100 research outputs found
The Riemannium
The properties of a fictitious, fermionic, many-body system based on the
complex zeros of the Riemann zeta function are studied. The imaginary part of
the zeros are interpreted as mean-field single-particle energies, and one fills
them up to a Fermi energy . The distribution of the total energy is shown
to be non-Gaussian, asymmetric, and independent of in the limit
. The moments of the limit distribution are computed
analytically. The autocorrelation function, the finite energy corrections, and
a comparison with random matrix theory are also discussed.Comment: 10 pages, 2 figures, 1 tabl
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
Nodal domain distributions for quantum maps
The statistics of the nodal lines and nodal domains of the eigenfunctions of
quantum billiards have recently been observed to be fingerprints of the
chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett.,
Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88
(2002), 114102). These statistics were shown to be computable from the random
wave model of the eigenfunctions. We here study the analogous problem for
chaotic maps whose phase space is the two-torus. We show that the distributions
of the numbers of nodal points and nodal domains of the eigenvectors of the
corresponding quantum maps can be computed straightforwardly and exactly using
random matrix theory. We compare the predictions with the results of numerical
computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction
Average ground-state energy of finite Fermi systems
Semiclassical theories like the Thomas-Fermi and Wigner-Kirkwood methods give
a good description of the smooth average part of the total energy of a Fermi
gas in some external potential when the chemical potential is varied. However,
in systems with a fixed number of particles N, these methods overbind the
actual average of the quantum energy as N is varied. We describe a theory that
accounts for this effect. Numerical illustrations are discussed for fermions
trapped in a harmonic oscillator potential and in a hard wall cavity, and for
self-consistent calculations of atomic nuclei. In the latter case, the
influence of deformations on the average behavior of the energy is also
considered.Comment: 10 pages, 8 figure
Quantum thermodynamic fluctuations of a chaotic Fermi-gas model
We investigate the thermodynamics of a Fermi gas whose single-particle energy
levels are given by the complex zeros of the Riemann zeta function. This is a
model for a gas, and in particular for an atomic nucleus, with an underlying
fully chaotic classical dynamics. The probability distributions of the quantum
fluctuations of the grand potential and entropy of the gas are computed as a
function of temperature and compared, with good agreement, with general
predictions obtained from random matrix theory and periodic orbit theory (based
on prime numbers). In each case the universal and non--universal regimes are
identified.Comment: 23 pages, 4 figures, 1 tabl
Thermodynamics of small Fermi systems: quantum statistical fluctuations
We investigate the probability distribution of the quantum fluctuations of
thermodynamic functions of finite, ballistic, phase-coherent Fermi gases.
Depending on the chaotic or integrable nature of the underlying classical
dynamics, on the thermodynamic function considered, and on temperature, we find
that the probability distributions are dominated either (i) by the local
fluctuations of the single-particle spectrum on the scale of the mean level
spacing, or (ii) by the long-range modulations of that spectrum produced by the
short periodic orbits. In case (i) the probability distributions are computed
using the appropriate local universality class, uncorrelated levels for
integrable systems and random matrix theory for chaotic ones. In case (ii) all
the moments of the distributions can be explicitly computed in terms of
periodic orbit theory, and are system-dependent, non-universal, functions. The
dependence on temperature and number of particles of the fluctuations is
explicitly computed in all cases, and the different relevant energy scales are
displayed.Comment: 24 pages, 7 figures, 5 table
Wavefunctions, Green's functions and expectation values in terms of spectral determinants
We derive semiclassical approximations for wavefunctions, Green's functions
and expectation values for classically chaotic quantum systems. Our method
consists of applying singular and regular perturbations to quantum
Hamiltonians. The wavefunctions, Green's functions and expectation values of
the unperturbed Hamiltonian are expressed in terms of the spectral determinant
of the perturbed Hamiltonian. Semiclassical resummation methods for spectral
determinants are applied and yield approximations in terms of a finite number
of classical trajectories. The final formulas have a simple form. In contrast
to Poincare surface of section methods, the resummation is done in terms of the
periods of the trajectories.Comment: 18 pages, no figure
Wavepacket Dynamics in Nonlinear Schr\"odinger Equations
Coherent states play an important role in quantum mechanics because of their
unique properties under time evolution. Here we explore this concept for
one-dimensional repulsive nonlinear Schr\"odinger equations, which describe
weakly interacting Bose-Einstein condensates or light propagation in a
nonlinear medium. It is shown that the dynamics of phase-space translations of
the ground state of a harmonic potential is quite simple: the centre follows a
classical trajectory whereas its shape does not vary in time. The parabolic
potential is the only one that satisfies this property. We study the time
evolution of these nonlinear coherent states under perturbations of their
shape, or of the confining potential. A rich variety of effects emerges. In
particular, in the presence of anharmonicities, we observe that the packet
splits into two distinct components. A fraction of the condensate is
transferred towards uncoherent high-energy modes, while the amplitude of
oscillation of the remaining coherent component is damped towards the bottom of
the well
Geometric characterization of nodal domains: the area-to-perimeter ratio
In an attempt to characterize the distribution of forms and shapes of nodal
domains in wave functions, we define a geometric parameter - the ratio
between the area of a domain and its perimeter, measured in units of the
wavelength . We show that the distribution function can
distinguish between domains in which the classical dynamics is regular or
chaotic. For separable surfaces, we compute the limiting distribution, and show
that it is supported by an interval, which is independent of the properties of
the surface. In systems which are chaotic, or in random-waves, the
area-to-perimeter distribution has substantially different features which we
study numerically. We compare the features of the distribution for chaotic wave
functions with the predictions of the percolation model to find agreement, but
only for nodal domains which are big with respect to the wavelength scale. This
work is also closely related to, and provides a new point of view on
isoperimetric inequalities.Comment: 22 pages, 11 figure
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