Analysis and Data-Driven Computation for Nonequilibrium Thermodynamic Models

统计力学的作用是研究微观实体(如原子和分子)的大型组装，并弥合微观实体与宏观属性(如温度分布和导热系数)之间的差距。非平衡统计力学中的许多问题的数学证明，例如，傅立叶定律关于热通量与温度梯度成正比的声明，仍然具有很大的挑战性。该项目旨在开发新的分析工具和数据驱动的计算方法，以研究由统计物理和波动湍流引起的一系列非平衡热力学模型。这些非平衡模型不仅与统计物理有关，而且与无数本质上不可逆和多尺度的应用科学问题有关，如化学反应和神经动力学。该项目还为研究生和高级本科生提供了研究培训的机会。在这个项目中，首席研究员(PI)将使用分析和计算相结合的方法来研究如何从一类微观能量转移模型推导热力学性质，这些模型包括经典的台球系统和来自非平衡统计物理和波动湍流的非线性振子链模型。一般的方法是用最少的计算工作绕过一些困难，得到数学上容易处理的随机模型。从这些随机模型发展热力学定律通常要容易得多。数据驱动计算方法的开发和应用是拟议研究的一个组成部分。PI发展了一系列新的计算方法，将传统的蒙特卡罗模拟与数值偏微分方程组求解器、耦合方法和人工神经网络等工具相结合。它们克服了传统的基于离散化的算法的几个缺点，特别是对于高维问题。在研究这个项目中的许多高维问题时，我们需要高维不变概率度量及其遍历性的计算结果，以绕过严格方法无法处理的困难。这一奖项反映了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，被认为值得支持。

The role of statistical mechanics is to study large assemblies of microscopic entities (such as atoms and molecules) and to bridge the gap between microscopic entities and macroscopic properties like temperature profile and thermal conductivity. Mathematical justifications of numerous problems in nonequilibrium statistical mechanics, such as, for example, Fourier’s law statement that the heat flux is proportional to the temperature gradient, remain highly challenging. This project aims to develop novel analytical tools and data-driven computational methods to study a series of nonequilibrium thermodynamic models arising from statistical physics and wave turbulence. These nonequilibrium models are relevant not only for the statistical physics, but also for countless applied scientific problems that are intrinsically irreversible and multiscale, such as chemical reactions and neural dynamics. The project also provides research training opportunities for graduate students and advanced undergraduate students.In this project, the principal investigator (PI) will use a combination of analytical and computational approaches to investigate how thermodynamic properties are derived from a class of microscopic energy transfer models, including classical billiards-like systems and nonlinear oscillator chain models coming from nonequilibrium statistical physics and wave turbulence. The general approach is to use minimum computational work to bypass some difficulties and to derive mathematically tractable stochastic models. Developing thermodynamic laws from those stochastic models are usually much easier. The development and application of data-driven computational methods is a constitutive element of the proposed research. The PI has developed a series of novel computational methods that combines traditional Monte Carlo simulation with tools like numerical partial differential equation solver, coupling method, and artificial neural network. They overcome several disadvantages of traditional, discretization-based algorithms, especially for high-dimensional problems. When studying many high-dimensional problems in this project, we need computational results of high-dimensional invariant probability measure and its ergodicity to bypass difficulties that are beyond the reach of rigorous methods.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.