Dynamics and Kinetics

确定性系统，如双摆，可能会表现出混沌或不可预测的行为。主要研究人员将研究混沌确定性系统的有限时间演化。传统上认为，只有当时间趋于无穷大时(即当混沌波动稳定下来时)，才能分析这类系统的微妙动力学。该项目旨在将以前对这类系统的有限时间演化的研究扩展到更大的系统类别，其中将包括一些具有实际应用的模型。这项研究的另一部分涉及这样的系统，其中点粒子的运动被非零大小的粒子代替。对这类系统的研究导致了量子混沌的一些新发现，并对实验数据做出了新的解释。这项研究将把这一理论扩展到二维以上，这将允许应用于对物理系统上更大类别的数值和实验数据的解释。这项研究的第三个主题涉及理解统计力学的概率时间不可逆定律是否以及如何严格地从经典力学的确定性时间可逆定律推导出来。该项目包括研究生的培训。严格的有限时间动力学理论使混沌和随机动力系统的相空间输运分析成为可能。也就是说，它允许预测哪些事件可能首先发生，例如，轨道更有可能访问相空间的哪个子集。到目前为止，这一理论已经发展为状态概率相等的伯努利移位。类似的结果将被证明为具有不等概率的伯努利移位和马尔可夫移位。构建混沌(双曲)粒子流的原则将被重新考虑，并将创造新的原则。当有限大小的硬球运动而不是点粒子时，将分析维度大于2的物理粒子。我们将研究一个简单的模型，其中两个圆盘定位在“细胞”中，但允许相互碰撞，并带有两个活塞。在该模型中，每个电池都连接到恒温器(由恒温器约束)，恒温器具有不同的温度，活塞不允许圆盘长时间停止或长时间非常缓慢地移动。一个目标是证明，平均(在时间上)，温度更高的电池中的圆盘将具有更高的能量。这将是证明傅里叶定律的主要一步。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

A deterministic system, such as a double pendulum, can exhibit chaotic or unpredictable behavior. The principal investigator will study the finite time evolution of chaotic deterministic systems. Traditionally it was assumed that the subtle dynamics of such systems can be analyzed only in the limit when time tends to infinity (that is to say when chaotic fluctuations settle down). This project aims to extend previous studies of finite time evolution of such systems to a much larger class of systems, which will include some models with practical applications. Another part of this research deals with systems where the motion of a point particle is substituted by a particle of nonzero size. The study of such systems has led to some new findings in quantum chaos and new interpretations of experimental data. The research will extend this theory to dimensions greater than two, which would allow application to interpretations of a much larger class of numerical and experimental data on physical systems. A third topic of the research deals with understanding whether and how probabilistic time-irreversible laws of statistical mechanics could be rigorously derived from deterministic time-reversible laws of classical mechanics. The project includes the training of graduate students. A rigorous theory of finite time dynamics makes possible the analysis of transport in the phase space of chaotic and random dynamical systems. Namely, it allows the prediction of which events will likely occur first, e. g., which subset of the phase space the orbits will more likely visit. Thus far, this theory has been developed for Bernoulli shifts with equal probabilities of states. Similar results will be proved for Bernoulli shifts with non-equal probabilities and for Markov shifts. The principles for constructing chaotic (hyperbolic) particle-like flows will be reconsidered, and new ones will be created. Physical particles where a finite size hard sphere moves instead of a point particle will be analyzed in dimensions higher than two. A simple model with two discs localized in "cells" but allowed to collide with each other and with two pistons will be studied. In the model, each cell is connected to (bounded by) a thermostat, the thermostats have different temperatures, and the pistons do not allow the discs to stop for a long time or move very slowly for a long time. A goal is to prove that on average (in time) the disc in a cell with a higher temperature thermostat will have a higher energy. This will be a main step to a proof of the Fourier law.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.