HYPERBOLIC DYNAMICS IN PHYSICAL MODELS

该项目致力于研究统计力学中的数学模型。它涉及双曲动力系统，其起源于电流，导电性，扩散和分子气体的物理研究。双曲性是用来推导电动力学和热力学定律的一个关键特征，或者用来描述底层粒子系统的宏观行为。该项目中要处理的具体模型包括硬球气体（特别是那些模仿布朗运动的气体），洛伦兹气体（模仿金属中的电子）和高尔顿板（弹球机的物理模拟）。该建议还涉及由Navier-Stokes方程和准地转方程控制的流体动力学。建议的研究的主要工具是双曲动力学，平均理论和随机过程的方法。该提案涉及的问题起源于物理学，电动力学，热力学和统计力学。它的主要目标是通过精确的数学来研究复杂的物理现象，并推导出精确的自然定律。在许多情况下，主要研究者希望获得物理过程的严格数学描述，这些物理过程仅在经验上或定性上已知。在某些情况下，数学分析揭示了复杂进化模型的新特征，这些特征在实验研究中被忽视或忽视。首席研究员与物理学家和统计学家在几个联合研究项目上密切合作。

The project is devoted to the study of mathematical models in statistical mechanics. It deals with hyperbolic dynamical systems that have their origins in physical studies of electrical current, conductivity, diffusion, and gases of molecules. Hyperbolicity is a key feature used to derive laws of electrodynamics and thermodynamics, or otherwise to describe the macroscopic behavior of the underlying systems of particles. Specific models to be treated in the project include gases of hard balls (in particular, those emulating Brownian motion), Lorentz gases (imitating electrons in metals), and the Galton board (a physics analogue of a pinball machine). The proposal also deals with the dynamics of fluids governed by the Navier-Stokes equations and quasi-geostrophic equations. The main tools of the proposed studies are methods of hyperbolic dynamics, averaging theories, and stochastic processes. The proposal deals with problems originating in physics, electrodynamics, thermodynamics, and statistical mechanics. Its main goal is to study complex physical phenomena by exact mathematics and derive precise laws of nature. In many cases the principal investigator hopes to obtain rigorous mathematical descriptions of physical processes that have been only known empirically or qualitatively. In some cases the mathematical analysis reveals new features of complex evolution models that have been overlooked or unnoticed in experimental studies. The principal investigator closely collaborates with physicists and statisticians on several join research projects.