From Deterministic Dynamics to Thermodynamic Laws

统计力学试图解释大系统的热力学行为，并在微观和宏观世界之间架起一座桥梁。平衡统计力学比较发达，但非平衡(即不可逆)统计力学中的许多问题还没有得到很好的回答。特别是，从微观确定性动力学推导宏观输运定律是一个世纪以来的挑战(例如，从微观热传导推导傅立叶热传导定律)。在这个项目中，研究人员从气体的微观颗粒描述出发，研究了不处于热平衡的气体的热力学性质的推导。所研究的微观热传导问题是许多物理系统的代表，这些系统本质上是高维、多尺度、噪声和非平衡的。在这个问题上的数学进展将对非平衡动力学是重要特征的其他科学领域具有价值，包括材料科学、神经科学、分子动力学、流体动力学和生物化学网络。这个项目为本科生提供了研究经验的机会。在这个项目中，研究人员研究了热力学性质是如何从非平衡微观热传导模型中得到的。其目的是发展一系列将微观确定性粒子系统、介观随机微分方程和宏观热力学定律联系起来的结果。该项目从一个确定性动能粒子系统出发，对两端不同温度的管内气体颗粒之间的热传导进行建模，从数值上寻找一个数学上容易处理的随机能量交换过程，该过程保留了确定性问题的主要动力学特征，检查了其随机稳定性和当局部动力学规模增加时的介观极限问题，并推导出与导热系数、长期关联、局部热力学平衡和涨落定理有关的宏观热力学性质。该奖项反映了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，认为值得支持。

Statistical mechanics attempts to explain the thermodynamic behavior of large systems and to bridge the gap between the microscopic and macroscopic worlds. Equilibrium statistical mechanics is relatively well-developed, but many problems in non-equilibrium (i.e., irreversible) statistical mechanics have not been well answered yet. In particular, the derivation of macroscopic transport laws from microscopic deterministic dynamics is a century-old challenge (for example, Fourier's law of heat conduction from microscopic heat conduction). In this project, the investigator studies the derivation of thermodynamic properties of a gas that is not at thermal equilibrium, starting from a microscopic particle description of the gas. The microscopic heat conduction problem under study is representative of numerous physical systems that are intrinsically high-dimensional, multiscale, noisy, and non-equilibrium. Mathematical advances on this problem will be of value for other areas of science where non-equilibrium dynamics is an important feature, including materials science, neuroscience, molecular dynamics, fluid dynamics, and bio-chemical networks. The project includes research experience opportunities for undergraduate students.In this project, the investigator studies how thermodynamic properties are derived from non-equilibrium microscopic heat conduction models. The aim is to develop a series of results that connect microscopic deterministic particle systems, mesoscopic stochastic differential equations, and macroscopic thermodynamic laws. Starting from a deterministic kinetic particle system that models the heat conduction among gas particles in a tube with different temperatures at two ends, the project looks numerically for a mathematically tractable stochastic energy exchange process that preserves the main dynamical features of the deterministic problem, examines its stochastic stability and the mesoscopic limit problem when the scale of local dynamics increases, and derives macroscopic thermodynamic properties related to thermal conductivity, long-range correlation, the local thermodynamic equilibrium, and the fluctuation theorem.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.