Many Particles' Systems: Theory and Applications

多粒子系统：理论与应用

• 批准号: 1312142
• 负责人: Pierre-Emmanuel Jabin
• 金额: $ 34.13万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312142&HistoricalAwards=false
• 关键词: Many; Particles; Systems; Theory; Applications

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.

该提案从理论和数字上研究了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，这些粒子可能很大，在某些设置中最多可达10^{25}。这使它们太复杂了，无法分析，并且太昂贵了，无法在数字上解决。然而，这些大型系统的关键特征是它们的多尺寸性质​​：它使动态在单个粒子的微观层面上变得极为复杂。但是，它也可以通过在较低维度的介质或宏观水平上近似介质或宏观水平来大大降低系统的复杂性。在这种情况下，主要难度是研究和控制颗粒之间的相关性。一个经典的例子是统计物理学中动力学方程的通常平均场极限问题，研究者将开发新技术以处理奇异电位。还将研究新的扩展，以超越经典框架，包括细菌悬架，平均现场游戏模型和对社会科学的应用（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视。研究者和合作者介绍了一种新的数值方案来计算这些相关性。在平均场游戏的情况下，除了粒子或代理之间的“经典”相互作用之外，中央代理对动力学的控制进行了优化。例如，通常的技术是在联合法律上的混乱传播，不再适用，我们基于N和1代理能量的最小化开发了一种新的方法。大量颗粒的系统在科学和工程中无处不在。实际上，根据上下文和模型，术语“粒子”可能代表非常不同的对象。在该项目的范围内，例如，粒子可以像电子或离子（等离子体中的颗粒）一样“简单”，或者像细菌，经济或社会药物一样复杂，甚至像星系（Astrophrophysics in Astrophysics中的簇的动态）一样复杂。例如，通过部分微分方程近似它们是能够使用它们的关键步骤（数值模拟...）是一个关键的步骤。 该项目将有助于理解这种现象的一些重要应用：统计物理学的经典框架，具有关键密度的细菌的悬浮液，经济中的多机构系统和共识形成。该项目还直接影响研究生的教育和未来职业，还将涉及本科生。