22 research outputs found
A Paradox of State-Dependent Diffusion and How to Resolve It
Consider a particle diffusing in a confined volume which is divided into two
equal regions. In one region the diffusion coefficient is twice the value of
the diffusion coefficient in the other region. Will the particle spend equal
proportions of time in the two regions in the long term? Statistical mechanics
would suggest yes, since the number of accessible states in each region is
presumably the same. However, another line of reasoning suggests that the
particle should spend less time in the region with faster diffusion, since it
will exit that region more quickly. We demonstrate with a simple microscopic
model system that both predictions are consistent with the information given.
Thus, specifying the diffusion rate as a function of position is not enough to
characterize the behaviour of a system, even assuming the absence of external
forces. We propose an alternative framework for modelling diffusive dynamics in
which both the diffusion rate and equilibrium probability density for the
position of the particle are specified by the modeller. We introduce a
numerical method for simulating dynamics in our framework that samples from the
equilibrium probability density exactly and is suitable for discontinuous
diffusion coefficients.Comment: 21 pages, 6 figures. Second round of revisions. This is the version
that will appear in Proc Roy So
Statistical properties of Lorenz like flows, recent developments and perspectives
We comment on mathematical results about the statistical behavior of Lorenz
equations an its attractor, and more generally to the class of singular
hyperbolic systems. The mathematical theory of such kind of systems turned out
to be surprisingly difficult. It is remarkable that a rigorous proof of the
existence of the Lorenz attractor was presented only around the year 2000 with
a computer assisted proof together with an extension of the hyperbolic theory
developed to encompass attractors robustly containing equilibria. We present
some of the main results on the statisitcal behavior of such systems. We show
that for attractors of three-dimensional flows, robust chaotic behavior is
equivalent to the existence of certain hyperbolic structures, known as
singular-hyperbolicity. These structures, in turn, are associated to the
existence of physical measures: \emph{in low dimensions, robust chaotic
behavior for flows ensures the existence of a physical measure}. We then give
more details on recent results on the dynamics of singular-hyperbolic
(Lorenz-like) attractors.Comment: 40 pages; 10 figures; Keywords: sensitive dependence on initial
conditions, physical measure, singular-hyperbolicity, expansiveness, robust
attractor, robust chaotic flow, positive Lyapunov exponent, large deviations,
hitting and recurrence times. Minor typos corrected and precise
acknowledgments of financial support added. To appear in Int J of Bif and
Chaos in App Sciences and Engineerin
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Dynamical equations are formulated and a numerical study is provided for
self-oscillatory model systems based on the triple linkage hinge mechanism of
Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic
mechanical constraint of three rotators as well as systems, where three
rotators interact by potential forces. We present and discuss some quantitative
characteristics of the chaotic regimes (Lyapunov exponents, power spectrum).
Chaotic dynamics of the models we consider are associated with hyperbolic
attractors, at least, at relatively small supercriticality of the
self-oscillating modes; that follows from numerical analysis of the
distribution for angles of intersection of stable and unstable manifolds of
phase trajectories on the attractors. In systems based on rotators with
interacting potential the hyperbolicity is violated starting from a certain
level of excitation.Comment: 30 pages, 18 figure
Structure of shocks in Burgers turbulence with L\'evy noise initial data
We study the structure of the shocks for the inviscid Burgers equation in
dimension 1 when the initial velocity is given by L\'evy noise, or equivalently
when the initial potential is a two-sided L\'evy process . When
is abrupt in the sense of Vigon or has bounded variation with
, we prove that the set
of points with zero velocity is regenerative, and that in the latter case this
set is equal to the set of Lagrangian regular points, which is non-empty. When
is abrupt we show that the shock structure is discrete. When
is eroded we show that there are no rarefaction intervals.Comment: 22 page
Escape Rates and Physically Relevant Measures for Billiards with Small Holes
We study the billiard map corresponding to a periodic Lorentz gas in
2-dimensions in the presence of small holes in the table. We allow holes in the
form of open sets away from the scatterers as well as segments on the
boundaries of the scatterers. For a large class of smooth initial
distributions, we establish the existence of a common escape rate and
normalized limiting distribution. This limiting distribution is conditionally
invariant and is the natural analogue of the SRB measure of a closed system.
Finally, we prove that as the size of the hole tends to zero, the limiting
distribution converges to the smooth invariant measure of the billiard map.Comment: 39 pages, 4 figure