Do bosons obey Bose–Einstein distribution: Two iterated limits of Gentile distribution

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.

A new method for derivation of statistical weight of the Gentile Statistics

“…Moreover, Dai and Xie showed that Gentile statistics does not reduce to Bose-Einstein statistics generally but only if fugacity z = e βµ < 1 [14]. In addition to that, they show that one can obtain Bose-Einstein distribution f BE , from Gentile distribution f G , in thermodynamic limit when one takes two limits, maximum occupation number q → ∞, and the total number of particles (N → ∞) in a given order (f BE ≡ lim <N >→∞ lim n→∞ f G ) [15]. Furthermore, Donald and Zly derived thermodynamic properties for a harmonically confined gas obeying Gentile statistics in d-dimensions and compared these results with a similar system obeying Haldene-Wu statistics [16].…”

Singular vacuum solutions as singular matter solutions: Where do spacetime singularities come from?

“…The Einstein equation, a differential equation, has no definition at this singular point. When encountering singularities, the calculation sensitively depends on the order of the operation [29]. The above two approaches are indeed an operation with two different orders: one takes the limit operation at the beginning while the other takes the limit operation at the end.…”

An exactly solvable phase transition model: generalized statistics and generalized Bose-Einstein condensation

“…We first consider a generalized statistics in which the maximum occupation numbers of different quantum states can take on different values. When the maximum occupation numbers of all the states are the same, e.g., equaling ∞, 1, or an arbitrary integer, the generalized statistics returns to Bose-Einstein, Fermi-Dirac, or Gentile statistics [5,6,12], respectively. The model constructed in this paper is an ideal gas obeying the generalized statistics in which the maximum occupation number of only one state is infinite, but of all other states is finite.…”