30 research outputs found
Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards
The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives
sufficient conditions under which a phase point has an open neighborhood that
belongs (mod 0) to one ergodic component. This theorem is a key ingredient of
many proofs of ergodicity for billiards and, more generally, for smooth
hyperbolic maps with singularities. However the proof of that theorem relies
upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check
for some physically relevant models, including gases of hard balls. Here we
give a proof of the Local Ergodic Theorem for two dimensional billiards without
using the Ansatz.Comment: 17 pages, 2 figure
The characteristic exponents of the falling ball model
We study the characteristic exponents of the Hamiltonian system of () point masses freely falling in the vertical half line
under constant gravitation and colliding with each other and
the solid floor elastically. This model was introduced and first studied
by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic
(Lyapunov) exponents of the above dynamical system are nonzero, provided that
(i. e. the masses do not increase as we go up) and
Heat conduction and diffusion of hard disks in a narrow channel
Using molecular dynamics we study heat conduction and diffusion of hard disks
in one dimensional narrow channels. When collisions preserve momentum the heat
conduction diverges with the number of disks as . Such a behaviour is seen both when the
ordering of disks is fixed ('pen-case' model), and when they can exchange their
positions. Momentum conservation results also in sound-wave effects that
enhance diffusive behaviour and on an intermediate time scale (that diverges in
the thermodynamic limit) normal diffusion takes place even in the 'pen-case'
model. When collisions do not preserve momentum, remains finite and
sound-wave effects are absent.Comment: 4 pages, accepted in Phys.Rev.
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
On the complexity of curve fitting algorithms
We study a popular algorithm for fitting polynomial curves to scattered data
based on the least squares with gradient weights. We show that sometimes this
algorithm admits a substantial reduction of complexity, and, furthermore, find
precise conditions under which this is possible. It turns out that this is,
indeed, possible when one fits circles but not ellipses or hyperbolas.Comment: 8 pages, no figure
Stable regimes for hard disks in a channel with twisting walls
We study a gas of hard disks in a box with semi-periodic boundary
conditions. The unperturbed gas is hyperbolic and ergodic (these facts are
proved for N=2 and expected to be true for all ). We study various
perturbations by twisting the outgoing velocity at collisions with the walls.
We show that the dynamics tends to collapse to various stable regimes, however
we define the perturbations and however small they are.Comment: 30 pages, final version to appear in "Chaos
Approximating multi-dimensional Hamiltonian flows by billiards
Consider a family of smooth potentials , which, in the limit
, become a singular hard-wall potential of a multi-dimensional
billiard. We define auxiliary billiard domains that asymptote, as
to the original billiard, and provide asymptotic expansion of
the smooth Hamiltonian solution in terms of these billiard approximations. The
asymptotic expansion includes error estimates in the norm and an
iteration scheme for improving this approximation. Applying this theory to
smooth potentials which limit to the multi-dimensional close to ellipsoidal
billiards, we predict when the separatrix splitting persists for various types
of potentials
On the work distribution for the adiabatic compression of a dilute classical gas
We consider the adiabatic and quasi-static compression of a dilute classical
gas, confined in a piston and initially equilibrated with a heat bath. We find
that the work performed during this process is described statistically by a
gamma distribution. We use this result to show that the model satisfies the
non-equilibrium work and fluctuation theorems, but not the
flucutation-dissipation relation. We discuss the rare but dominant realizations
that contribute most to the exponential average of the work, and relate our
results to potentially universal work distributions.Comment: 4 page
Ethical living: relinking ethics and consumption through care in Chile and Brazil
Mainstream conceptualizations of ‘ethical consumption’ equate the notion with conscious, individual, market-mediated choices motivated by ethical or political aims that transcend ordinary concerns. Drawing on recent sociology and anthropology of consumption literature on the links between ordinary ethics and ethical consumption, this article discusses some of the limitations of this conceptualization. Using data from 32 focus groups conducted in Chile and Brazil, we propose a conceptualization of ethical consumption that does not centre on individual, market-mediated choices but understands it at the level of practical outcomes, which we refer to as different forms of ‘ethical living’. To do that, we argue, we need to depart from the deontological understanding of ethics that underpins mainstream approaches to ethical consumption and adopt a more consequentialist view focusing on ethical outcomes. We develop these points through describing one particular ordinary moral regime that seemed to be predominant in participants’ accounts of ethics and consumption in both Chile and Brazil: one that links consumption and ethics through care. We show that the moral regime of care leads to ‘ethical outcomes’, such as energy saving or limiting overconsumption, yet contrary to the mainstream view of ethical consumption emphasizing politicized choice expressed through markets, these result from following ordinary ethics, often through routines of practices
Rotation sets of billiards with one obstacle
We investigate the rotation sets of billiards on the -dimensional torus
with one small convex obstacle and in the square with one small convex
obstacle. In the first case the displacement function, whose averages we
consider, measures the change of the position of a point in the universal
covering of the torus (that is, in the Euclidean space), in the second case it
measures the rotation around the obstacle. A substantial part of the rotation
set has usual strong properties of rotation sets