Mathematical and Computational Methods for Non-Equilbrium Systems

本课题研究的是受随机影响的复杂系统在时间上的行为。具有大量参数和主体的复杂系统无处不在——例如，高速公路网络上的交通或生物化学系统，它们涉及许多类型的分子和许多化学反应。对于这样的系统，敏感性分析的问题提出了特殊的挑战：敏感性分析包括理解参数对系统未来行为的影响——例如，在繁忙的高速公路上关闭一条车道如何影响交通，可能是几英里外，几小时后。理解这些影响是预测建模领域的核心，并提出了艰巨的挑战，这往往超出了现代计算机的能力。首席研究员将开发概念和计算工具，以互补的方式利用来自不同科学领域的技术来解决这些问题。这包括信息论（量化信息在物理过程中是如何丢失的）、统计力学（允许人们理解非常大的系统的整体行为）和数值分析（允许人们在计算机上模拟和再现复杂系统行为的科学）的方法。理解由外力、边界效应和多物理场驱动的非平衡系统带来了许多数学和计算挑战，特别是当系统很大且涉及许多参数时。一个典型的例子是生化反应网络，它通常涉及多个时间尺度、反馈回路、数百种物种和数百个参数。该项目的核心是开发用于分析和模拟这种非平衡高维随机系统的数学和数值工具。该研究融合了信息论、统计力学、数值分析和概率论的概念，以开发新的算法，以及新的工具来评估复杂系统的现有算法。敏感性分析问题在工作中起着核心作用。这里的一个新颖之处是系统地使用和发展了用于随机系统的分析和不确定性量化的信息论工具，特别强调了系统的长期行为。像相对熵和费雪信息矩阵这样的概念，特别适用于系统的时间历史，是特别合适的。该项目的另一个重要特征涉及非平衡系统在时间反转下不尊重不变性的事实。定量测量这种对称性破缺是现代非平衡统计力学的主要主题之一，主要研究人员将使用该项目中开发的概念来创建有效的算法，以实现复杂系统的对称性破缺。最后，研究小组将探讨记忆或延迟效应的作用，这是在多空间或时间尺度的复杂系统中不可避免的。

This research project is concerned with the behavior in time of complex systems subject to random influences. Complex systems with a large number of parameters and agents are ubiquitous -- think, for example, of the traffic on a network of highways or of bio-chemical systems, which involve many types of molecules and many chemical reactions. For such systems, the issue of sensitivity analysis presents special challenges: sensitivity analysis consists in understanding the effects of parameters on the future behavior of the system -- for example, how the closure of one lane on a busy highway influences the traffic, possibly many miles away and several hours later. Understanding these effects is at the heart of the field of predictive modeling and poses formidable challenges, which often exceed the capabilities of modern computers. The Principal Investigators will develop tools, both conceptual and computational, to tackle such issues by using, in a complementary fashion, techniques from different fields of science. This includes methods from information theory (which quantifies how information is lost in physical processes), from statistical mechanics (which allows one to understand the global behavior of very large systems), and from numerical analysis (which is the science that allows one to simulate and reproduce the behavior of complex systems on a computer).Understanding non-equilibrium systems driven by external forces, boundary effects, and multi-physics presents numerous mathematical and computational challenges, especially if the system is large and involves many parameters. A paradigmatic example is a bio-chemical reaction network, which often involves multiple time scales, feedback loops, hundreds of species, and hundreds of parameters. At the center of this project is the development of mathematical and numerical tools for the analysis and simulation of such non-equilibrium high-dimensional stochastic systems. The research blends together concepts from information theory, statistical mechanics, numerical analysis, and probability theory in order to develop novel algorithms, as well as novel tools to assess existing algorithms for complex systems. The issue of sensitivity analysis plays a central role in the work. One of the novelties here is the systematic use and development of information-theoretic tools for the analysis and uncertainty quantification of stochastic systems, with a strong emphasis on the long-time behavior of the systems. Concepts such as the relative entropy and the Fisher information matrix, especially applied to the time histories of the system, are particularly well suited. Another important feature of the project concerns the fact that non-equilibrium systems do not respect invariance under time reversal. Measuring quantitatively this symmetry breaking is one the main themes in modern non-equilibrium statistical mechanics, and the Principal Investigators will use concepts developed in this project to create efficient algorithms to do this for complex systems. Finally, the research team will investigate the role of memory or delay effects, which are unavoidable in complex systems with multiple spatial or temporal scales.