András Krámli

A “transversal” fundamental theorem for semi-dispersing billiards

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A “Transversal” Fundamental Theorem has recently been suggested by the present authors to proveglobal ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems ofN≧3 elastic hard balls conjectured by the celebratedBoltzmann-Sinai ergodic hypothesis. (In fact, the suggested “Transversal” Fundamental Theorem has been successfully applied by the authors in the casesN=3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtainlocal ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersing-billiards. “Transversality” means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.

The K-property of three billiard balls

Sinais strengthened version of the ergodic hypothesis is proved for three billiard balls on the v-dimensional torus: On connected components of the submanifold of the phase space specified by the trivial conservation laws of the energy and of the trajectory of the center of mass, the system is a K-flow. To cope with the difficulty that in the isomorphic one-particle-billiard system the scatterers are not strictly convex, geometric-algebraic, ergodic-theoretic and topological methods are elaborated

The

A further step is achieved toward establishing the celebrated Boltzmann-Sinai ergodic hypothesis: for systems of four hard balls on the ν-torus (ν>2) it is shown that, on the submanifold of the phase specified by the trivial conservation laws, the system is aK-flow. All parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened. The main novelties are: (i) A refinement of the geometric-algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three-billiards. As a matter of fact, it is this part of our arguments, where further understanding and new ideas are necessary before attacking the general ergodic problem; (ii) In the “pasting” part of the proof, which is a sophisticated version of Hopfs classical device, the arguments are so general that it is hoped they work in the general case, too. This is achieved for four balls, in particular, by a version of the Transversal Fundamental Theorem which, on one hand, is simpler and more suitable for applications than the previous one and, on the other hand, as we have discovered earlier, is the main tool to prove global ergodicity of semi-dispersing billiards; (iii) The verification of the Chernov-Sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case.

K

Between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift. By using a functional-analytic approach based on the spectral theory of matrix polynomials, the asymptotics of the first-hitting probabilities is obtained when the distance of the barriers tends to infinity.

-property of four billiard balls

Billiards are considered on two-dimensional, smooth, compact Riemannian manifolds with dispersing scatterers. We prove that these billiards are ergodic if only Vetiers conditions for the absence of focal points hold.

Random walks with internal degrees of freedom

The Markov partition of the Sinai billiard allows the following heuristic interpretation for the Lorentz process with a ℤ2-periodic configuration of scatterers: while executing a (non-Markovian) random walk on ℤ2, and particle changes its internal state according to the symbolic dynamics defined by the Markov partition. This picture can be formalized and then the Lorentz process appears as the limit of a sequence of (Markovian!) random walks with a finite but increasing number of internal states and the central limit theorem can be proved for it by perturbational expansions with uniformly bounded — in a sence related to the Perron-Frobenius theorem — coefficients and uniform remainder terms.

Dispersing billiards without focal points on surfaces are ergodic

We prove that, for the planar Lorentz process with a periodic configuration of scatterers, the quasi-local CLT of the gaussian {logϱn} type holds for any ϱ>1. Consequently, for arbitrary ϱ>3/2, the probabilities that, at the moment of thenth reflection, this process lies in a square of size logϱn are asymptotically gaussian. This implies that these events occur for infinitely many values ofn (i.e. a weaker form of recurrence).

Central limit theorem for the Lorentz process via perturbation theory

Heat transport coefficients are calculated for various random walks with internal states (the Markov partition of the Sinai billiard connects these walks with the Lorentz gas among a periodic configuration of scatterers). Models with reflecting or absorbing barriers and also those without or with local thermal equilibrium are investigated. The method is unified and is based on the Keldysh expansion of the resolvent of a matrix polynomial.

The problem of recurrence for Lorentz processes

An exposition of some methods of proving exponential (stretched exponential) decay of correlations is given. One-dimensional strictly hyperbolic and quadratic maps and two-dimensional piecewise smooth, uniformly hyperbolic maps are considered. The emphasis is on the fundamental constructions of the Markov sieve method due to Bunimovich-Chernov-Sinai and those of Liveranis Hilbert metric method.

Heat conduction in caricature models of the Lorentz gas

For a degenerate random walk in a 2D Bernoulli environment without local traps, computer results show a non-Wiener behavior. For a better exploitation of the memory, the analysis is based on the statistics of the first exit time from a square.