14 research outputs found
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 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 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