On the quantum density of states and partitioning an integer

Abstract

This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For N bosons in a one-dimensional harmonic oscillator potential, it is well known that the asymptotic ( N→∞) density of states is identical to the Hardy–Ramanujan formula for the partitions p( n), of a number n into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for p s ( n), the latter being the number of partitions of n into a sum of sth powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain d s ( n) for distinct partitions. We find that the distinct square partitions d 2( n) show pronounced oscillations as a function of n about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdos–Lehner formula for restricted partitions for the s=1 case, we use the modified technique to obtain a new formula for distinct restricted partitions.