Entropy of an extreme Reissner-Nordström black hole

The definition of entropy consistent with the Nernst theorem of ordinary thermodynamics contradicts the area theorem, which means the breakdown of weak energy condition in an adiabatic process. Such has never occurred in the ordinary thermodynamics. It implies that the extreme black hole is not so alike as the ordinary system and it cannot be treated as the limit of the non-extreme case. In consideration of the Bose-Einstein condensation of the scalar field, the quantum entropy of an extreme RNBH is proportional to the logarithm of the horizon area plus the logarithmic divergence of ln 1/ε, ε is a cutoff near the horizon. It is satisfying that the thermodynamic limit of the quantum entropy approaches zero even if ε → 0.