Ground state fluctuations in finite Fermi and Bose systems

We consider a small and fixed number of fermions (bosons) in a trap. The ground state of the system is defined at T=0. For a given excitation energy, there are several ways of exciting the particles from this ground state. We formulate a method for calculating the number fluctuation in the ground state using microcanonical counting, and implement it for small systems of noninteracting fermions as well as bosons in harmonic confinement. This exact calculation for fluctuation, when compared with canonical ensemble averaging, gives considerably different results, specially for fermions. This difference is expected to persist at low excitation even when the fermion number in the trap is large.Comment: 20 pages (including 1 appendix), 3 postscript figures. An error was found in one section of the paper. The corrected version is updated on Sep/05/200

On the Microcanonical Entropy of a Black Hole

It has been suggested recently that the microcanonical entropy of a system may be accurately reproduced by including a logarithmic correction to the canonical entropy. In this paper we test this claim both analytically and numerically by considering three simple thermodynamic models whose energy spectrum may be defined in terms of one quantum number only, as in a non-rotating black hole. The first two pertain to collections of noninteracting bosons, with logarithmic and power-law spectra. The last is an area ensemble for a black hole with equi-spaced area spectrum. In this case, the many-body degeneracy factor can be obtained analytically in a closed form. We also show that in this model, the leading term in the entropy is proportional to the horizon area A, and the next term is ln A with a negative coefficient.Comment: 15 pages, 1 figur

On the Quantum Density of States and Partitioning an Integer

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

Number Fluctuation in an interacting trapped gas in one and two dimensions

It is well-known that the number fluctuation in the grand canonical ensemble, which is directly proportional to the compressibility, diverges for an ideal bose gas as T -> 0. We show that this divergence is removed when the atoms interact in one dimension through an inverse square two-body interaction. In two dimensions, similar results are obtained using a self-consistent Thomas-Fermi (TF) model for a repulsive zero-range interaction. Both models may be mapped on to a system of non-interacting particles obeying the Haldane-Wu exclusion statistics. We also calculate the number fluctuation from the ground state of the gas in these interacting models, and compare the grand canonical results with those obtained from the canonical ensemble.Comment: 11 pages, 1 appendix, 3 figures. Submitted to J. Phys. B: Atomic, Molecular & Optica

Number Fluctuation and the Fundamental Theorem of Arithmetic

We consider N bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counter example, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations. The use of the standard canonical or grand canonical ensembles, on the other hand, gives substantial number fluctuation for the ground state. This difference between the microcanonical and canonical results cannot be accounted for within the framework of equilibrium statistical mechanics.Comment: 4 pages, 4 figures. To be submitted to Phys. Rev. Let

Feature/model selection by the linear programming SVM combined with state-of-the-art classifiers: What we can learn about the data

NRC publication: Ye