Canonical Distribution of the Occupancy Numbers of Bosonic Systems

TL;DR

This paper derives the canonical distribution of bosonic occupancy numbers, demonstrating that it converges to a multinomial distribution under typicality assumptions, and challenges existing notions of occupancy distributions.

Contribution

It provides a detailed derivation of the canonical distribution for bosonic systems and links it to the concept of canonical typicality, offering new insights into occupancy statistics.

Findings

Canonical distribution of bosonic occupancy numbers derived.

Asymptotic convergence to multinomial distribution shown.

Current occupancy distribution models are incompatible with canonical typicality.

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.