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 -> 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.

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

References in corpus (5)

• Quantum Geometry and Black Hole Entropy
• General Logarithmic Corrections to Black Hole Entropy
• Condensate Fluctuations in Trapped Bose Gases: Canonical vs. Microcanonical Ensemble
• Semiclassical Trace Formulas for Noninteracting Identical Particles
• Exact ground state number fluctuations of trapped ideal and interacting fermions

Cited by in corpus (22)

• Level Density of a Bose Gas and Extreme Value Statistics
• Fractional statistics and finite bosonic system: A one-dimensional case
• Generating All Partitions: A Comparison Of Two Encodings
• Gentile statistics and restricted partitions
• Statistical mechanics approach in the counting of integer partitions
• On the Microcanonical Entropy of a Black Hole
• Fermi-Dirac statistics and the number theory
• Thermodynamics of low-dimensional trapped Fermi gases
• A statistical mechanical approach to restricted integer partition functions
• Asymptotic formulas for integer partitions within the approach of microcanonical ensemble
• Holography and entropy bounds in the plane wave matrix model
• Level density of a Fermi gas and integer partitions: a Gumbel-like finite-size correction
• Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes
• On the asymptotic prime partitions of integers
• What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I
• Developments in the Khintchine-Meinardus probabilistic method for asymptotic enumeration
• Enumeration of plane partitions with a restricted number of parts
• Semiclassical analysis of distinct square partitions
• Study of the vacuum matrix element of products of parafields
• A glimpse into the Ultrametric spectrum
• The curious case of operators with spectral density increasing as $\Omega (E)\sim e^{Const.E^2}$
• A general asymptotic formula for distinct partitions