In the microcanonical ensemble, the ideal Bose system is divided into a normal part and a condensed part, with the weight of each part determined by the condition of maximum entropy. Using this binary approximation, the characteristic temperature of the ideal Bose system and the relationship between the ground state occupation number and temperature can be obtained. Taking the ideal Bose gas in three-dimensional free space and in a one-dimensional harmonic oscillator potential as examples, this paper demonstrates the solution process by this binary approximation, and the results are very close to the exact solutions.

The statement on the thickness of the energy shell in the microcanonical ensemble varies from literature to literature. Based on the particular derivation of the entropy of the ideal gas, we conclude that the thickness of the energy shell of the ideal gas system must be larger than the energy per particle, which is quite different from the previous literatures.