Quantifying Chaos, Correlations, and Oscillations in Multi-Agent Systems and Advection Equations

该项目研究在生物科学、统计物理和流体力学中应用的输运和平流方程。该研究的第一个指导例子包括物理学中长期使用的相互作用的“媒介”或“粒子”的大型系统：在天体物理学中预测星系或星系团的时间演化，试图了解它们是如何形成的，在等离子体中描述离子和电子的动力学，模拟水中气泡或流体中其他小物体的行为。此外，这种大型系统现在被广泛应用于生物学，以模拟微生物的运动（“粒子”可以是人体内的细胞），金融和经济以及其他社会科学。第二组指导示例由各种流体模型（液体或气体）组成，这些模型可以表现出在地球物理（海洋或大气）或生物环境中常见的复杂压缩效应。该项目旨在推导和验证有效模型：通过所谓的平均场极限来降低大型代理/粒子系统的复杂性，或者通过控制流体力学中先验不稳定的有效状态定律的行为。本研究中研究的模型均采用输运方程：可压缩流体的低维非线性对流系统，简单多智能体系统的大维线性Liouville方程，以及控制系统的更复杂的非线性方程（如Hamilton-Jacobi-Bellman）。该项目特别关注不稳定或单一的系统，并期望产生或放大相关性和振荡。研究中一个关键的统一问题是确定模型中的临界尺度，以量化这些相关性或振荡可能如何发展。对于大的粒子系统，该项目利用涉及重标相对熵的显式估计，重新规范化以包括吉布斯平衡之间的比较。对于对流模型，该项目引入了新的半规范，可以精确地跟踪密度在临界对数或对数尺度上的可能振荡。该奖项反映了美国国家科学基金会的法定使命，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 project studies transport and advection equations with applications in the Biosciences, Statistical Physics and Fluid Mechanics. A first guiding example for the research includes large systems of interacting "agents" or "particles" that have long been employed in Physics: in Astrophysics to predict the evolution in time of galaxies or galaxy clusters to try to understand how they are formed, in plasmas to describe the dynamics of ions and electrons in plasmas, to model the behavior of air bubbles in water or other small objects in a fluid. In addition, such large systems are now widely used in Biology to model the motion of micro-organisms ("particles" can then be cells in the human body), in Finance and Economics, and other Social Sciences. A second guiding set of examples is composed of various fluid models (liquid or gas) which can exhibit complex compressive effects commonly found in geophysical (oceans or atmosphere) or biological settings. The project aims at deriving and validating effective models: By reducing the complexity of large systems of agents/particles through the so-called mean-field limit, or by controlling the behavior of effective state laws in Fluid Mechanics that are a priori unstable.The models investigated in this research all employ transport equations: non-linear convection systems in low dimension for compressible fluids, the linear Liouville equation in very large dimension for simple multi-agent systems but more also more complex non-linear equations (such as Hamilton-Jacobi-Bellman) for systems with control. The project focuses in particular on systems that are unstable or singular and are expected to create or amplify correlations and oscillations. A key unifying question in the research is to identify the critical scale in the models to quantify how those correlations or oscillations may develop. For large systems of particles, the project makes use of explicit estimates involving the rescaled relative entropy, renormalized to include the comparisons between the Gibbs equilibria. For convective models, the project introduces new semi-norms which precisely track the possible oscillations in the density at the critical log or log log scales.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.