Canonical Distribution of the Occupancy Numbers of Bosonic Systems

The paper works out the canonical probability distribution of the occupancy numbers of a bosonic system and shows that canonical typicality applies to the canonical density operator of the occupancy numbers. The result is that, if, as it is today standard, the canonical system’s mixed state is obtained by tracing out the environment from any typical pure state of the universe, then asymptotically the canonical probability distribution of system’s occupancy numbers tends in probability to the multinomial distribution. The paper also shows that the currently accepted probability distribution of the occupancy numbers of a system with fixed number of particles is not compatible with the commonly accepted notion of canonical system.