The proposed research addresses some long-standing problems in the theory of dynamical systems and statistical mechanics, as well as some new natural questions that are important from a general point of view and for applications. The project develops a new approach to the design of hyperbolic billiards that will allow one to prove hyperbolicity for more a general class of billiards. This approach is based on a new general characterization of absolutely focusing curves, which are the only admissible focusing components of hyperbolic billiards, in terms of continued fractions. A long-standing problem on whether one can smooth the boundary of a stadium billiard will be resolved. The project will also shed new light on the question of where the border lies between completely chaotic billiards in convex domains and billiards with divided phase spaces into chaotic and regular components. Two natural questions raised by the principal investigator on the dynamics of open systems will be addressed. The first one asks how the escape through a hole depends on the position of a hole in phase space. The second question is about the relationship between escape through one hole and escape through multiple holes. These questions reveal some subtle connections between combinatorics and number theory. The dynamics of a finite-size billiard particle in nonconvex polygon will be shown to be hyperbolic. This will be applied to the classical Ehrenfest periodic wind-tree model in statistical mechanics and demonstrate that, from a natural physical point of view, this model surpasses the periodic Lorentz model in the richness of its dynamics.
The project will provide new visual and relatively simple models of billiard dynamical systems with chaotic as well as with mixed (coexisting regions with chaotic and with regular dynamics) behavior. (N.B."Billiards" is a technical mathematical concept that does not refer to the parlor game of that name.) Moreover, some of the models introduced by the principal investigator will be (and some already have been) used in physics by both theoreticians and experimentalists, who have actually built such devices, and therefore will foster interdisciplinary collaborations. A problem of finding an optimal (to ensure the fastest/slowest escape) placement of a hole will have a potentially large variety of applications for open systems. This question, as well as the one on escape through multiple holes, was inspired by experiments on atomic billiards. Moreover, this approach opens up the possibility of making finite-time (rather than asymptotic in time) predictions of dynamics (e.g., predicting a moment after which escape through a specific hole is more likely than escape through any other hole of the same size). The analysis of the wind-tree model with a finite-size particle will have applications in statistical mechanics. The project will enhance the infrastructure for research and education through collaborations with researchers in the US, Mexico, Canada, and Europe. Graduate students are already involved in this research, and the involvement of undergraduates is anticipated. The results of the project will be broadly disseminated to enhance scientific and technological understanding via participation of the principal investigator (often with plenary talks), his collaborators, and his students in interdisciplinary conferences with a broad participation of physicists, biologists, and engineers.
The results of this project contain possibly transformable discovery. It is indeed a discovery, it was not planed or proposed but the Principle Investigator run into it while trying to answer a new question in the mathematical theory of open system;"How escape rate depends on position of a hole/". A discovery says that it is possible to analyse rigorously a finite time behavior of observables rather than traditionally analized their asymptotic in time (when time tends to infinity) behavior or average behavior (where average is taken over infinite time). Two suvh important obeservables were determined so far which are responsible for absorption and transmittion of perturbation ("information") by different parts of the state spaces of the dynamical or stochastic system understudy. It refers only to dynamical systems with chaotic behavior while for dynamical systems with regular behavior such finite time qualitative effects probably do not exist. One of immediate applications of this discovery is possibility to characterize elements of networks dynamically (rather than traditional static approach),e.g. with respect to the elements of networks ability to absorb or transmit information. It was proven that focusing components of boundaries of chaotic billiard systems must be absolutely focusing. The last means that any parallel beam of rays which id fallen on a focusing mirror after a complette series of reflections off it must be focusing (rather than disperising or parallel). It was constructed a new class of billiards which, besides its own surprising properties, may have applications in nanotechnology. Thesenew type of billiards were called track billiards. In track billiards orbits are chaotic (with probability one) but they keep an orientation, i.e. half of them moves clockwise but another half counterclockwise. Besides, in this project were obtained many technical mathematical results which advance knowledge in the area of chaotic dynamics and statistical mechanics.
