From the Kac Model to Non-Equilibrium Statistical Mechanics

热力学学科产生于工程需要，即在新发明的蒸汽机中设计更好的将热转化为机械功的机制。它很快就变得清楚，这是必要的，以了解统计力学，即，大量原子和分子的机械运动和相互作用如何决定气体的宏观热力学参数，如压力或温度。从那时起，最初提出的理解热量做功转换的想法和方法已经扩展到流体运动，导电，化学反应和更多的设置。在过去的30年里，它们已被应用于由大量相互作用的基本组成部分形成的各种系统，如神经网络，动物种群，大型人类网络和经济系统。尽管取得了这些成功，统计力学的深层概念和数学基础仍然是不完整的，特别是对于非平衡情况。马克·卡茨（Mark Kac）介绍了一个非常简化的模型，用于研究碰撞硬球的稀气体中有效宏观方程（玻尔兹曼方程）向平衡态的收敛和出现。由于该模型的简单性，人们可以精确地计算演化的谱隙，并证明单粒子分布一致收敛于玻尔兹曼-卡茨方程的解。这是以失去所有空间结构为代价的，因此也失去了研究局部平衡和接近非平衡稳态等关键现象的能力。该项目将通过重新引入空间结构并将其与热量和粒子库连接来扩展Kac模型，以获得更接近真实的气体但仍然可以严格处理的模型。通过改进为研究大型系统的复杂动力学而开发的数学技术，如缩放极限，双曲动力学和函数不等式，首席研究员将把Kac系统的已知结果扩展到这个更丰富，更现实的模型。虽然大部分工作将是分析性质的，数值模拟将被用来作为更严格的理解指导。最终目标是建立一个模型，能够严格描述从微观碰撞动力学到介观玻尔兹曼方程，最终到宏观热力学/流体动力学流体演化的转变。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The discipline of thermodynamics arose from the engineering need to design better mechanisms of the transformation of heat into mechanical work in the newly invented steam engine. It soon became clear that it was necessary to understand statistical mechanics, i.e., how the mechanical motion and interactions of the extremely large number of atoms and molecules determine macroscopic thermodynamic parameters of the gas, such as the pressure or temperature. Since then, the ideas and methods initially set forth to understand the heat to work transformation have been extended to fluid motion, electrical conduction, chemical reactions and many more settings. In the last 30 years, they have been applied to a diverse variety of system formed by a large number of interacting elementary constituents like neural network, animal populations, large human networks and economic systems. These successes notwithstanding, the deep conceptual and mathematical foundations of statistical mechanics are incomplete, especially so for the non-equilibrium case. This project is aimed to improve understanding of these foundations.Mark Kac introduced a very simplified model to study convergence to equilibrium and emergence of effective macroscopic equations (the Boltzmann equation) in a dilute gas of colliding hard spheres. Thanks to the simplicity of this model one can compute exactly the spectral gap of the evolution and prove uniform convergence of the one-particle distribution to the solution of the Boltzmann-Kac equation. This comes at the price of losing all spatial structure and thus the ability to investigate crucial phenomena such as local equilibrium and approach to non-equilibrium steady states. This project will extend the Kac model by reintroducing a space structure and connecting it with heat and particle reservoirs in such a way as to obtain a model that is closer to a real gas but still is rigorously treatable. By improving the mathematical techniques developed to study complex dynamics of large systems, like scaling limits, hyperbolic dynamics, and functional inequalities, the Principal Investigator will extend the results known for the Kac system to this richer and more realistic model. Although most of the work will be of analytic nature, numerical simulation will be used as a guide toward more rigorous understanding. The final goal is to have a model where one can rigorously describe the transition from microscopic collision dynamics to a mesoscopic Boltzmann type equation and finally to a macroscopic thermodynamic/hydrodynamic fluid evolution.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.