Chaotic Dynamics, Attractors and Repellers

【摘要】chernov该项目致力于平衡和非平衡统计力学中的数学问题，重点研究确定性混沌动力系统的遍历和统计性质。具体模型包括外场中的洛伦兹气体、硬球的粘性气体、带逃逸孔的开放哈密顿系统。我们将严格研究离散时间和连续时间外场中多粒子洛伦兹气体的几种物理定律（包括欧姆定律、格林-久保公式和爱因斯坦关系）。我们期望证明相关的指数衰减，伯努利性质和各种概率极限定理。本文还将研究硬球气体的遍历性质和熵。另一个研究方向是研究开放系统（含逃逸孔）中混沌驱避器的条件不变测度、逃逸率形式主义和分形结构。在过去的40年里，混沌和动力系统的数学理论得到了巨大的发展。它始于玻尔兹曼和麦克斯韦创建的统计物理学基础。它涵盖了基本的生物模型（人口动力学），工程中的各种问题，地震预测，交通控制，计算机数据处理的神经网络，制造和管理任务的优化，以及无数其他复杂系统长期行为问题的应用。自然界和社会中各种过程所表现出的混沌的普遍特征，使得用一个理论来描述和预测这种现象成为可能。特别是，吸引子和排斥子的概念（作为决定复杂系统长期演化的结构或模式）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相对年轻，与古老的经典数学有很大不同。

AbstractChernov The project is devoted to mathematical problems in equilibrium and nonequilibrium statistical mechanics, focusing on the ergodic and statistical properties of deterministic chaotic dynamical systems. Specific models include Lorentz gases in external fields, viscous gases of hard balls, open Hamiltonian systems with escape holes. We are going to study rigorously several laws of physics (including Ohm's law, Green-Kubo formula and Einstein relation) for multiparticle Lorentz gases in external fields with discrete and continuous times. We expect to prove exponential decay of correlations, Bernoulli property and various probabilistic limit theorems. The ergodic properties and entropy of hard ball gases will also be studied. Another direction of research is to investigate conditionally invariant measures, escape-rate formalism, and fractal structure of chaotic repellers in open systems (with escape holes). 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 and Maxwell. 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 evolution of complex systems) are basic categories in the theory of chaos. Mathematical theory of chaotic attractors and repellers is still relatively young and quite different from the old, classical mathematics.