More Accurate Theory for Bose-Einstein Condensation Fraction

In the thermodynamic limit the ratio of system size to thermal de Broglie wavelength tends to infinity and the volume per particle of the system is constant. Our familiar Bose-Einstein statistics is absolutely valid in the thermodynamic limit. For finite thermodynamical system this ratio as well as the number of particles is much greater than 1. However, according to the experimental setup of Bose-Einstein condensation of harmonically trapped Bose gas of alkali atoms this ratio near the condensation temperature($T_c$) typically is $\sim 32$ and at ultralow temperatures well below $T_c$ a large fraction of particles come down to the single particle ground state, and this ratio becomes comparable to 1. We justify the finite size as well as ultralow temperature correction to Bose-Einstein statistics. From this corrected statistics we plot condensation fraction versus temperature graph. This theoretical plot satisfies well with the experimental plot(A. Griesmaier et al..,Phys.Rev.Lett. {\bf{{94}}}{(2005){160401}}).