatures below the Bose–Einstein condensation temperature, the lowest-energy state is not properly accounted for if one simply replaces sums by integrals, and it must be included explicitly. The statistical distribution function is discussed in Sec. 2.1, as is the single-particle density of states, which is a key ingredient in the calculations of thermodynamic properties. Calculations of the transition temperature and the fraction of particles in the condensate are described in Sec. 2.2. In Sec. 2.3 the semi-classical distribution function is introduced. From this we obtain the density profile and the velocity distribution of particles, and use these to determine the shape of an anisotropic cloud after free expansion. Thermodynamic properties of Bose gases are calculated as functions of the temperature in Sec. 2.4. Finally, Sec. 2.5 treats corrections to the transition temperature due to a finite particle number.
The condensation of a non-relativistic ideal Bose gas in an Einstein universe is investigated. Explicit expressions for the condensate fraction N 0 /N and the specific heat are obtained by using the Poisson summation formula to express the summations as integrations plus corrections. It is shown that the finiteness of the system smoothes out the cusp-like singularity of the infinite system. A rigorous asymptotic analysis of the critical temperature and the specific heat maximum are given, and the relation with the scaling theory of finite size effects is briefly discussed.
This chapter focuses on the applications involving finite temperature theories. Technically, the grand canonical partition function is easiest to handle and in general it is the one used for the analysis of the thermodynamical properties. In fact, for properties like the energy and the specific heat of the gas this is justified because in the thermodynamical limit the different ensembles are known to give the same answer. The chapter uses the grand canonical ensemble to obtain properties like critical temperature, energy and specific heat. It summarizes the grand canonical description of ideal Bose gases trapped by magnetic fields and describes the formulation of Bose-Einstein condensation as a symmetry-breaking phenomenon.
A simple pedagogical introduction to the effective action method of quantum field theory is given at a level suitable for beginning postgraduate students. It is shown how to obtain the effective potential at zero temperature from a regularized zero-point energy. The results are applicable to curved as well as to flat space. The generalization to finite temperatures is also given. It is shown how to obtain high temperature expansions of the thermodynamic potential for the neutral free Bose gas and the charged Bose gas in both the relativistic and nonrelativistic limits. The results are obtained for an arbitrary spatial dimension and in curved space. Results are also obtained for the self-interacting relativistic gas in three spatial dimensions. A detailed discussion of how the formalism may be applied to study Bose-Einstein condensation is given. The interpretation of Bose-Einstein condensation as symmetry breaking is discussed. Application is given to the study of charged bosons, both relativistic and nonrelativistic, in a constant magnetic field. The Meissner effect is obtained for the nonrelativistic gas in three spatial dimensions. The final application is to the study of nonrelativistic bosons in a harmonic oscillator confing potential trap. A number of analytical approaches to this are discussed.