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）的大集合，并弥合微观实体与宏观性质（如温度分布和热导率）之间的差距。非平衡统计力学中的许多问题的数学证明，例如傅立叶定律中热通量与温度梯度成比例的说法，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，以研究一系列由统计物理和波动湍流引起的非平衡热力学模型。这些非平衡态模型不仅与统计物理学有关，而且与无数本质上不可逆和多尺度的应用科学问题有关，例如化学反应和神经动力学。本项目的主要研究者（PI）将结合分析和计算方法，研究如何从一类微观能量传递模型中推导出热力学性质，这些模型包括经典的类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。一般的方法是使用最小的计算工作，以绕过一些困难，并推导出数学上易于处理的随机模型。从这些随机模型中发展热力学定律通常要容易得多。数据驱动的计算方法的开发和应用是拟议研究的组成部分。PI开发了一系列新颖的计算方法，将传统的Monte Carlo模拟与数值偏微分方程求解器，耦合方法和人工神经网络等工具相结合。他们克服了传统的，基于离散化的算法，特别是高维问题的几个缺点。在研究该项目中的许多高维问题时，我们需要高维不变概率测度及其遍历性的计算结果，以绕过严格方法无法达到的困难。该奖项反映了NSF的法定使命，通过使用基金会的智力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 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.