Calculation of the entropy of an ideal Bose Einstein Condensate (BEC) in a three dimensional trap reveals unusual, previously unrecognized, features of the Canonical Ensemble. It is found that, for any temperature, the entropy of the Bose gas is equal to the entropy of the excited particles although the entropy of the particles in the ground state is nonzero. We explain this by considering the correlations between the ground state particles and particles in the excited states. These correlations lead to a correlation entropy which is exactly equal to the contribution from the ground state. The correlations themselves arise from the fact that we have a fixed number of particles obeying quantum statistics. We present results for correlation functions between the ground and excited states in Bose gas, so to clarify the role of fluctuations in the system. We also report the sub-Poissonian nature of the ground state fluctuations.
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