Leonid A. Bunimovich

On the ergodic properties of nowhere dispersing billiards

For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.

Statistical properties of lorentz gas with periodic configuration of scatterers

In our previous paper Markov partitions for some classes of dispersed billiards were constructed. Using these partitions we estimate the decay of velocity auto-correlation function and prove the central limit theorem of probability theory and Donskers Invariance Principle for Lorentz Gas with periodic configuration of scatterers.

Spacetime chaos in coupled map lattices

Coupled map lattices have been introduced for studying systems with spatial complexity. The authors consider simple examples of such systems generated by expanding maps of the unit interval (or circle) with some kind of diffusion coupling. It is shown that such systems have a symbolic representation by two-dimensional lattice models of statistical mechanics. The main result states that the Z2 dynamical system generated by space translations and dynamics has a unique invariant mixing Gibbs measure with absolutely continuous finite-dimensional projections. This measure is an analogy of the BRS measure constructed for finite-dimensional hyperbolic transformations.

Markov partitions for dispersed billiards

Markov Partitions for some classes of billiards in two-dimensional domains on ℝ2 or two-dimensional torus are constructed. Using these partitions we represent the microcanonical distribution of the corresponding dynamical system in the form of a limit Gibbs state and investigate the character of its approximations by finite Markov chains.

On the Boltzmann equation for the Lorentz gas

We consider the Boltzmann-Grad limit for the Lorentz, or wind-tree, model. We prove that if ω is a fixed configuration of scatterer centers belonging to a set of full measure with respect to the Poisson distribution with parameter λ>0, then the evolution of an initial a.c. particle density tends in the Boltzmann-Grad limit to the solution of the Boltzmann equation for the model. As an intermediate step we prove that the process of the free path lengths and impact parameters induced by the Lebesgue measure on a small region tends to a limiting independent process.

Coupled map lattices: some topological and ergodic properties

Abstract Lattice dynamical systems (LDSs) form the class of extended systems that is the intermediate one between partial differential equations (PDEs) and cellular automata. The most popular class of LDSs is formed by coupled map lattices (CMLs). While being introduced rather recently LDSs allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio-temporal dynamics and to obtain some new, and even rigorous results on space-time chaos, intermittency and pattern formation. We discuss the rigorous results that were obtained in this area, but eventually give also some applications.

One-Dimensional Dynamical Systems and Benford's Law

One-dimensional projections of (at least) almost all orbits of many multi- dimensional dynamical systems are shown to follow Benfords law, i.e. their (base b) mantissa distribution is asymptotically logarithmic, typically for all bases b. As a generalization and uniflcation of known results it is proved that under a (generic) non-resonance condition on A 2C d£d , for every z 2C d real and imaginary part of each non-trivial component of (A n z)n2N0 and (e At z)t‚0 follow Benfords law. Also, Benford behavior is found to be ubiquitous for several classes of non-linear maps and difierential equations. In particular, emergence of the logarithmic mantissa distribution turns out to be generic for complex analytic maps T with T(0) = 0, jT0(0)j < 1. The results signiflcantly extend known facts obtained by other, e.g. number-theoretical methods, and also generalize recent flndings for one-dimensional systems.

Short- and long-term forecast for chaotic and random systems (50 years after Lorenz's paper)

We briefly review a history of the impact of the famous 1963 paper by E Lorenz on hydrodynamics, physics and mathematics communities on both sides of the iron curtain. This paper was an attempt to apply the ideas and methods of dynamical systems theory to the problem of weather forecast. Its major discovery was the phenomenon of chaos in dissipative dynamical systems which makes such forecasts rather problematic, if at all possible. In this connection we present some recent results which demonstrate that both a short-term and a long-term forecast are actually possible for the most chaotic dynamical (as well as for the most random, like IID and Markov chain) systems. Moreover, there is a sharp transition between the time interval where one may use a short-term forecast and the times where a long-term forecast is applicable. Finally we discuss how these findings could be incorporated into the forecast strategy outlined in the Lorenzs paper.

Recurrence properties of Lorentz lattice gas cellular automata

Recurrence properties of a point particle moving on a regular lattice randomly occupied with scatterers are studied for strictly deterministic, nondeterministic, and purely random scattering rules.

Coupled map lattices: one step forward and two steps back

Abstract We discuss the general notion of Lattice Dynamical Systems and some recent progress in their studies. We suggest also the approach that allows to extract some information on the dynamics of finitely extended (“real”) systems from the results obtained on the infinitely extended (“artificial”) ones.