Many Particles' Systems: Theory and Applications

该提案从理论上和数值上研究了在各种应用中发现的广泛的粒子系统。这些系统的尺寸与所涉及的粒子数量成正比，在某些情况下可达10^{25}。这使得它们太复杂而无法分析，并且在数值上求解成本太高。然而，这些大型系统的一个关键特征是它们的多尺度性质：这使得单个粒子的微观水平上的动力学极其复杂。但它也可以导致在介观或宏观水平上的系统的复杂性的大幅降低与较低维的偏微分方程近似；在这种情况下，主要的困难是研究和控制粒子之间的相互关系。一个典型的例子是统计物理中动力学方程的平均场极限问题，研究者将开发新的技术来处理奇异势。新的扩展也将被调查，以推动超越经典框架，包括细菌悬浮液，平均场博弈模型和应用到社会科学（共识形成…）。所研究的细菌悬浮液具有相对较高的数量密度，因此颗粒或细菌之间的相关性不可忽略。研究者和合作者引入了一种新的数值方案来计算这些相关性。在平均场博弈的情况下，除了粒子或代理之间的“经典”相互作用外，对动力学的控制由中心代理进行优化。通常的技术，例如混沌在联合律上的传播，不再适用，我们开发了一种新的方法，基于N和1个代理能量的最小化。具有大量粒子的系统在科学和工程中无处不在。事实上，根据上下文和模型的不同，术语“粒子”可能代表非常不同的对象；在这个项目的范围内，一个粒子可以像电子或离子（等离子体中的粒子）一样“简单”，也可以像细菌、经济或社会代理人那样复杂，甚至可以像星系那样巨大的结构（天体物理学中的星系团动力学）。降低这些大型系统的复杂性，例如通过偏微分方程近似它们，是能够使用它们（数值模拟……）的关键一步。该项目将有助于理解这一现象的几个重要应用：统计物理的经典框架，临界密度的细菌悬浮液，经济中的多主体系统和共识形成。该项目对研究生的教育和未来的职业生涯也有直接影响，也将涉及本科生。

The proposal investigates, theoretically and numerically, a wide spectrum of particle's systems found in various applications. The dimension of those systems is proportional to the number of particles involved which can be quite large, up to 10^{25} in some settings. This makes them too complex to analyze and too costly to solve numerically. However a key feature of those large systems is their multiscale nature: It makes the dynamics extremely complex at the microscopic level of an individual particle. But it can also lead to a drastic reduction in the complexity of the system by approximating it at the mesoscopic or macroscopic level with PDE's in lower dimensions; the main difficulty in that case is the study and control of the correlations between particles. A classical example is the usual mean field limit problem for kinetic equations in statistical physics for which the investigator will develop novel techniques in order to handle singular potentials. New extensions will be investigated as well in order to push beyond the classical framework, including bacterial suspensions, mean field games models and applications to social sciences (consensus formation...). The bacterial suspensions under study have a relatively high number density with hence non negligible correlations between the particles or bacteria. The investigator and collaborators introduce a new numerical scheme to calculate those correlations. In the case of mean field games, in addition of "classical" interactions between the particles or agents, a control on the dynamics is optimized by a central agent. The usual techniques, propagation of chaos on the joint law for instance, are no more applicable and we develop a new approach, based on the minimization of N and 1 agent energies. Systems with a very large number of particles are ubiquitous in science and engineering. Indeed depending on the context and the model, the term "particle" may represent very different objects; in the scope of this project, a particle could for instance be as "simple" as an electron or ion (particles in a plasma), or more complex like a bacteria, an economical or social agent or even a very large structure like a galaxy (dynamics of clusters in astrophysics). Reducing the complexity of such large systems, for instance by approximating them by Partial Differential Equations, is a critical step to be able to use them (numerical simulations...). The project will contribute to the understanding of this phenomenon for a few important applications: the classical framework of statistical physics, suspensions of bacteria with critical density, multi-agent systems in economy and consensus formation. The project also has a direct impact on the education and future careers of graduate students and will also involve undergraduate students.