On the Quantum Density of States and Partitioning an Integer

Details On the Quantum Density of States and Partitioning an Integer On the Quantum Density of States and Partitioning an Integer

2003-09-08

arxiv.org/abs/math-ph/0309020v1

Annals Phys. 311 (2004) 204-219

Physics

Mathematical Physics

17 pages including figure captions. 6 figures. To be submitted to J. Phys. A: Math. Gen

Scientific paper

10.1016/j.aop.2003.12.004

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.

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