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On Bose-Einstein Condensation in Any Dimension

Hugo Pérez Rojas

Published Jan 1, 1996• 12 citations
OpenAlex

Abstract

Arbitrarily large ground state population is a general property of any ideal bose gas when conditions of degeneracy are satisfied; it occurs at any dimension D. For D = 1, the condensation is diffuse, at D = 2 it is a sort of quasi-condensate. The discussion is made by following a microscopic approach and for finite systems. Some astrophysical consequences are discussed, as well as the temperature-dependent mass case.

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Journal of Zhangzhou Normal University•Jan 1, 2001•0 citations
Arbitrarily large ground state population is a general property of any ideal bose gas when conditions of degeneracy are satisfied; it occurs at any dimension D. For D = 1, the condensation is diffuse, at D = 2 it is a sort of quasi-condensate. The discussion is made by following a microscopic approach and for finite systems. Some astrophysical consequences are discussed, as well as the temperature-dependent mass case.
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.
The calculation of the thermodynamic properties in the Bose-Einstein distribution is presented.The behaviour of the Bose-gas in the Bose-Einstein distribution is analysed based on high and low temperature dependence.At > all particles are in the excited state and the ground state is essentially occupied.But at B T T < particles gradually falls to the ground state and eventually contains all the particles in the system as .0 → TThe analytical expressions for some important parameters such as Bose-Einstein condensation of an ideal Bose gas, the population density of the excited and ground state particles are also presented.Heart capacity of an ideal Bose gas and total internal energy are also calculated and the results presented graphically.The results obtained are consistent with those found in the literatures.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.
Bose–Einstein condensation can be understood as a macroscopic occupation of a single-particle state. But an alternative definition, which requires only a macroscopic occupation of an arbitrarily small energy band of single-particle states, is also known, referred to as generalized Bose–Einstein condensation, and considered to be thermodynamically more stable. Depending on the number of macroscopically occupied single-particle states in the condensate, three types are then distin-guished. If the number of macroscopically occupied single-particle states is finite but at least one, then the condensation is said to be of type-I. A type-II condensate occurs if infinitely many single-particle states are macroscopically occupied. If a generalized Bose–Einstein condensation without any macroscopically occupied single-particle states is present, then one speaks of a type-III con-densation. In noninteracting Bose gases, it is generally easier to prove the occurrence of generalized Bose–Einstein condensation than to determine its type. While a main step for the former task is to show that the critical density is finite, quite detailed knowledge of the energy gaps of the cor-responding single-particle Hamiltonian at the lower edge of the spectrum is needed for the latter one. In this dissertation, we study the occurrence of a Bose–Einstein condensate in Bose gases that are placed in metrically transitive random potentials. We mainly consider the Poisson random potential. The thermodynamic limit and Dirichlet boundary conditions are used throughout this work. We explore noninteracting Bose gases in arbitrary dimension at positive temperatures and a one-dimensional interacting Bose gas at zero temperature. We provide fairly general sufficient conditions for the random potential that imply the almost sure occurrence of generalized Bose–Einstein condensation in noninteracting Bose gases. These conditions are fulfilled by a Poisson random potential, depending on its single-impurity potential. They are also met by the one-dimensional Luttinger–Sy model, which is a central system in the research area of Bose gases in random potentials, and a generalization thereof. Moreover, we state sufficient energy gap conditions such that one can conclude a type-I condensation in Poisson random potentials. We consider different probabilistic notions here, such as convergence in the rth mean, r P N, in probability, and with probability almost one. In particular, we prove that in the Luttinger–Sy model with infinitely strong random potential a type-I condensation occurs in the rth mean, r P N, if the particle density is greater than the critical one. For the Luttinger–Sy model with a finitely strong random potential, we show the same statement, but with probability almost one. Finally, we discuss the Luttinger–Sy model with a repulsive contact interaction between the particles at zero temperature. We provide sufficient requirements such that a generalized Bose–Einstein condensate almost surely occurs. We also show that the type of the condensate is altered by the interparticle interaction. In particular, we demonstrate a transition from a type-I to a type-III condensate when the strength of the contact interaction is increased. Furthermore, we prove that the interparticle interaction reduces the local particle density.