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 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.
The chemical potential and its differential coefficients as a function of temperature of an ideal system trapped in three dimensions isotropic harmonic traps with an finite number of particles were calcu- lated from the statistical theory.The results show that,there is no phase transition in a system with an finite number of particles,but the Bose-Einstein condensation occurs at a finite temperature.The critical temperatures were well defined by the minimum temperatures of the second derivative of chemical potential, which were in good agreement with the experimental results.
We have investigated the thermodynamic behaviour of ideal Bose gases with an arbitrary number of particles confined in a harmonic potential. By taking into account the conservation of the total number N of particles and using a saddle-point approximation, we derive analytically the simple explicit expression of mean occupation number in any state of the finite system. The temperature dependence of the chemical potential, specific heat, and condensate fraction for the trapped finite-size Bose system is obtained numerically. We compare our results with the usual treatment which is based on the grand canonical ensemble. It is shown that there exists a considerable difference between them at sufficiently low temperatures, especially for the relative small numbers of Bose atoms. The finite-size scaling at the transition temperature for the harmonically trapped systems is also discussed. We find that the scaled condensate fractions for various system sizes and temperatures collapse onto a single scaled form.
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 consider a finite number of noninteracting bosons trapped in an isotropic three-dimensional harmonic oscillator. Using the grand canonical ensemble, we calculate the heat capacity of the system as a function of temperature, for several values of the number of particles. We find that a new definition of critical temperature is necessary for a finite number of trapped particles, and present a definition that is experimentally convenient and in good agreement with a recent definition given by W. Ketterle and N. J. van Druten [Phys. Rev. A 54, 656 (1996)].
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number
N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature
Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of
x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of
Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with
104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.