Statistical mechanics and partitions into non-integral powers of integers

Abstract

1. The problem of the partition of numbers, first investigated in detail by Hardy and Ramanujan (1), has in recent years assumed importance on account of its application by Bohr and Kalckar (2) in evaluating the density of energy levels in heavy nuclei. A ‘physical approach’ to the partition theory has been made by Auluck and Kothari (3), who have studied the properties of quantal statistical assemblies corresponding to the partition functions familiar in the theory of numbers. The thermodynamical approach to the partition theory, apart from its intrinsic interest, draws attention to aspects and generalizations of the partition problem that would, otherwise, perhaps go unnoticed. Thus we are led to consider restricted partitions such as: partitions where the summands are repeated not more than a specified number of times; partitions where the summands are all different; partitions into summands which must not be less than a specified value; partitions into a prescribed number of summands, and so on. The generalization that seemed to us to be the most interesting is the extension of the partition concept to include partitions into non-integral powers of integers.