Charting a New Paradigm for Large Non-Exchangeable Multi-Agent and Many-Particle Systems

该项目将开发一个新的框架，用于研究相互作用的“代理”或“粒子”的大型系统。主要目标是通过用平均场的概念取代精确的交互来降低具有大量智能体的系统的复杂性。现有的理论，为了证明这种近似，通常假设代理人是相同的，或者至少是不可区分的。当考虑物理学中的经典应用时，例如静电势下电子的动力学，这个假设是有效的。研究者将引入新的方法来消除难以区分的粒子或媒介的假设，同时仍然涉及现实的奇异相互作用：这将允许考虑更复杂的应用，如物理学中的多物种模型和生物神经元网络动力学模型。本项目为研究生提供培训研究机会。从数学上讲，拥有不相同的粒子或媒介会改变系统的结构。首先，它使它们变得非对称和不可交换，这在试图定义所谓的可观察性时具有深刻的后果，例如子系统的联合定律。它还打破了一些抵消效应，例如相互作用的反对称性，许多当前的方法依赖于处理奇异相互作用。该项目引入了新的方法来绕过这些问题，如新的二元性公式和扩展的边际概念。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 will develop a new framework for the study of large systems of interacting "agents" or "particles". The main goal is to obtain a reduction in complexity for systems with a very large number of agents, by replacing exact interactions with a notion of mean field. Existing theories, to justify this approximation, typically assume that the agents are identical or at least indistinguishable. This assumption can be valid when considering classical applications in Physics, such as the dynamics of electrons under an electrostatic potential. The investigator will introduce new approaches to remove the assumption of indistinguishable particles or agents while still involving realistic singular interactions: This will allow to consider more complex applications such as multi-species models in Physics and models for the dynamics of networks of biological neurons. The project provides training research opportunities for graduate students. Mathematically speaking having non-identical particles or agents changes the structure of the systems. First, it makes them non-symmetric and non-exchangeable, which has deep consequences when trying to define so-called observables, such as the joint law of a sub-system. It also breaks some of the cancellation effects, for example the anti-symmetry of the interaction, that many current methods relied on to handle singular interactions. The project introduces novel methods to bypass those issues such as new duality formulations and an extended notion of marginal.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.