Dynamics and Kinetics

摘要提出的研究涉及有限维和无限维动力系统以及一些混合（介于动力和随机之间）系统的研究。将特别讨论的混合系统是由随机环境中的确定性行走形成的。最近已经证明，这类系统的一个大子类（在刚性环境中行走）在一维中是完全可解的。本文研究的是变刚度环境的情况。该建议的另一部分涉及最近构造的具有分相空间的哈密顿系统的第一个自然例子的进一步发展。现在的目标是证明在任何（有限）维度上，一类相当一般的台球都具有正的度量熵。统计力学考虑的问题涉及开放系统中动力学系数和逃逸率公式的严格推导。在处理无穷维系统的问题中有晶格动力系统的混沌阶跃迁和非线性波动方程的混沌。其他问题涉及各种系统，从遍历理论（本福德定律）到大脑动力学和物流的一些简单模型。传统上，真实系统的数学模型要么是纯确定性的，要么是纯随机的。这两类模型都有丰富的理论，研究人员对它们的行为有很好的直觉，也就是说，他们基本上知道会发生什么。这种直觉本质上是基于一个丰富的完全可解决（即完全理解）的模型集合。然而，大多数真实系统既不是纯确定性的，也不是纯随机的。相反，它们同时具有（确定性和随机性）特征。本项目涉及一大类此类模型，这些模型是在通信理论、统计物理、化学动力学、人工智能理论等领域独立引入的。

PI: Leonid BunimovichProposal Number: 0140165 ABSTRACTThe proposed research deals with the studies of finite- andinfinite-dimensional dynamical systems as well as of somehybrid (intermediate between dynamical and stochastic) systems.The class of hybrid systems which will be specifically addressedis formed by deterministic walks in random environments. It has been shownrecently that a large subclass of such systems (walksin rigid environments) is completely solvable in 1D. The proposedstudy deals with the case of environments with variable rigidity.Another part of the proposal deals with the farther development ofrecently constructed first natural examples of Hamiltonian systemswith divided phase space. The goal now is to show that a rather general class of such billiards has positive metric entropy in any (finite) dimension. The considered problems of statistical mechanics deal with the rigorous derivation of formulas for kinetic coefficients and for escape rate in open systems. Among the problems which deal with infinite-dymensional systems are chaos-order transitions in Lattice Dynamical Systems and chaos in nonlinear wave equations. Other problems deal with various systems ranging from the Ergodic Theory (Benford's law)to some simple models of brain dynamics and logistics. Traditionally the mathematical models of real systems are either purely deterministic or purely stochastic. These two classes of models enjoy having a rich theory, and researchers have quite good intuition on their behavior, i.e. they basically know what to expect. This intuition is essentially based on a rich collection of completely solvable (i.e. completely understood) models. However, a majority of real systems are neither purely deterministic nor purely stochastic. Instead they have both (deterministic and stochastic) features. This project deals with a big class of such models which were independently introduced in communication theory, statistical physics,chemical kinetics, theory of artificiall intellect, etc.