580 research outputs found
Fluctuations in the level density of a Fermi gas
We present a theory that accurately describes the counting of excited states
of a noninteracting fermionic gas. At high excitation energies the results
reproduce Bethe's theory. At low energies oscillatory corrections to the
many--body density of states, related to shell effects, are obtained. The
fluctuations depend non-trivially on energy and particle number. Universality
and connections with Poisson statistics and random matrix theory are
established for regular and chaotic single--particle motion.Comment: 4 pages, 1 figur
Rademacher's infinite partial fraction conjecture is (almost certainly) false
In his book \emph{Topics in Analytic Number Theory}, Hans Rademacher
conjectured that the limits of certain sequences of coefficients that arise in
the ordinary partial fraction decomposition of the generating function for
partitions of integers into at most parts exist and equal particular values
that he specified. Despite being open for nearly four decades, little progress
has been made toward proving or disproving the conjecture, perhaps in part due
to the difficulty in actually computing the coefficients in question.
In this paper, we provide a fast algorithm for calculating the Rademacher
coefficients, a large amount of data, direct formulas for certain collections
of Rademacher coefficients, and overwhelming evidence against the truth of the
conjecture. While the limits of the sequences of Rademacher coefficients do not
exist (the sequences oscillate and attain arbitrarily large positive and
negative values), the sequences do get very close to Rademacher's conjectured
limits for certain (predictable) indices in the sequences
Level density of a Fermi gas: average growth and fluctuations
We compute the level density of a two--component Fermi gas as a function of
the number of particles, angular momentum and excitation energy. The result
includes smooth low--energy corrections to the leading Bethe term (connected to
a generalization of the partition problem and Hardy--Ramanujan formula) plus
oscillatory corrections that describe shell effects. When applied to nuclear
level densities, the theory provides a unified formulation valid from
low--lying states up to levels entering the continuum. The comparison with
experimental data from neutron resonances gives excellent results.Comment: 4 pages, 1 figur
Riemann solvers and undercompressive shocks of convex FPU chains
We consider FPU-type atomic chains with general convex potentials. The naive
continuum limit in the hyperbolic space-time scaling is the p-system of mass
and momentum conservation. We systematically compare Riemann solutions to the
p-system with numerical solutions to discrete Riemann problems in FPU chains,
and argue that the latter can be described by modified p-system Riemann
solvers. We allow the flux to have a turning point, and observe a third type of
elementary wave (conservative shocks) in the atomistic simulations. These waves
are heteroclinic travelling waves and correspond to non-classical,
undercompressive shocks of the p-system. We analyse such shocks for fluxes with
one or more turning points.
Depending on the convexity properties of the flux we propose FPU-Riemann
solvers. Our numerical simulations confirm that Lax-shocks are replaced by so
called dispersive shocks. For convex-concave flux we provide numerical evidence
that convex FPU chains follow the p-system in generating conservative shocks
that are supersonic. For concave-convex flux, however, the conservative shocks
of the p-system are subsonic and do not appear in FPU-Riemann solutions
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
An Exact Black Hole Entropy Bound
We show that a Rademacher expansion can be used to establish an exact bound
for the entropy of black holes within a conformal field theory framework. This
convergent expansion includes all subleading corrections to the
Bekenstein-Hawking term.Comment: 6 pages, Latex, v2 minor re-wording, additional reference, to appear
in Phyical Review D (title changed in journal
Finsler Conformal Lichnerowicz-Obata conjecture
We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture
showing that a complete and essential conformal vector field on a
non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski
metric.Comment: 13 pages, 2 figures; the new version has only minor changes with
respect to v1, and is the version that will be published in Annales de
L'Institut Fourie
An Investigation of Aircraft Heaters XXXI : Summary of Laboratory Testing of Several Exhaust-gas and Air Heat Exchangers
Cardy and Kerr
The Kerr/CFT correspondence employs the Cardy formula to compute the entropy
of the left moving CFT states. This computation, which correctly reproduces the
Bekenstein--Hawking entropy of the four-dimensional extremal Kerr black hole,
is performed in a regime where the temperature is of order unity rather than in
a high-temperature regime. We show that the comparison of the entropy of the
extreme Kerr black hole and the entropy in the CFT can be understood within the
Cardy regime by considering a D0-D6 system with the same entropic properties.Comment: 20 pages; LaTeX; JHEP format; v.2 references added, v.3 Section 4
adde
On the energy functional on Finsler manifolds and applications to stationary spacetimes
In this paper we first study some global properties of the energy functional
on a non-reversible Finsler manifold. In particular we present a fully detailed
proof of the Palais--Smale condition under the completeness of the Finsler
metric. Moreover we define a Finsler metric of Randers type, which we call
Fermat metric, associated to a conformally standard stationary spacetime. We
shall study the influence of the Fermat metric on the causal properties of the
spacetime, mainly the global hyperbolicity. Moreover we study the relations
between the energy functional of the Fermat metric and the Fermat principle for
the light rays in the spacetime. This allows us to obtain existence and
multiplicity results for light rays, using the Finsler theory. Finally the case
of timelike geodesics with fixed energy is considered.Comment: 23 pages, AMSLaTeX. v4 matches the published versio
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