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 investigate the equilibrium phase-coherence properties of Bose-condensed particle systems, focusing on their shape dependence and finite-size scaling (FSS). We consider three-dimensional (3D) homogeneous systems confined to anisotropic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{a}$ boxes, below the Bose-Einstein-condensate (BEC) transition temperature
Tc. We show that the phase correlations develop peculiar anisotropic FSS for any
T<Tc, in the large-
L limit keeping the ratio $\ensuremath{\lambda}\ensuremath{\equiv}{L}_{a}/{L}^{2}$ and the particle density fixed. This phenomenon is effectively described by the 3D spin-wave theory. Its universality is confirmed by quantum Monte Carlo simulations of the 3D Bose-Hubbard model in the BEC phase. The phase-coherence correlations of very elongated BEC systems, $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$, are characterized by the coherence length ${\ensuremath{\xi}}_{a}\ensuremath{\sim}{A}_{t}{\ensuremath{\rho}}_{s}/T$, where
At is the transverse area and ${\ensuremath{\rho}}_{s}$ is the superfluid density.