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Journal of Xiamen University

Numerical Analysis of Bose-Einstein Condensation with Finite Number of Particles in a Two-Dimensional Harmonic Oscillator Potential

Zhou Zhao

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The properties of Bose Einstein condensation with finite number of particles in a two dimensional harmonic oscillator potential are studied by using numerical analysis. In this way, it can be seen that the fraction N 0/N is changing gently with the function of T/T C and the heat capacity in all the temperature area is continuous. Considering the energy at ground state may effect the fugacity in this system, but it does not have any effect on the properties of the system.

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The properties of Bose Einstein condensation with finite number of particles in a two dimensional harmonic oscillator potential are studied by using numerical analysis. In this way, it can be seen that the fraction N 0/N is changing gently with the function of T/T C and the heat capacity in all the temperature area is continuous. Considering the energy at ground state may effect the fugacity in this system, but it does not have any effect on the properties of the system.
Bose–Einstein condensation is the anomalous accumulation of particles in the ground state of a system of bosons, when compared with the population in a system of particles obeying classical statistics. I use undergraduate-level statistical mechanics and a symbolic algebra computer program to study the occupation numbers of the energy levels of a finite number of noninteracting particles confined in a three-dimensional harmonic oscillator potential. I also calculate the heat capacity of the gas. The harmonic oscillator potential simplifies the calculations and approximates the conditions of the recent experiments achieving Bose–Einstein condensation in laser-cooled alkali vapors.
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.
By defining the characteristic length, the boundary effects of Bose-Einstein condensation in a three-dimensional harmonic trap are investigated using the Euler-MacLaurin formula. Results show that the condensed fraction of particles reduces due to the finite-size effects and the effects of finite particle number; the corrections of the condensation fraction and the condensation temperature have, respectively, a maximum value due to the boundary effect, hence it is very effective to optimize the parameters of the harmonic traps for improving the condensation fraction and the condensation temperature. In the jump of heat capacity exist the boundary effects and the effects of finite particle number, and the jump of heat capacity has a minimum because the parameters of harmonic traps are selected to be reasonable. The equation of state is derived for a finite ideal Bose gas system in a three-dimensional harmonic trap; the anisotropy (or isotropy) of the pressure is determined by the anisotropy (or isotropy) of the frequency of the harmonic potential.
We study the Bose-Einstein condensation phenomenon in a two-dimensional (2D) system of bosons subjected to a harmonic-oscillator-type confining potential. The interaction among the 2D bosons is described by a \ensuremath{\delta} function in configuration space. Solving the Gross-Pitaevskii equation within the two-fluid model we calculate the condensate fraction, ground-state energy, and specific heat of the system. Our results indicate that interacting bosons have similar behavior to those of an ideal system for weak interactions.
The Bose-Einstein condensation of a dilute gas of rubidium-87 atoms was achieved by cooling a small number of atoms in a magnetic trap. The effective potential of the trap is to lowest order harmonic and under these conditions we estimate the critical temperature and the heat capacity of an ideal gas of Bose particles. Previously this has been estimated within the semiclassical approximation. However, because of the small number of atoms, the resulting TcT_cTc​ deviates slightly from our result obtained by exact numerical calculations. For the gap of the heat capacity at the transition point the deviation is 20%20\%20%.
Abstract The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be εi≡εs1,....,s10=constanth2ms1α−1a12+...+swαaw2(1≤α≤2) the quantum numbers being s1 ,w= 1, 2, ..., and a1, ..., aw being certain characteristic lengths. (For α = 2, the potential field is that of the w-dimensional rectangular box; for α = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w/α determines the appearance of an Einstein transition temperature T0·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ- per particle and the specific heat dϵ-/dT are continuous at T = T0· For q > 2, the specific heat is discontinuous at T = T0, giving rise to a A λ-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T0 is finite only if the quantity v = N/(a1 .... aw)2/ α¯is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, then T0 likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T 0' there is a finite fraction N0 /N of the particles, given by N 0/N = 1-(T/T0)q, in the lowest state. London’s formula (1938 b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London’s continuous spectrum approximation.