Dynamics in Physical Models

该项目致力于物理起源的动力系统，包括洛伦兹气体（平衡和外力作用下）、硬球系统、台球、具有大质量测试粒子的理想气体和开放哈密顿系统。其中大多数已知（或预期）具有混沌行为。由于D. Dolgopyat， D. Ruelle, Ya。西奈半岛,L.-S。Young和其他人，双曲和混沌动力系统理论中的数学工具似乎已经发展到足以解决许多迄今为止仅由物理学家启发式或数值研究的开放性问题，如果有的话。特别是，我们计划利用sinnai - ruelle - bowen测度、出现在输运定律和扩散方程中的时间相关函数、允许条件不变测度的开放哈密顿系统、使用适当的时空极限在理想气体中的大质量粒子的运动等手段来研究非平衡稳态的性质。在每种情况下，我们的目标是获得精确的结果，并提供坚实严谨的证明。该项目的总体目标是对物理界关注的事实和现象进行数学研究，并在数学之外有应用。特别是，这些结果将有助于统计力学和热力学的数学基础，并可以加强动力系统理论与物理学和其他科学之间的联系。

The project is devoted to dynamical systems of physical origin, including Lorentz gases (both in equilibrium and under external forces), hard ball systems, billiards, ideal gases with a massive test particle, and open Hamiltonian systems. Most of these are known (or expected) to have chaotic behavior. Due to recent works of D. Dolgopyat, D. Ruelle, Ya. Sinai, L.-S. Young and others, mathematical tools in the theory of hyperbolic and chaotic dynamical systems appear to be developed far enough to attack many open problems that have been so far only studied heuristically or numerically by physicists, if at all. In particular, we plan to investigate the nature of nonequilibrium steady states by means of Sinai-Ruelle-Bowen measures, time correlation functions that appear in transport laws and diffusion equations, open Hamiltonian systems that admit conditionally invariant measures, the motion of a massive particle in an ideal gas by using an appropriate space-time limit, etc. In each case we aim at obtaining exact results and providing solid rigorous proofs. The general goal of the project is to conduct mathematical studies of facts and phenomena that have attracted attention in physical community and have applications outside of mathematics. In particular, the results would contribute to the mathematical foundation of statistical mechanics and thermodynamics and could strengthen the link between the theory of dynamical systems and physics and other sciences.