Transition temperatures of Bose-Einstein condensation in traps

A careful study is made of the definitions of the transition temperature T-c of an ideal Bose system in traps. We review several physical quantities that are used to define the transition temperature, which corresponds to different kinds of T-c. The different definitions give different values for T-c and the differences are quite large for finite systems. This makes the comparisons of theoretical predictions of the transition temperature shift with experimental results quite difficult. We also find that the derivative of chemical potential with respect to temperature is nearly discontinuous. This implies that in the thermodynamic limit the system might undergo a first-order phase transition.