987 research outputs found
Regularity and chaos in the nuclear masses
Shell effects in atomic nuclei are a quantum mechanical manifestation of the
single--particle motion of the nucleons. They are directly related to the
structure and fluctuations of the single--particle spectrum. Our understanding
of these fluctuations and of their connections with the regular or chaotic
nature of the nucleonic motion has greatly increased in the last decades. In
the first part of these lectures these advances, based on random matrix
theories and semiclassical methods, are briefly reviewed. Their consequences on
the thermodynamic properties of Fermi gases and, in particular, on the masses
of atomic nuclei are then presented. The structure and importance of shell
effects in the nuclear masses with regular and chaotic nucleonic motion are
analyzed theoretically, and the results are compared to experimental data. We
clearly display experimental evidence of both types of motionComment: 40 pages, 10 figures, Lectures delivered at the VIII Hispalensis
International Summer School, Sevilla, Spain, June 2003 (to appear in Lecture
Notes in Physics, Springer--Verlag, Eds. J. M. Arias and M. Lozano
Random matrices, random polynomials and Coulomb systems
It is well known that the joint probability density of the eigenvalues of
Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We
review these classical results for hermitian and complex random matrices, with
special attention devoted to electrostatic analogies. We also discuss the joint
probability density of the zeros of polynomials whose coefficients are complex
Gaussian variables. This leads to a new two-dimensional solvable gas of
interacting particles, with non-trivial interactions between particles.Comment: 8 pages, to appear in the Proceedings of the International Conference
on Strongly Coupled Coulomb Systems, Saint-Malo, 199
Correlations in Nuclear Masses
It was recently suggested that the error with respect to experimental data in
nuclear mass calculations is due to the presence of chaotic motion. The theory
was tested by analyzing the typical error size. A more sensitive quantity, the
correlations of the mass error between neighboring nuclei, is studied here. The
results provide further support to this physical interpretation.Comment: 4 pages, 2 figure
Normal forms and complex periodic orbits in semiclassical expansions of Hamiltonian systems
Bifurcations of periodic orbits as an external parameter is varied are a
characteristic feature of generic Hamiltonian systems. Meyer's classification
of normal forms provides a powerful tool to understand the structure of phase
space dynamics in their neighborhood. We provide a pedestrian presentation of
this classical theory and extend it by including systematically the periodic
orbits lying in the complex plane on each side of the bifurcation. This allows
for a more coherent and unified treatment of contributions of periodic orbits
in semiclassical expansions. The contribution of complex fixed points is find
to be exponentially small only for a particular type of bifurcation (the
extremal one). In all other cases complex orbits give rise to corrections in
powers of and, unlike the former one, their contribution is hidden in
the ``shadow'' of a real periodic orbit.Comment: better ps figures available at http://www.phys.univ-tours.fr/~mouchet
or on request to [email protected]
Semiclassical Theory of Bardeen-Cooper-Schrieffer Pairing-Gap Fluctuations
Superfluidity and superconductivity are genuine many-body manifestations of
quantum coherence. For finite-size systems the associated pairing gap
fluctuates as a function of size or shape. We provide a parameter free
theoretical description of pairing fluctuations in mesoscopic systems
characterized by order/chaos dynamics. The theory accurately describes
experimental observations of nuclear superfluidity (regular system), predicts
universal fluctuations of superconductivity in small chaotic metallic grains,
and provides a global analysis in ultracold Fermi gases.Comment: 4 pages, 2 figure
On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates
We consider a single particle spectrum as given by the eigenvalues of the
Wigner-Dyson ensembles of random matrices, and fill consecutive single particle
levels with n fermions. Assuming that the fermions are non-interacting, we show
that the distribution of the total energy is Gaussian and its variance grows as
n^2 log n in the large-n limit. Next to leading order corrections are computed.
Some related quantities are discussed, in particular the nearest neighbor
spacing autocorrelation function. Canonical and gran canonical approaches are
considered and compared in detail. A semiclassical formula describing, as a
function of n, a non-universal behavior of the variance of the total energy
starting at a critical number of particles is also obtained. It is illustrated
with the particular case of single particle energies given by the imaginary
part of the zeros of the Riemann zeta function on the critical line.Comment: 28 pages in Latex format, 5 figures, submitted for publication to
Physica
Spectral spacing correlations for chaotic and disordered systems
New aspects of spectral fluctuations of (quantum) chaotic and diffusive
systems are considered, namely autocorrelations of the spacing between
consecutive levels or spacing autocovariances. They can be viewed as a
discretized two point correlation function. Their behavior results from two
different contributions. One corresponds to (universal) random matrix
eigenvalue fluctuations, the other to diffusive or chaotic characteristics of
the corresponding classical motion. A closed formula expressing spacing
autocovariances in terms of classical dynamical zeta functions, including the
Perron-Frobenius operator, is derived. It leads to a simple interpretation in
terms of classical resonances. The theory is applied to zeros of the Riemann
zeta function. A striking correspondence between the associated classical
dynamical zeta functions and the Riemann zeta itself is found. This induces a
resurgence phenomenon where the lowest Riemann zeros appear replicated an
infinite number of times as resonances and sub-resonances in the spacing
autocovariances. The theoretical results are confirmed by existing ``data''.
The present work further extends the already well known semiclassical
interpretation of properties of Riemann zeros.Comment: 28 pages, 6 figures, 1 table, To appear in the Gutzwiller
Festschrift, a special Issue of Foundations of Physic
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