It is practically impossible the predict the future behavior of one trajectory in a chaotic dynamical system as any small measurement error leads to huge uncertainty in relatively short amount of time. Many such chaotic systems have been proposed in both the mathematics and physics literature to model some real-life phenomena. For example, a deterministic system of interacting particles could model the microscopic motion of electrons. This research project studies such systems. The goal is to prove, in a mathematically rigorous way, that these systems on the long run behave as if they were random. Consequently, ideas from the theory of random processes can be applied to derive the emergence of statistical macroscopic order from microscopic disorder, that is when the system size or the time of observation is large and the initial state is typical. For the above-mentioned example, this approach provides a derivation of the heat equation from some microscopic deterministic models, which is a central question in the mathematical theory of statistical physics.
Specifically, three research projects will be studied. In the first one, some large system of deterministic interacting balls is considered. Motivated by the common separation of time scale phenomenon is physics, a so called rare interaction limit will be studied, that is when the system can be approximated by independent Sinai billiard particles between any pair interactions. The second project is about advanced statistical properties of hyperbolic systems in both finite and infinite measure case. For example, the local central limit theorem, as a very useful tool in many applications including project one, as well as its connections to infinite measure mixing will be studied for hyperbolic maps and flows. The third project studies stochastic systems of interacting particles. The problems to be studied include the emergence of local equilibrium for systems forced out of equilibrium and joint transport of mass and energy.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
