92 research outputs found
Absolute Continuity Theorem for Random Dynamical Systems on
In this article we provide a proof of the so called absolute continuity
theorem for random dynamical systems on which have an invariant
probability measure. First we present the construction of local stable
manifolds in this case. Then the absolute continuity theorem basically states
that for any two transversal manifolds to the family of local stable manifolds
the induced Lebesgue measures on these transversal manifolds are absolutely
continuous under the map that transports every point on the first manifold
along the local stable manifold to the second manifold, the so-called
Poincar\'e map or holonomy map. In contrast to known results, we have to deal
with the non-compactness of the state space and the randomness of the random
dynamical system.Comment: 46 page
A simple piston problem in one dimension
We study a heavy piston that separates finitely many ideal gas particles
moving inside a one-dimensional gas chamber. Using averaging techniques, we
prove precise rates of convergence of the actual motions of the piston to its
averaged behavior. The convergence is uniform over all initial conditions in a
compact set. The results extend earlier work by Sinai and Neishtadt, who
determined that the averaged behavior is periodic oscillation. In addition, we
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.Comment: Accepted by Nonlinearity. 27 pages, 2 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
A strong pair correlation bound implies the CLT for Sinai Billiards
For Dynamical Systems, a strong bound on multiple correlations implies the
Central Limit Theorem (CLT) [ChMa]. In Chernov's paper [Ch2], such a bound is
derived for dynamically Holder continuous observables of dispersing Billiards.
Here we weaken the regularity assumption and subsequently show that the bound
on multiple correlations follows directly from the bound on pair correlations.
Thus, a strong bound on pair correlations alone implies the CLT, for a wider
class of observables. The result is extended to Anosov diffeomorphisms in any
dimension.Comment: 13 page
Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
This paper deals with the problem of analytically computing the largest
Lyapunov exponent for many degrees of freedom Hamiltonian systems. This aim is
succesfully reached within a theoretical framework that makes use of a
geometrization of newtonian dynamics in the language of Riemannian geometry. A
new point of view about the origin of chaos in these systems is obtained
independently of homoclinic intersections. Chaos is here related to curvature
fluctuations of the manifolds whose geodesics are natural motions and is
described by means of Jacobi equation for geodesic spread. Under general
conditions ane effective stability equation is derived; an analytic formula for
the growth-rate of its solutions is worked out and applied to the
Fermi-Pasta-Ulam beta model and to a chain of coupled rotators. An excellent
agreement is found the theoretical prediction and the values of the Lyapunov
exponent obtained by numerical simulations for both models.Comment: RevTex, 40 pages, 8 PostScript figures, to be published in Phys. Rev.
E (scheduled for November 1996
Liquation phenomena in the system Al2O3-SiO2. Communication 2. Microliquation and its depiction on the phase diagram of the binary system
Additivity of the Crystal Structure and Properties of Solid Solutions of Rare Earth Metal Hexaborides
- …