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European Journal of Physics

On the Bose - Einstein condensation and phase transition of ann-dimensional ideal Bose gas

Sun Chang-yong, Wang Jisuo

Published Jul 1, 1997• 3 citations
Publisher

Abstract

The general expressions of energy, heat capacity and Bose - Einstein condensation (BEC) temperature are given for an n-dimensional ideal Bose gas with a dispersion relation of , with which the possibility of occurrence of BEC can be determined. The curve of against temperature of the gas with is continuous at , while that of the gas with has a finite jump . These are two different kinds of phase transitions.

DOI
10.1088/0143-0807/18/4/006

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The general expressions of energy, heat capacity and Bose - Einstein condensation (BEC) temperature are given for an n-dimensional ideal Bose gas with a dispersion relation of , with which the possibility of occurrence of BEC can be determined. The curve of against temperature of the gas with is continuous at , while that of the gas with has a finite jump . These are two different kinds of phase transitions.
The properties of a trapped ideal Bose gas in n -dimensional space are studied. General analytic expressions of the critical temperature of Bose-Einstein condensation, the jump of heat capacity at and the fraction of condensation at temperatures below have been derived. How these physical quantities depend on the external potential, particle characteristics and space dimensionality are discussed. We find that when a proper external potential is taken, Bose-Einstein condensation may occur in any dimensional space.
Bose-Einstein condensation in an ideal (i.e. interactionless) boson gas can be studied analytically, at university-level statistical and solid state physics, in any positive dimensionality ( d >0) for identical bosons with any positive-exponent ( s >0) energy-momentum (i.e. dispersion) relation. Explicit formulae with arbitrary d / s are discussed for: the critical temperature (non-zero only if d / s >1); the condensate fraction; the internal energy; and the constant-volume specific heat (found to possess a jump discontinuity only if d / s >2). Classical results are recovered at sufficiently high temperatures. Applications to `ordinary' Bose-Einstein condensation, as well as to photons, phonons, ferro- and antiferromagnetic magnons, and (very specially) to Cooper pairs in superconductivity, are mentioned.
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