Do bosons obey Bose-Einstein distribution: two iterated limits of Gentile distribution

TL;DR

This paper investigates the conditions under which bosons obey the Bose-Einstein distribution, emphasizing the importance of the order of limits in the derivation involving maximum occupation number and total particles.

Contribution

It reveals that achieving the Bose-Einstein distribution requires careful consideration of the order of taking limits for maximum occupation number and total particles.

Findings

Different limit orders lead to different distributions.

Infinite maximum occupation number alone does not guarantee Bose-Einstein distribution.

Both limits must be taken to infinity in a specific order for the distribution to hold.

Abstract

It is a common impression that by only setting the maximum occupation number to infinity, which is the demand of the indistinguishability of bosons, one can achieve the statistical distribution that bosons obey -- the Bose-Einstein distribution. In this letter, however, we show that only with an infinite maximum occupation number one cannot uniquely achieve the Bose-Einstein distribution, since in the derivation of the Bose-Einstein distribution, the problem of iterated limit is encountered. For achieving the Bose-Einstein distribution, one needs to take both the maximum occupation number and the total number of particles to infinities, and, then, the problem of the order of taking limits arises. Different orders of the limit operations will lead to different statistical distributions. For achieving the Bose-Einstein distribution, besides setting the maximum occupation number, we also need to state the order of the limit operations.