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arXiv:quant-ph/9808006 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Finite number and finite size effects in relativistic Bose-Einstein condensation

K. Shiokawa, B. L. Hu

Published Oct 22, 1999• 14 citations
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Abstract

Bose-Einstein condensation of a relativistic ideal Bose gas in a rectangular cavity is studied. Finite size corrections to the critical temperature are obtained by the heat kernel method. Using zeta-function regularization of the one-loop effective potential, lower-dimensional critical temperatures are calculated. In the presence of strong anisotropy, the condensation is shown to occur in multisteps. The criteria of this behavior is that critical temperatures corresponding to lower-dimensional systems are smaller than the three-dimensional critical temperature.

DOI
10.1103/physrevd.60.105016

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Bose-Einstein condensation of a relativistic ideal Bose gas in a rectangular cavity is studied. Finite size corrections to the critical temperature are obtained by the heat kernel method. Using zeta-function regularization of the one-loop effective potential, lower-dimensional critical temperatures are calculated. In the presence of strong anisotropy, the condensation is shown to occur in multisteps. The criteria of this behavior is that critical temperatures corresponding to lower-dimensional systems are smaller than the three-dimensional critical temperature.
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Multistep Bose-Einstein condensation of an ideal Bose gas in anisotropic harmonic atom traps is studied. In the presence of strong anisotropy realized by the different trap frequency in each direction, finite size effect dictates a series of dimensional crossovers into lower-dimensional excitations. Two-step condensation and the dynamical reduction of the effective dimension can appear in three separate steps. When the multistep behavior occurs, the occupation number of atoms excited in each dimension is shown to behave similarly as a function of the temperature. Multistep behaviors can be easily controlled by changing the degree of anisotropy. 1
Multistep Bose-Einstein condensation of an ideal Bose gas in anisotropic harmonic atom traps is studied. In the presence of strong anisotropy realized by the different trap frequency in each direction, the finite size effect dictates a series of dimensional crossovers into lower-dimensional excitations. Two-step condensation and the dynamical reduction of the effective dimension can appear in three separate steps. When the multistep behaviour occurs, the occupation number of atoms excited in each dimension is shown to behave similarly as a function of the temperature. Multistep behaviours can be easily controlled by changing the degree of anisotropy.
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.
We show various aspects of finite size effects on Bose-Einstein condensation(BEC). In the first section we introduce very briefly the BEC of harmonically trapped ideal Bose gas. In the second section we theoretically argued that Bose-Einstein(B-E) statistics needs a correction for finite system at ultralow temperatures. As a corrected statistics we introduced a Tsallis type of generalized B-E statistics. The condensate fraction calculated with this generalized B-E statistics, is satisfied well with the experimental result. In the third section we show how to apply the scaling theory in an inhomogeneous system like harmonically trapped Bose condensate at finite temperatures. We calculate the temperature dependence of the critical number of particles by a scaling theory within the Hartree-Fock approximation and find that there is a dramatic increase in the critical number of particles as the condensation point is approached. Our results support the experimental result which was obtained well below the condensation temperature. In the fourth section we concentrate on the thermodynamic Casimir force on the Bose-Einstein condensate. We explored the temperature dependence of the Casimir force.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.