13,662 research outputs found
METASTABLE SYSTEMS AS RANDOM MAPS
Metastable dynamical systems were recently studied [GonzĂĄlez-Tokman et al., 2011] in the framework of one-dimensional piecewise expanding maps on two disjoint invariant sets, each possessing its own ergodic absolutely continuous invariant measure (acim). Under small deterministic perturbations, holes between the two disjoint systems are created, and the two ergodic systems merge into one. The long term dynamics of the newly formed metastable system is defined by the unique acim on the combined ergodic sets. The main result of [GonzĂĄlez-Tokman et al., 2011] proves that this combined acim can be approximated by a convex combination of the disjoint acims with weights depending on the ratio of the respective measures of the holes. In this note we present an entirely different approach to metastable systems. We consider two piecewise expanding maps: one is the original map, Ï1, defined on two disjoint invariant sets of âN and the other is a deterministically perturbed version of Ï1, Ï2, which allows passage between the two disjoint invariant sets of Ï1. We model this system by a position dependent random map based on Ï1 and Ï2, to which we associate position dependent probabilities that reflect the switching between the maps. A typical orbit spends a long time in one of the ergodic sets but eventually switches to the other. Such behavior can be attributed to physical holes as between adjoining billiard tables or more abstract situations where balls can "leap" from one table to the other. Using results for random maps, a result similar to the one-dimensional main result of [GonzĂĄlez-Tokman et al., 2011] is proved in N dimensions. We also consider holes in more than two invariant sets. A number of examples are presented
Escape Rates Formulae and Metastability for Randomly perturbed maps
We provide escape rates formulae for piecewise expanding interval maps with
`random holes'. Then we obtain rigorous approximations of invariant densities
of randomly perturbed metabstable interval maps. We show that our escape rates
formulae can be used to approximate limits of invariant densities of randomly
perturbed metastable systems.Comment: Appeared in Nonlinearity, May 201
Pseudo-Orbits, Stationary Measures and Metastability
We study random perturbations of multidimensional piecewise expanding maps.
We characterize absolutely continuous stationary measures (acsm) of randomly
perturbed dynamical systems in terms of pseudo-orbits linking the ergodic
components of absolutely invariant measures (acim) of the unperturbed system.
We focus on those components, called least-elements, which attract
pseudo-orbits. We show that each least element admits a neighbourhood which
supports exactly one ergodic acsm of the random system. We use this result to
identify random perturbations that exhibit a metastable behavior.Comment: To appear in Dynamical System
Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators
This paper presents a diffusion based probabilistic interpretation of
spectral clustering and dimensionality reduction algorithms that use the
eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency
matrix of all points, we define a diffusion distance between any two data
points and show that the low dimensional representation of the data by the
first few eigenvectors of the corresponding Markov matrix is optimal under a
certain mean squared error criterion. Furthermore, assuming that data points
are random samples from a density p(\x) = e^{-U(\x)} we identify these
eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck
operator in a potential 2U(\x) with reflecting boundary conditions. Finally,
applying known results regarding the eigenvalues and eigenfunctions of the
continuous Fokker-Planck operator, we provide a mathematical justification for
the success of spectral clustering and dimensional reduction algorithms based
on these first few eigenvectors. This analysis elucidates, in terms of the
characteristics of diffusion processes, many empirical findings regarding
spectral clustering algorithms.Comment: submitted to NIPS 200
Ubiquity of metastable-to-stable crossover in weakly chaotic dynamical systems
We present a comparative study of several dynamical systems of increasing
complexity, namely, the logistic map with additive noise, one, two and many
globally-coupled standard maps, and the Hamiltonian Mean Field model (i.e., the
classical inertial infinitely-ranged ferromagnetically coupled XY spin model).
We emphasize the appearance, in all of these systems, of metastable states and
their ultimate crossover to the equilibrium state. We comment on the underlying
mechanisms responsible for these phenomena (weak chaos) and compare common
characteristics. We point out that this ubiquitous behavior appears to be
associated to the features of the nonextensive generalization of the
Boltzmann-Gibbs statistical mechanics.Comment: Communication at next2003, Second Sardinian International Conference
on News and Expectations in Thermostatistics, Villasimius (Cagliari) Italy,
21st-28th September 2003. Submitted to Physica A. Elsevier Latex, 17 pages, 8
figure
Noise Enhanced Stability
The noise can stabilize a fluctuating or a periodically driven metastable
state in such a way that the system remains in this state for a longer time
than in the absence of white noise. This is the noise enhanced stability
phenomenon, observed experimentally and numerically in different physical
systems. After shortly reviewing all the physical systems where the phenomenon
was observed, the theoretical approaches used to explain the effect are
presented. Specifically the conditions to observe the effect: (a) in systems
with periodical driving force, and (b) in random dichotomous driving force, are
discussed. In case (b) we review the analytical results concerning the mean
first passage time and the nonlinear relaxation time as a function of the white
noise intensity, the parameters of the potential barrier, and of the
dichotomous noise.Comment: 18 pages, 6 figures, in press Acta Physica Polonica (2004
Formation of Two Glass Phases in Binary Cu-Ag Liquid
The glass transition is alternatively described as either a dynamic transition in which there is a dramatic slowing down of the kinetics, or as a thermodynamic phase transition. To examine the physical origin of the glass transition in fragile Cu-Ag liquids, we employed molecular dynamics (MD) simulations on systems in the range of 32,000 to 2,048,000 atoms. Surprisingly, we identified a 1st order freezing transition from liquid (L) to metastable heterogenous solid-like phase, denoted as the G-glass, when a supercooled liquid evolves isothermally below its melting temperature at deep undercooling. In contrast, a more homogenous liquid-like glass, denoted as the L-glass, is achieved when the liquid is quenched continuously to room temperature with a fast cooling rate of âŒ10ÂčÂč K/sec. We report a thermodynamic description of the L-G transition and characterize the correlation length of the heterogenous structure in the G-glass. The shear modulus of the G-glass is significantly higher than the L-glass, suggesting that the first order L-G transition is linked fundamentally to long-range elasticity involving elementary configurational excitations in the G-glass
Some open questions in "wave chaos"
The subject area referred to as "wave chaos", "quantum chaos" or "quantum
chaology" has been investigated mostly by the theoretical physics community in
the last 30 years. The questions it raises have more recently also attracted
the attention of mathematicians and mathematical physicists, due to connections
with number theory, graph theory, Riemannian, hyperbolic or complex geometry,
classical dynamical systems, probability etc. After giving a rough account on
"what is quantum chaos?", I intend to list some pending questions, some of them
having been raised a long time ago, some others more recent
Rare events, escape rates and quasistationarity: some exact formulae
We present a common framework to study decay and exchanges rates in a wide
class of dynamical systems. Several applications, ranging form the metric
theory of continuons fractions and the Shannon capacity of contrained systems
to the decay rate of metastable states, are given
Topological Defects and Non-homogeneous Melting of Large 2D Coulomb Clusters
The configurational and melting properties of large two-dimensional clusters
of charged classical particles interacting with each other via the Coulomb
potential are investigated through the Monte Carlo simulation technique. The
particles are confined by a harmonic potential. For a large number of particles
in the cluster (N>150) the configuration is determined by two competing
effects, namely in the center a hexagonal lattice is formed, which is the
groundstate for an infinite 2D system, and the confinement which imposes its
circular symmetry on the outer edge. As a result a hexagonal Wigner lattice is
formed in the central area while at the border of the cluster the particles are
arranged in rings. In the transition region defects appear as dislocations and
disclinations at the six corners of the hexagonal-shaped inner domain. Many
different arrangements and type of defects are possible as metastable
configurations with a slightly higher energy. The particles motion is found to
be strongly related to the topological structure. Our results clearly show that
the melting of the clusters starts near the geometry induced defects, and that
three different melting temperatures can be defined corresponding to the
melting of different regions in the cluster.Comment: 7 pages, 11 figures, submitted to Phys. Rev.
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