Two-step BEC coming from a temperature dependent energy gap

We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap $\Delta (T)$ at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature $T_B$, as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature $T_c$ is equal to that of the ideal Bose gas $T_0$, for all $T_B \leq T_0$; surprisingly, the condensate fraction presents two different filling rates of the ground state at $T_c = T_0$ and at $T_B < T_0$; while the specific heat shows a finite jump at $T_c$ as well as a divergence at $T_B$. Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero.Comment: 10 pages, 17 figure