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Journal of Yunyang Teachers College

Low-dimensional Bose-einstein Condensation under External Potential

Liu Chuan-yong

Published Jan 1, 2009• 0 citations
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Abstract

The discussion is made on the relation between the state density of systems and the dimensionality of Bose-Einstein Condensation(BEC),with the further stress laid on the analysis of low dimensional BEC under the influence of power-law potential and harmonic potential.The result is the non-production of low dimensional BEC with no external potential and the production of one-dimensional and two-dimensional BEC under the power of external potential.

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•Jan 1, 2001•7 citations
The discussion is made on the relation between the state density of systems and the dimensionality of Bose-Einstein Condensation(BEC),with the further stress laid on the analysis of low dimensional BEC under the influence of power-law potential and harmonic potential.The result is the non-production of low dimensional BEC with no external potential and the production of one-dimensional and two-dimensional BEC under the power of external potential.
The relationship of the critical temperature versus the total number of particles in different dimensional spaces,as well as the effect of the first-excited-state energy of particles on the critical temperature of low-dimensional Bose-Einstein condensation,are discussed in this paper.It is shown that the critical temperature in low-dimensional space increases more quickly than that in high-dimensional space with density of particle number increasing,and the critical temperature of system has close relation to the first-excited-state energy.In addition,even under the condition of absent special external potential,such as harmonic potential and power-law potential,the Bose-Einstein condensation can be realized at several K0 for excitons gas trapped in GaAs quantum well.
In this thesis, the properties of the Bose-Einstein condensation (BEC) in low dimensions are reviewed. Three dimensional weakly interacting Bose systems are examined by the variational method. The effects of both the attractive and the repulsive interatomic forces are studied. Thomas-Fermi approximation is applied to find the ground state energy and the chemical potential. The occurrence of the BEC in low dimensional systems, is studied for ideal gases confined by both harmonic and power-law potentials. The properties of BEC in highly anisotropic trap are investigated and the conditions for reduced dimensionality are derived.
In a recent paper [Int. J. Mod. Phys. B 14, 405 (2000)] we discussed the Bose-Einstein condensation (BEC) in the framework of Tsallis’s nonextensive statistical mechanics. In particular, we studied an ideal gas of bosons in a confining harmonic potential. In this memoir we generalize our previous analysis by investigating an ideal Bose gas in a generic power-law external potential. We derive analytical formulas for the energy of the system, the BEC transition temperature and the condensed fraction. PACS numbers: 05.30-d; 03.75.Fi 1 Recently we analyzed the consequences of weak nonextensivity on Bose-Einstein Condensation for an ideal Bose gas 1 by using Tsallis’s nonextensive statistical mechanics (NSM). 2 From the nonextensive Bose-Einstein distribution 3−5 we derived the BEC transition temperature, the condensed fraction and the energy per particle in three different cases: the homogeneous gas, the gas in a harmonic trap and the relativistic homogenous gas. In this short contribution we generalize our previous results by considering the ideal Bose gas in a generic power-law potential. All the calculations are performed by assuming a D-dimensional space. The NSM predicts that, for an ideal quantum gas of identical bosons in the grand canonical ensemble, the weak nonextensivity formula of the average number of particles with energy ǫ is given by 〈n(ǫ)〉q = 1