On the Quantum Density of States and Partitioning an Integer

2003/09/08 by Muoi N. Tran, M. V. N. Murthy, Rajat K. Bhaduri · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP #math.NT #nucl-th

paper · pdf · doi:10.1016/j.aop.2003.12.004

published as Annals Phys. 311 (2004) 204-219 · 17 pages including figure captions. 6 figures. To be submitted to J. Phys. A: Math. Gen

arxiv created 2003/09/08 · arxiv updated 2009/12/01

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 d²(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.

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