715 research outputs found
Universal diffusion near the golden chaos border
We study local diffusion rate in Chirikov standard map near the critical
golden curve. Numerical simulations confirm the predicted exponent
for the power law decay of as approaching the golden curve via principal
resonances with period (). The universal
self-similar structure of diffusion between principal resonances is
demonstrated and it is shown that resonances of other type play also an
important role.Comment: 4 pages Latex, revtex, 3 uuencoded postscript figure
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
Quantum Arnol'd Diffusion in a Simple Nonlinear System
We study the fingerprint of the Arnol'd diffusion in a quantum system of two
coupled nonlinear oscillators with a two-frequency external force. In the
classical description, this peculiar diffusion is due to the onset of a weak
chaos in a narrow stochastic layer near the separatrix of the coupling
resonance. We have found that global dependence of the quantum diffusion
coefficient on model parameters mimics, to some extent, the classical data.
However, the quantum diffusion happens to be slower that the classical one.
Another result is the dynamical localization that leads to a saturation of the
diffusion after some characteristic time. We show that this effect has the same
nature as for the studied earlier dynamical localization in the presence of
global chaos. The quantum Arnol'd diffusion represents a new type of quantum
dynamics and can be observed, for example, in 2D semiconductor structures
(quantum billiards) perturbed by time-periodic external fields.Comment: RevTex, 11 pages including 12 ps-figure
Disruption of the three-body gravitational systems: Lifetime statistics
We investigate statistics of the decay process in the equal-mass three-body
problem with randomized initial conditions. Contrary to earlier expectations of
similarity with "radioactive decay", the lifetime distributions obtained in our
numerical experiments turn out to be heavy-tailed, i.e. the tails are not
exponential, but algebraic. The computed power-law index for the differential
distribution is within the narrow range, approximately from -1.7 to -1.4,
depending on the virial coefficient. Possible applications of our results to
studies of the dynamics of triple stars known to be at the edge of disruption
are considered.Comment: 13 pages, 2 tables, 3 figure
Quantum Ergodicity and Localization in Conservative Systems: the Wigner Band Random Matrix Model
First theoretical and numerical results on the global structure of the energy
shell, the Green function spectra and the eigenfunctions, both localized and
ergodic, in a generic conservative quantum system are presented. In case of
quantum localization the eigenfunctions are shown to be typically narrow and
solid, with centers randomly scattered within the semicircle energy shell while
the Green function spectral density (local spectral density of states) is
extended over the whole shell, but sparse.Comment: 4 pages in RevTex and 4 Postscript figures; presented to Phys. Lett.
Kepler-16b: safe in a resonance cell
The planet Kepler-16b is known to follow a circumbinary orbit around a system
of two main-sequence stars. We construct stability diagrams in the "pericentric
distance - eccentricity" plane, which show that Kepler-16b is in a hazardous
vicinity to the chaos domain - just between the instability "teeth" in the
space of orbital parameters. Kepler-16b survives, because it is close to the
stable half-integer 11/2 orbital resonance with the central binary, safe inside
a resonance cell bounded by the unstable 5/1 and 6/1 resonances. The
neighboring resonance cells are vacant, because they are "purged" by
Kepler-16b, due to overlap of first-order resonances with the planet. The newly
discovered planets Kepler-34b and Kepler-35b are also safe inside resonance
cells at the chaos border.Comment: 17 pages, including 5 figure
Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model with Gauss Heat Bath
Large entropy fluctuations in a nonequilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple 2-freedom
model with the so-called Gauss time-reversible thermostat. The local
fluctuations (on a set of fixed trajectory segments) from the average heat
entropy absorbed in thermostat were found to be non-Gaussian. Approximately,
the fluctuations can be discribed by a two-Gaussian distribution with a
crossover independent of the segment length and the number of trajectories
('particles'). The distribution itself does depend on both, approaching the
single standard Gaussian distribution as any of those parameters increases. The
global time-dependent fluctuations turned out to be qualitatively different in
that they have a strict upper bound much less than the average entropy
production. Thus, unlike the equilibrium steady state, the recovery of the
initial low entropy becomes impossible, after a sufficiently long time, even in
the largest fluctuations. However, preliminary numerical experiments and the
theoretical estimates in the special case of the critical dynamics with
superdiffusion suggest the existence of infinitely many Poincar\'e recurrences
to the initial state and beyond. This is a new interesting phenomenon to be
farther studied together with some other open questions. Relation of this
particular example of nonequilibrium steady state to a long-standing persistent
controversy over statistical 'irreversibility', or the notorious 'time arrow',
is also discussed. In conclusion, an unsolved problem of the origin of the
causality 'principle' is touched upon.Comment: 21 pages, 7 figure
Big Entropy Fluctuations in Statistical Equilibrium: The Macroscopic Kinetics
Large entropy fluctuations in an equilibrium steady state of classical
mechanics were studied in extensive numerical experiments on a simple
2--freedom strongly chaotic Hamiltonian model described by the modified Arnold
cat map. The rise and fall of a large separated fluctuation was shown to be
described by the (regular and stable) "macroscopic" kinetics both fast
(ballistic) and slow (diffusive). We abandoned a vague problem of "appropriate"
initial conditions by observing (in a long run)spontaneous birth and death of
arbitrarily big fluctuations for any initial state of our dynamical model.
Statistics of the infinite chain of fluctuations, reminiscent to the Poincar\'e
recurrences, was shown to be Poissonian. A simple empirical relation for the
mean period between the fluctuations (Poincar\'e "cycle") has been found and
confirmed in numerical experiments. A new representation of the entropy via the
variance of only a few trajectories ("particles") is proposed which greatly
facilitates the computation, being at the same time fairly accurate for big
fluctuations. The relation of our results to a long standing debates over
statistical "irreversibility" and the "time arrow" is briefly discussed too.Comment: Latex 2.09, 26 pages, 6 figure
Manifestation of the Arnol'd Diffusion in Quantum Systems
We study an analog of the classical Arnol'd diffusion in a quantum system of
two coupled non-linear oscillators one of which is governed by an external
periodic force with two frequencies. In the classical model this very weak
diffusion happens in a narrow stochastic layer along the coupling resonance,
and leads to an increase of total energy of the system. We show that the
quantum dynamics of wave packets mimics, up to some extent, global properties
of the classical Arnol'd diffusion. This specific diffusion represents a new
type of quantum dynamics, and may be observed, for example, in 2D semiconductor
structures (quantum billiards) perturbed by time-periodic external fields.Comment: RevTex, 4 pages including 7 ps-figures, corrected forma
Scar Intensity Statistics in the Position Representation
We obtain general predictions for the distribution of wave function
intensities in position space on the periodic orbits of chaotic ballistic
systems. The expressions depend on effective system size N, instability
exponent lambda of the periodic orbit, and proximity to a focal point of the
orbit. Limiting expressions are obtained that include the asymptotic
probability distribution of rare high-intensity events and a perturbative
formula valid in the limit of weak scarring. For finite system sizes, a single
scaling variable lambda N describes deviations from the semiclassical N ->
infinity limit.Comment: To appear in Phys. Rev. E, 10 pages, including 4 figure
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