On the Quantum Density of States and Partitioning an Integer

TL;DR

This paper connects quantum many-particle density of states with integer partition functions, deriving new formulas for partitions into powers and distinct parts, and analyzing oscillations in partition counts.

Contribution

It introduces a statistical mechanics approach to derive formulas for partitions into powers and distinct parts, including new generalizations and insights into oscillatory behaviors.

Findings

Derived formulas for p^s(n), the number of partitions into s-th powers.

Obtained d^s(n), the number of distinct partitions into s-th powers.

Identified oscillations in the number of distinct square partitions d^2(n).

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 -> infinity) 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 s-th 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 by our method, we generalize it to obtain a new formula for distinct restricted partitions.