1,444 research outputs found
Singularities and nonhyperbolic manifolds do not coincide
We consider the billiard flow of elastically colliding hard balls on the flat
-torus (), and prove that no singularity manifold can even
locally coincide with a manifold describing future non-hyperbolicity of the
trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli
mixing property) of all such systems, i.e. the verification of the
Boltzmann-Sinai Ergodic Hypothesis.Comment: Final version, to appear in Nonlinearit
Limiting Distribution of Frobenius Numbers for
The purpose of this paper is to give a complete derivation of the limiting
distribution of large Frobenius numbers outlined in earlier work of J. Bourgain
and Ya. Sinai and fill some gaps formulated there as hypotheses.Comment: 13 page
Non-ergodicity of the motion in three dimensional steep repelling dispersing potentials
It is demonstrated numerically that smooth three degrees of freedom
Hamiltonian systems which are arbitrarily close to three dimensional strictly
dispersing billiards (Sinai billiards) have islands of effective stability, and
hence are non-ergodic. The mechanism for creating the islands are corners of
the billiard domain.Comment: 6 pages, 8 figures, submitted to Chao
Proving The Ergodic Hypothesis for Billiards With Disjoint Cylindric Scatterers
In this paper we study the ergodic properties of mathematical billiards
describing the uniform motion of a point in a flat torus from which finitely
many, pairwise disjoint, tubular neighborhoods of translated subtori (the so
called cylindric scatterers) have been removed. We prove that every such system
is ergodic (actually, a Bernoulli flow), unless a simple geometric obstacle for
the ergodicity is present.Comment: 24 pages, AMS-TeX fil
Velocity-strengthening friction significantly affects interfacial dynamics, strength and dissipation
Frictional interfaces are abundant in natural and manmade systems and their
dynamics still pose challenges of fundamental and technological importance. A
recent extensive compilation of multiple-source experimental data has revealed
that velocity-strengthening friction, where the steady-state frictional
resistance increases with sliding velocity over some range, is a generic
feature of such interfaces. Moreover, velocity-strengthening friction has very
recently been linked to slow laboratory earthquakes and stick-slip motion. Here
we elucidate the importance of velocity-strengthening friction by theoretically
studying three variants of a realistic rate-and-state friction model. All
variants feature identical logarithmic velocity-weakening friction at small
sliding velocities, but differ in their higher velocity behaviors. By
quantifying energy partition (e.g. radiation and dissipation), the selection of
interfacial rupture fronts and rupture arrest, we show that the presence or
absence of velocity-strengthening friction can significantly affect the global
interfacial resistance and the total energy released during frictional
instabilities ("event magnitude"). Furthermore, we show that different forms of
velocity-strengthening friction (e.g. logarithmic vs. linear) may result in
events of similar magnitude, yet with dramatically different dissipation and
radiation rates. This happens because the events are mediated by interfacial
rupture fronts with vastly different propagation velocities, where stronger
velocity-strengthening friction promotes slower rupture. These theoretical
results may have significant implications on our understanding of frictional
dynamics.Comment: 9 pages, 6 figure
Instabilities at Frictional Interfaces: Creep Patches, Nucleation and Rupture Fronts
The strength and stability of frictional interfaces, ranging from
tribological systems to earthquake faults, are intimately related to the
underlying spatially-extended dynamics. Here we provide a comprehensive
theoretical account, both analytic and numeric, of spatiotemporal interfacial
dynamics in a realistic rate-and-state friction model, featuring both
velocity-weakening and strengthening behaviors. Slowly extending, loading-rate
dependent, creep patches undergo a linear instability at a critical nucleation
size, which is nearly independent of interfacial history, initial stress
conditions and velocity-strengthening friction. Nonlinear propagating rupture
fronts -- the outcome of instability -- depend sensitively on the stress state
and velocity-strengthening friction. Rupture fronts span a wide range of
propagation velocities and are related to steady state fronts solutions.Comment: Typos and figures corrected. Supplementary information at:
http://www.weizmann.ac.il/chemphys/bouchbinder/frictional_instabilities.htm
On the velocity-strengthening behavior of dry friction
The onset of frictional instabilities, e.g. earthquakes nucleation, is
intimately related to velocity-weakening friction, in which the frictional
resistance of interfaces decreases with increasing slip velocity. While this
frictional response has been studied extensively, less attention has been given
to steady-state velocity-strengthening friction, in spite of its potential
importance for various aspects of frictional phenomena such as the propagation
speed of interfacial rupture fronts and the amount of stored energy released by
them. In this note we suggest that a crossover from steady-state
velocity-weakening friction at small slip velocities to steady-state
velocity-strengthening friction at higher velocities might be a generic feature
of dry friction. We further argue that while thermally activated rheology
naturally gives rise to logarithmic steady-state velocity-strengthening
friction, a crossover to stronger-than-logarithmic strengthening might take
place at higher slip velocities, possibly accompanied by a change in the
dominant dissipation mechanism. We sketch a few physical mechanisms that may
account for the crossover to stronger-than-logarithmic steady-state
velocity-strengthening and compile a rather extensive set of experimental data
available in the literature, lending support to these ideas.Comment: Updated to published version: 2 Figures and a section adde
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