Canonical Distribution of the Occupancy Numbers of Bosonic Systems

TL;DR

This paper solves the canonical occupancy-number problem for bosonic systems by showing that the canonical density operator, when restricted to the symmetric subspace, yields a multinomial distribution over occupancy numbers: $P_n = W_n\prod_c P_c^{n_c}$ with $W_n=N!/\prod_c n_c!$. It then proves a bosonic canonical typicality result: for a universe state chosen uniformly from the strongly typical set, the system’s occupancy statistics concentrate around the canonical form, with Levy-type bounds extending to the symmetric subspace. The Boltzmann distribution is shown to be incompatible with the canonical multinomial structure in small-$N$ regimes, and the Ziff–Uhlenbeck–Kac-type occupancy proposals are argued to arise from grand-canonical considerations rather than strict canonical sampling. Overall, the work reconciles canonicality for bosonic occupancy numbers and provides a rigorous link between typicality, symmetry, and multinomial statistics, with implications for few-particle condensates and entropy considerations.

Abstract

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.