236 research outputs found
Typicality of recurrence for Lorentz gases
It is a safe conjecture that most (not necessarily periodic) two-dimensional
Lorentz gases with finite horizon are recurrent. Here we formalize this
conjecture by means of a stochastic ensemble of Lorentz gases, in which i.i.d.
random scatterers are placed in each cell of a co-compact lattice in the plane.
We prove that the typical Lorentz gas, in the sense of Baire, is recurrent,
and give results in the direction of showing that recurrence is an almost sure
property (including a zero-one law that holds in every dimension). A few toy
models illustrate the extent of these results.Comment: 22 pages, 5 figure
Ergodic Properties of the Quantum Ideal Gas in the Maxwell-Boltzmann Statistics
It is proved that the quantization of the Volkovyski-Sinai model of ideal gas
(in the Maxwell-Boltzmann statistics) enjoys at the thermodynamical limit the
properties of mixing and ergodicity with respect to the quantum canonical Gibbs
state. Plus, the average over the quantum state of a pseudo-differential
operator is exactly the average over the classical canonical measure of its
Weyl symbol.Comment: 35 pages, LaTe
Uniformly expanding Markov maps of the real line: exactness and infinite mixing
We give a fairly complete characterization of the exact components of a large
class of uniformly expanding Markov maps of . Using this result,
for a class of -invariant maps and finite modifications thereof, we
prove certain properties of infinite mixing recently introduced by the author.Comment: Final version to be published in Discrete and Continuous Dynamical
Systems A. 47 pages, 5 figures. Labeling of appendices (and related wording)
may differ from published versio
Exactness, K-property and infinite mixing
We explore the consequences of exactness or K-mixing on the notions of mixing
(a.k.a. infinite-volume mixing) recently devised by the author for
infinite-measure-preserving dynamical systems.Comment: Corrected reference to published version and fixed some typos, 15
page
Infinite-horizon Lorentz tubes and gases: recurrence and ergodic properties
We construct classes of two-dimensional aperiodic Lorentz systems that have
infinite horizon and are 'chaotic', in the sense that they are (Poincar\'e)
recurrent, uniformly hyperbolic, ergodic, and the first-return map to any
scatterer is -mixing. In the case of the Lorentz tubes (i.e., Lorentz gases
in a strip), we define general measured families of systems (\emph{ensembles})
for which the above properties occur with probability 1. In the case of the
Lorentz gases in the plane, we define families, endowed with a natural metric,
within which the set of all chaotic dynamical systems is uncountable and dense.Comment: Final version, to appear in Physica D (2011
Localization in infinite billiards: a comparison between quantum and classical ergodicity
Consider the non-compact billiard in the first quandrant bounded by the
positive -semiaxis, the positive -semiaxis and the graph of , . Although the Schnirelman Theorem holds,
the quantum average of the position is finite on any eigenstate, while
classical ergodicity entails that the classical time average of is
unbounded.Comment: 9 page
Large deviations in quantum lattice systems: one-phase region
We give large deviation upper bounds, and discuss lower bounds, for the
Gibbs-KMS state of a system of quantum spins or an interacting Fermi gas on the
lattice. We cover general interactions and general observables, both in the
high temperature regime and in dimension one.Comment: 30 pages, LaTeX 2
More ergodic billiards with an infinite cusp
In a previous paper (nlin.CD/0107041) the following class of billiards was
studied: For convex,
sufficiently smooth, and vanishing at infinity, let the billiard table be
defined by , the planar domain delimited by the positive -semiaxis, the
positive -semiaxis, and the graph of .
For a large class of we proved that the billiard map was hyperbolic.
Furthermore we gave an example of a family of that makes this map ergodic.
Here we extend the latter result to a much wider class of functions.Comment: 13 pages, 4 figure
Global-local mixing for the Boole map
In the context of 'infinite-volume mixing' we prove global-local mixing for
the Boole map, a.k.a. Boole transformation, which is the prototype of a
non-uniformly expanding map with two neutral fixed points. Global-local mixing
amounts to the decorrelation of all pairs of global and local observables. In
terms of the equilibrium properties of the system it means that the evolution
of every absolutely continuous probability measure converges, in a certain
precise sense, to an averaging functional over the entire space.Comment: 15 pages, 2 figures. Final version to be published in Chaos, Solitons
& Fractal
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