Brownian motion in partially hyperbolic systems

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The project deals with multiparticle systems that are mixtures of particles of very different masses and/or sizes. Such mixed gases are commonly studied in statistical mechanics, and they present many challenging mathematical problems. One example is a heavy ball interacting with light point-like atoms moving in a closed container (this is a classical model of Brownian motion). Another example is a heavy disk (a piston) moving back and forth in a cylinder filled with an ideal gas. The dynamics in these models is partially hyperbolic, which makes the application of the modern methods of ergodic theory possible. The PI plans to investigate the behavior of the heavy object (the ball or the piston) in the thermodynamical limit, as its mass grows to infinity, so that its trajectory converges to a di@usion process, which is either a Brownian motion, or an integral of Brownian motion, or an Ornstein-Uhlenbeck process, or a variation thereof (depending on the details of the model). To prove the convergence, the PI will use the partial hyperbolicity and derivea fast decay of correlations and moment estimates for the relevant distribution functions.The broader impact of the proposal derives from its applied character. It is motivated by problems in physics and other sciences and its main goal is the development of mathematical methods for those applied disciplines. Brownian motion is a model for many natural processes where the evolution is subjected to many random factors. The piston in a cylinder models a car engine. The methods of this proposal can clearly apply to wide classes of problems coming from natural sciences.