Many Particles' Systems: Theory and Applications

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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.