71 research outputs found

### Entropy Production in Random Billiards

We introduce a class of random mechanical systems called random billiards to
study the problem of quantifying the irreversibility of nonequilibrium
macroscopic systems. In a random billiard model, a point particle evolves by
free motion through the interior of a spatial domain, and reflects according to
a reflection operator, specified in the model by a Markov transition kernel,
upon collision with the boundary of the domain. We derive a formula for entropy
production rate that applies to a general class of random billiard systems.
This formula establishes a relation between the purely mathematical concept of
entropy production rate and textbook thermodynamic entropy, recovering in
particular Clausius' formulation of the second law of thermodynamics. We also
study an explicit class of examples whose reflection operator, referred to as
the Maxwell-Smolukowski thermostat, models systems with boundary thermostats
kept at possibly different temperatures. We prove that, under certain mild
regularity conditions, the class of models are uniformly ergodic Markov chains
and derive formulas for the stationary distribution and entropy production rate
in terms of geometric and thermodynamic parameters.Comment: 30 pages, 9 figure

### Groups that do not act by automorphisms of codimension-one foliations

Let G be a finitely generated group having the property that any action of
any finite-index subgroup of G by homeomorphisms of the circle must have a
finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real
rank at least two have this property.) Suppose that such a G acts on a compact
manifold M by automorphisms of a codimension-one C2 foliation, F. We show that
if F has a compact leaf, then some finite-index subgroup of G fixes a compact
leaf of F. Furthermore, we give sufficient conditions for some finite-index
subgroup of G to fix each leaf of F.Comment: Latex2e file, 15 pages, no figure

### Multiple scattering in random mechanical systems and diffusion approximation

This paper is concerned with stochastic processes that model multiple (or
iterated) scattering in classical mechanical systems of billiard type, defined
below. From a given (deterministic) system of billiard type, a random process
with transition probabilities operator P is introduced by assuming that some of
the dynamical variables are random with prescribed probability distributions.
Of particular interest are systems with weak scattering, which are associated
to parametric families of operators P_h, depending on a geometric or mechanical
parameter h, that approaches the identity as h goes to 0. It is shown that (P_h
-I)/h converges for small h to a second order elliptic differential operator L
on compactly supported functions and that the Markov chain process associated
to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L
are selfadjoint (densely) defined on the space L2(H,{\eta}) of
square-integrable functions over the (lower) half-space H in R^m, where {\eta}
is a stationary measure. This measure's density is either (post-collision)
Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes
with infinitesimal generator L respectively correspond to what we call MB
diffusion and (generalized) Legendre diffusion. Concrete examples of simple
mechanical systems are given and illustrated by numerically simulating the
random processes.Comment: 34 pages, 13 figure

### Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature

We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure Î». Our main technical result is the following: If the Anosov foliations are Câˆž, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant Câˆž affine connection âˆ‡ on V such that âˆ‡R â‰¡ 0, where R is the curvature tensor of âˆ‡, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a Câˆž splitting of TV (defined everywhere on V) and for any invariant ergodic measure Î¼, there exists Ï‡_Î¼ > 0 such that the Lyapunov exponents are âˆ’2Ï‡_Î¼, âˆ’Ï‡_Î¼, 0, Ï‡_Î¼, and 2Ï‡_Î¼, Î¼-almost everywhere.
As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are Câˆž. Then, this flow is Câˆž conjugate to the geodesic flow on a manifold of constant negative curvature

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