158 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

### 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.

### 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

### 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 $\beta \approx 1.3$. This value is
smaller compared to the average exponent $\beta \approx 1.5$ 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 $\beta \approx 1.3$ 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

### Deterministic spin models with a glassy phase transition

We consider the infinite-range deterministic spin models with Hamiltonian
$H=\sum_{i,j=1}^N J_{i,j}\sigma_i\sigma_j$, where $J$ 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 $T\sim 0.8$.Comment: 8 pages, no figures, RevTex forma

### 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.

### Spectral statistics for quantized skew translations on the torus

We study the spectral statistics for quantized skew translations on the
torus, which are ergodic but not mixing for irrational parameters. It is shown
explicitly that in this case the level--spacing distribution and other common
spectral statistics, like the number variance, do not exist in the
semiclassical limit.Comment: 7 pages. One figure, include

### 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

### Linear dynamical entropy and free-independence for quantized maps on the torus

We study the relations between the averaged linear entropy production in
periodically measured quantum systems and ergodic properties of their classical
counterparts. Quantized linear automorphisms of the torus, both classically
chaotic and regular ones, are used as examples. Numerical calculations show
different entropy production regimes depending on the relation between the
Kolmogorov-Sinai entropy and the measurement entropy. The hypothesis of free
independence relations between the dynamics and measurement proposed to explain
the initial constant and maximal entropy production is tested numerically for
those models.Comment: 7 pages, 5 figure

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