AI 中文总结
探讨半经典方法在物理多体系统与解析数论整数划分中的应用，通过单粒子和多粒子系统的态密度与周期轨道联系，再现渐近数划分，在平方划分中有明显振荡，还讨论了素数划分新结果。
AI 中文摘要
在本综述中，我们讨论了传统上用于描述物理中多体系统的半经典方法，这些方法也可用于解析数论中描述整数划分。具体而言，我们探索统计力学方法与数的划分之间的联系。尽管这两个领域看似不同，但它们的基本问题极为相似。前者是给定温度下具有明确性质的系综中给定能量在粒子间的分布，后者是整数划分为其他整数的方式。我们首先讨论单粒子量子态密度，通过半经典迹公式说明态密度与经典周期轨道的联系，然后扩展到多粒子系统。我们表明，在自变量的离散整数值处，渐近数划分由态密度的平均（平滑）部分再现。在不同平方划分的特别有趣的情况下，周期轨道理论能很好地再现由勾股数三元组表征的少数轨道的明显振荡。我们推测与费马定理的联系，即为何这种规则振荡（虽渐近消失）仅在这种特殊情况下存在。最后，我们讨论了素数整数划分的一些新结果，包括无限制和不同的情况。
英文摘要
In this review we discuss semi-classical methods that are traditionally used to describe many-body systems in physics, but may also be used to describe partitions of integers in analytic number theory. Specifically, we explore the connection between the methods of statistical mechanics and number partitions. Though the two fields appear very different, their fundamental issues bear a close resemblance. In the former case it is the distribution of a given amount of energy among the particles in an ensemble at a given temperature with well defined properties, while in the latter case it is the way an integer is partitioned into other integers, with or without restrictions. We begin with a discussion of the single-particle quantum density of states, also called the level density, in which we illustrate the connection between the density of states and the classical periodic orbits through the semiclassical trace formula. This is then extended to many particle systems. We show that the asymptotic number partition is reproduced by the average (smooth) part of the level density at discrete integer values of the argument. In the especially interesting case of distinct square partitions, pronounced oscillations are well reproduced by the periodic orbit theory in terms of a few orbits characterised by Pythagorean number triples. We speculate on the connection to Fermat's theorem as to why such regular oscillations (though vanishing asymptotically) exist only in this special case. Finally, we discuss some new results for integer partitions of primes, both unrestricted and distinct.
Comments52 pages, 21 figures
