Mathematical Sciences: Chaotic Attractors and Repellers: Rigorous Results

切尔诺夫这个项目致力于研究混沌动力系统。它侧重于非平衡统计物理、混沌吸引子和排斥子。这个项目的一个主要目标是寻找具有奇异性的连续时间双曲动力系统(如周期洛伦兹气体)的关联函数的有效界。这将使严格推导相关非平衡热力学模型的输运定律成为可能。其中包括欧姆定律、爱因斯坦关系和周期洛伦兹气体中受小电场和磁场影响的带电粒子的格林-久保公式。其中还包括环面上两个硬盘模型的剪切粘度的牛顿定律。另一个主要目标是研究带有小孔的光滑混沌映射的排斥子和条件不变测度。这是为了解决混乱的有口袋的台球桌，然后解决1979年的皮亚尼吉亚尼-约克猜想。该项目包括在所谓的麦克斯韦恶魔边界条件下对盒子中硬盘的非平衡动力学进行某些数值实验。在过去的40年里，混沌和动力系统的数学理论有了巨大的发展。它始于十九世纪马克斯韦尔和波尔兹曼创立的统计物理学基础.它涵盖了基本的生物学模型(人口动力学)、工程中的各种问题、地震预测、交通控制、用于计算机数据处理的神经网络、生产和管理任务的优化，并在复杂系统的长期行为受到质疑的地方有无数其他应用。自然界和社会中各种过程所表现出的混沌的普遍特征，使得用一种理论来描述和预测这种现象成为可能。特别是，吸引子和排斥子(作为决定复杂系统长期演化的结构或模式)的概念是混沌理论中的基本范畴。混沌吸引子和排斥子的数学理论仍然非常年轻，与古老的古典数学有很大的不同。这是目前数学研究不仅最活跃，而且服务于其他现代科学和人类活动的直接需求的领域之一。

Abstract Chernov The project is devoted to chaotic dynamical systems. It focuses on nonequilibrium statistical physics, chaotic attractors, and repellers. A major goal of this project is to find effective bounds on correlation functions for continuous-time hyperbolic dynamical systems with singularities, such as periodic Lorentz gas. This will make it possible to derive rigorously transport laws for related models of nonequilibrium thermodynamics. These include Ohm's law, the Einstein relation and the Green-Kubo formula for a charged particle in the periodic Lorentz gas subject to a small electrical and magnetic field. These also include Newton's law for shear viscosity for the model of two hard disks on the torus. Another major goal is the study of repellers and conditionally invariant measures for smooth chaotic maps with small holes. This aims at chaotic billiard tables with pockets and then solving the Pianigiani-Yorke conjecture of 1979. The project includes certain numerical experiments on nonequilibrium dynamics of hard disks in a box with the so-called Maxwell demon boundary conditions. The mathematical theory of chaos and dynamical systems has grown dramatically over the past 40 years. It started with foundations of statistical physics created by Boltzmann, Maxwell in the XIX-th century. It has covered basic biological models (dynamics of population), various problems in engineering, earthquake prediction, traffic control, neural networks for computer data processing, optimization of manufacturing and management tasks, and has countless other applications where long-time behavior of complex systems is in question. Universal features of chaos exhibited by various processes in nature and society make it possible to describe and predict such phenomena by the means of one theory. In particular, the concepts of attractors and repellers (as structures or patterns determining long-time ev olution of complex systems) are basic categories in the theory of chaos. Mathematical theory of chaotic attractors and repellers is still very young and quite different from the old, classical mathematics. This is one of the areas where mathematical research is currently not only most active but serves the direct needs of other modern sciences and human activities.