Brownian motion in partially hyperbolic systems

该项目处理的是多粒子系统，这些系统是质量和/或大小非常不同的粒子的混合物。这种混合气体是统计力学中普遍研究的问题，它们提出了许多具有挑战性的数学问题。一个例子是一个重球与在封闭容器中运动的轻点状原子相互作用(这是布朗运动的经典模型)。另一个例子是一个沉重的圆盘(活塞)在装满理想气体的圆柱体中来回运动。这些模型中的动力学部分是双曲线的，这使得遍历理论的现代方法的应用成为可能。PI计划研究重物(球或活塞)在热力学极限下的行为，当它的质量增长到无穷大时，使其轨迹收敛到扩散过程，该过程要么是布朗运动，要么是布朗运动的积分，要么是Ornstein-Uhlenbeck过程，要么是其变体(取决于模型的细节)。为了证明收敛，PI将利用部分双曲性，并推导相关分布函数的相关性和矩估计的快速衰减。该方案的更广泛的影响源于其应用特性。它的动机是物理学和其他科学中的问题，它的主要目标是为那些应用学科发展数学方法。布朗运动是许多自然过程的模型，它的演化受到许多随机因素的影响。汽缸中的活塞模拟汽车发动机。这一建议的方法显然可以适用于来自自然科学的广泛类别的问题。

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.