2,485 research outputs found
Chaotic scattering in solitary wave interactions: A singular iterated-map description
We derive a family of singular iterated maps--closely related to Poincare
maps--that describe chaotic interactions between colliding solitary waves. The
chaotic behavior of such solitary wave collisions depends on the transfer of
energy to a secondary mode of oscillation, often an internal mode of the pulse.
Unlike previous analyses, this map allows one to understand the interactions in
the case when this mode is excited prior to the first collision. The map is
derived using Melnikov integrals and matched asymptotic expansions and
generalizes a ``multi-pulse'' Melnikov integral and allows one to find not only
multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps
derived exhibits singular behavior, including regions of infinite winding. This
problem is shown to be a singular version of the conservative Ikeda map from
laser physics and connections are made with problems from celestial mechanics
and fluid mechanics.Comment: 29 pages, 17 figures, submitted to Chaos, higher-resolution figures
available at author's website: http://m.njit.edu/goodman/publication
Cooperative surmounting of bottlenecks
The physics of activated escape of objects out of a metastable state plays a
key role in diverse scientific areas involving chemical kinetics, diffusion and
dislocation motion in solids, nucleation, electrical transport, motion of flux
lines superconductors, charge density waves, and transport processes of
macromolecules, to name but a few. The underlying activated processes present
the multidimensional extension of the Kramers problem of a single Brownian
particle. In comparison to the latter case, however, the dynamics ensuing from
the interactions of many coupled units can lead to intriguing novel phenomena
that are not present when only a single degree of freedom is involved. In this
review we report on a variety of such phenomena that are exhibited by systems
consisting of chains of interacting units in the presence of potential
barriers.
In the first part we consider recent developments in the case of a
deterministic dynamics driving cooperative escape processes of coupled
nonlinear units out of metastable states. The ability of chains of coupled
units to undergo spontaneous conformational transitions can lead to a
self-organised escape. The mechanism at work is that the energies of the units
become re-arranged, while keeping the total energy conserved, in forming
localised energy modes that in turn trigger the cooperative escape. We present
scenarios of significantly enhanced noise-free escape rates if compared to the
noise-assisted case.
The second part deals with the collective directed transport of systems of
interacting particles overcoming energetic barriers in periodic potential
landscapes. Escape processes in both time-homogeneous and time-dependent driven
systems are considered for the emergence of directed motion. It is shown that
ballistic channels immersed in the associated high-dimensional phase space are
the source for the directed long-range transport
Fractal Weyl laws in discrete models of chaotic scattering
We analyze simple models of quantum chaotic scattering, namely quantized open
baker's maps. We numerically compute the density of quantum resonances in the
semiclassical r\'{e}gime. This density satisfies a fractal Weyl law, where the
exponent is governed by the (fractal) dimension of the set of trapped
trajectories. This type of behaviour is also expected in the (physically more
relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we
are able to rigorously prove this Weyl law, and compute quantities related to
the "coherent transport" through the system, namely the conductance and "shot
noise". The latter is close to the prediction of random matrix theory.Comment: Invited article in the Special Issue of Journal of Physics A on
"Trends in Quantum Chaotic Scattering
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
Classical dynamics on graphs
We consider the classical evolution of a particle on a graph by using a
time-continuous Frobenius-Perron operator which generalizes previous
propositions. In this way, the relaxation rates as well as the chaotic
properties can be defined for the time-continuous classical dynamics on graphs.
These properties are given as the zeros of some periodic-orbit zeta functions.
We consider in detail the case of infinite periodic graphs where the particle
undergoes a diffusion process. The infinite spatial extension is taken into
account by Fourier transforms which decompose the observables and probability
densities into sectors corresponding to different values of the wave number.
The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a
Frobenius-Perron operator corresponding to a given sector. The diffusion
coefficient is obtained from the hydrodynamic modes of diffusion and has the
Green-Kubo form. Moreover, we study finite but large open graphs which converge
to the infinite periodic graph when their size goes to infinity. The lifetime
of the particle on the open graph is shown to correspond to the lifetime of a
system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure
Non-perturbative features of driven scattering systems
We investigate the scattering properties of one-dimensional, periodically and
non-periodically forced oscillators. The pattern of singularities of the
scattering function, in the periodic case, shows a characteristic hierarchical
structure where the number Nc of zeros of the solutions plays the role of an
order parameter marking the level of the observed self-similar structure. The
behavior is understood both in terms of the return map and of the intersections
pattern of the invariant manifolds of the outermost fixed points. In the
non-periodic case the scattering function does not provide a complete
development of the hierarchical structure. The singularities pattern of the
outgoing energy as a function of the driver amplitude is connected to the
arrangement of gaps in the fundamental regions. The survival probability
distribution of temporarily bound orbits is shown to decay asymptotically as a
power law. The "stickiness" of regular regions of phase space, given by KAM
surfaces and remnant of KAM curves, is responsible for this observation
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