Microcanonical approach to the entropy of a quantum field in a Kerr hole

It is shown that that the area la rv for the entropy of a quantum field in a Kerr space-time is due to the quantum statistics with the maximal entropy principle. The entropies for one particle, a Boltzmann gas, a quantum mech anical gas obeying Bose-Einstein or Fermi-Dirac statistics, and a quantum f ield in the Kerr space-time are calculated using the microcanonical ensembl e approach and the brick wall method. The area law holds only when the syst em follows the quantum statistics and it satisfies the maximal entropy prin ciple of a given energy.