6,069 research outputs found
On Recent Progress for the Stochastic Navier Stokes Equations
We give an overview of the ideas central to some recent developments in the
ergodic theory of the stochastically forced Navier Stokes equations and other
dissipative stochastic partial differential equations. Since our desire is to
make the core ideas clear, we will mostly work with a specific example: the
stochastically forced Navier Stokes equations. To further clarify ideas, we
will also examine in detail a toy problem. A few general theorems are given.
Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and
hypoellipticity are all discussed.Comment: Corrected version of Journees Equations aux derivees partielles
paper(June 2003). Original at
http://www.math.sciences.univ-nantes.fr/edpa/2003
On the relaxation to nonequilibrium steady states
The issue of relaxation has been addressed in terms of ergodic theory in the
past. However, the application of that theory to models of physical interest is
problematic, especially when dealing with relaxation to nonequilibrium steady
states. Here, we consider the relaxation of classical, thermostatted particle
systems to equilibrium as well as to nonequilibrium steady states, using
dynamical notions including decay of correlations. We show that the condition
known as {\Omega}T-mixing is necessary and sufficient to prove relaxation of
ensemble averages to steady state values. We then observe that the condition
known as weak T-mixing applied to smooth observables is sufficient for
relaxation to be independent of the initial ensemble. Lastly, weak T-mixing for
integrable functions makes relaxation independent of the ensemble member, apart
from a negligible set of members enabling the result to be applied to
observations from a single physical experiment. The results also allow us to
give a microscopic derivation of Prigogine's principle of minimum entropy
production in the linear response regime. The key to deriving these results
lies in shifting the discussion from characteristics of dynamical systems, such
as those related to metric transitivity, to physical measurements and to the
behaviour of observables. This naturally leads to the notion of physical
ergodicity.Comment: 44 pages, 1 figur
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure
Ergodicity for Infinite Periodic Translation Surfaces
For a Z-cover of a translation surface, which is a lattice surface, and which
admits infinite strips, we prove that almost every direction for the
straightline flow is ergodic
Multidimensional hyperbolic billiards
The theory of planar hyperbolic billiards is already quite well developed by
having also achieved spectacular successes. In addition there also exists an
excellent monograph by Chernov and Markarian on the topic. In contrast, apart
from a series of works culminating in Sim\'anyi's remarkable result on the
ergodicity of hard ball systems and other sporadic successes, the theory of
hyperbolic billiards in dimension 3 or more is much less understood. The goal
of this work is to survey the key results of their theory and highlight some
central problems which deserve particular attention and efforts
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