微正则、正则和巨正则系综中非相互作用玻色子和费米子的统计：综述
Statistics of non-interacting bosons and fermions in micro-canonical, canonical and grand-canonical ensembles: A survey
摘要：
当系统可以与一个大的储存库（巨正则系综）交换能量和粒子时，最容易获得限制在捕获势中的非相互作用玻色子和费米子的统计特性。然而，在某些情况下，所考虑的系统可以被认为是孤立的（微正则系综）。本文首先回顾了与微正则系综相关的研究成果。其中一些是很久以前获得的，特别是Khinchin在1950年。另一些是最近才获得的，通常是受到与原子限制有关的实验结果的启发。本文首次报道了一些公式。从微正则系综表达式导出了适用于系统可以与储库（正则系综）交换能量但不交换粒子的情况的公式。在所谓的热力学极限中，三个系综之间的差异趋于消失，即当粒子数和体积达到无穷大而粒子数密度保持不变时。但我们最感兴趣的是中等大小的系统，通常被称为介观，其中巨正则形式主义不适用。数学结果主要依赖于数的分区的枚举。
英文摘要：
The statistical properties of non-interacting bosons and fermions confined in trapping potentials are most easily obtained when the system may exchange energy and particles with a large reservoir (grand-canonical ensemble). There are circumstances, however, where the system under consideration may be considered as being isolated (micro-canonical ensemble). This paper first reviews results relating to micro-canonical ensembles. Some of them were obtained a long time ago, particularly by Khinchin in 1950. Others were obtained only recently, often motivated by experimental results relating to atomic confinement. A number of formulas are reported for the first time in the present paper. Formulas applicable to the case where the system may exchange energy but not particles with a reservoir (canonical ensemble) are derived from the micro-canonical ensemble expressions. The differences between the three ensembles tend to vanish in the so-called Thermodynamics limit, that is, when the number of particles and the volume go to infinity while the particle number density remains constant. But we are mostly interested in systems of moderate size, often referred to as being mesoscopic, where the grand-canonical formalism is not applicable. The mathematical results rest primarily on the enumeration of partitions of numbers.
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