177 research outputs found
Quantum chaos and the double-slit experiment
We report on the numerical simulation of the double-slit experiment, where
the initial wave-packet is bounded inside a billiard domain with perfectly
reflecting walls. If the shape of the billiard is such that the classical ray
dynamics is regular, we obtain interference fringes whose visibility can be
controlled by changing the parameters of the initial state. However, if we
modify the shape of the billiard thus rendering classical (ray) dynamics fully
chaotic, the interference fringes disappear and the intensity on the screen
becomes the (classical) sum of intensities for the two corresponding one-slit
experiments. Thus we show a clear and fundamental example in which transition
to chaotic motion in a deterministic classical system, in absence of any
external noise, leads to a profound modification in the quantum behaviour.Comment: 5 pages, 4 figure
Time-metric equivalence and dimension change under time reparameterizations
We study the behavior of dynamical systems under time reparameterizations,
which is important not only to characterize chaos in relativistic systems but
also to probe the invariance of dynamical quantities. We first show that time
transformations are locally equivalent to metric transformations, a result that
leads to a transformation rule for all Lyapunov exponents on arbitrary
Riemannian phase spaces. We then show that time transformations preserve the
spectrum of generalized dimensions D_q except for the information dimension
D_1, which, interestingly, transforms in a nontrivial way despite previous
assertions of invariance. The discontinuous behavior at q=1 can be used to
constrain and extend the formulation of the Kaplan-Yorke conjecture
Drift of particles in self-similar systems and its Liouvillian interpretation
We study the dynamics of classical particles in different classes of
spatially extended self-similar systems, consisting of (i) a self-similar
Lorentz billiard channel, (ii) a self-similar graph, and (iii) a master
equation. In all three systems the particles typically drift at constant
velocity and spread ballistically. These transport properties are analyzed in
terms of the spectral properties of the operator evolving the probability
densities. For systems (i) and (ii), we explain the drift from the properties
of the Pollicott-Ruelle resonance spectrum and corresponding eigenvectorsComment: To appear in Phys. Rev.
On the classical-quantum correspondence for the scattering dwell time
Using results from the theory of dynamical systems, we derive a general
expression for the classical average scattering dwell time, tau_av. Remarkably,
tau_av depends only on a ratio of phase space volumes. We further show that,
for a wide class of systems, the average classical dwell time is not in
correspondence with the energy average of the quantum Wigner time delay.Comment: 5 pages, 1 figur
Poincar\'e recurrences in Hamiltonian systems with a few degrees of freedom
Hundred twenty years after the fundamental work of Poincar\'e, the statistics
of Poincar\'e recurrences in Hamiltonian systems with a few degrees of freedom
is studied by numerical simulations. The obtained results show that in a
regime, where the measure of stability islands is significant, the decay of
recurrences is characterized by a power law at asymptotically large times. The
exponent of this decay is found to be . This value is
smaller compared to the average exponent found previously
for two-dimensional symplectic maps with divided phase space. On the basis of
previous and present results a conjecture is put forward that, in a generic
case with a finite measure of stability islands, the Poncar\'e exponent has a
universal average value being independent of number of
degrees of freedom and chaos parameter. The detailed mechanisms of this slow
algebraic decay are still to be determined.Comment: revtex 4 pages, 4 figs; Refs. and discussion adde
Steady-state conduction in self-similar billiards
The self-similar Lorentz billiard channel is a spatially extended
deterministic dynamical system which consists of an infinite one-dimensional
sequence of cells whose sizes increase monotonically according to their
indices. This special geometry induces a nonequilibrium stationary state with
particles flowing steadily from the small to the large scales. The
corresponding invariant measure has fractal properties reflected by the
phase-space contraction rate of the dynamics restricted to a single cell with
appropriate boundary conditions. In the near-equilibrium limit, we find
numerical agreement between this quantity and the entropy production rate as
specified by thermodynamics
Deterministic spin models with a glassy phase transition
We consider the infinite-range deterministic spin models with Hamiltonian
, where is the quantization of a
chaotic map of the torus. The mean field (TAP) equations are derived by summing
the high temperature expansion. They predict a glassy phase transition at the
critical temperature .Comment: 8 pages, no figures, RevTex forma
Singular continuous spectra in a pseudo-integrable billiard
The pseudo-integrable barrier billiard invented by Hannay and McCraw [J.
Phys. A 23, 887 (1990)] -- rectangular billiard with line-segment barrier
placed on a symmetry axis -- is generalized. It is proven that the flow on
invariant surfaces of genus two exhibits a singular continuous spectral
component.Comment: 4 pages, 2 figure
Resonances of the cusp family
We study a family of chaotic maps with limit cases the tent map and the cusp
map (the cusp family). We discuss the spectral properties of the corresponding
Frobenius--Perron operator in different function spaces including spaces of
analytic functions. A numerical study of the eigenvalues and eigenfunctions is
performed.Comment: 14 pages, 3 figures. Submitted to J.Phys.
The triangle map: a model of quantum chaos
We study an area preserving parabolic map which emerges from the Poincar\' e
map of a billiard particle inside an elongated triangle. We provide numerical
evidence that the motion is ergodic and mixing. Moreover, when considered on
the cylinder, the motion appear to follow a gaussian diffusive process.Comment: 4 pages in RevTeX with 4 figures (in 6 eps-files
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