Nonlocal Partial Differential Equations: entropies, gradient flows, phase transitions and applications

了解相互作用的大粒子系统的定性性质在物理和生物学中的许多应用中都是至关重要的，例如分子动力学模拟、浸泡在流体中的粒子、器官发生建模、社会科学中用于优化或聚集的群体方法以及意见形成的模型等等。这个项目将致力于通过热力学极限的途径，进一步促进我们对粒子描述和连续介质模型之间的联系的理解。我们的主要目标之一将是研究崎岖、多尺度能量场景中相互作用粒子系统的热力学(平均场)极限，这是人们在生物物理和化学物理等应用中经常遇到的类型。这种势能中的动力学在所有尺度上都表现出亚稳态行为。特别是，我们想要了解多尺度结构对平均场动力学定态的存在性和稳定性的影响。对于崎岖的能源环境，将分析由噪声驱动的相变，即行为的突然变化。然后，我们将使用控制理论中的工具来开发稳定不稳定稳定状态的算法。此外，我们将开发有效的数值技术来求解非局部、非线性平均场方程，并将它们应用于各种问题，如动力密度泛函理论和社会科学的数学模型，包括意见形成模型。此外，我们将使用适当的相互作用粒子系统及其相应的平均场限制来开发基于共识的全局优化算法，该算法可以应用于具有(无限)许多局部极小特征的多尺度结构的势。

Understanding the qualitative properties of large systems of interacting particles is of crucial importance in many applications in physics and biology such as molecular dynamic simulations, particles immersed in a fluid, organogenesis modelling, swarming methods for optimization or herding in the social sciences and models for opinion formation, to name a few. This project will be devoted to the further advancement in our understanding of the connection between particle descriptions and continuum models via the passage to the thermodynamic limit. One of our main goals will be to study the thermodynamic (mean field) limit for systems of interacting particles in rugged, multiscale energy landscapes, of the type that one frequently encounters in applications such as biophysics and chemical physics. The dynamics in such a potential exhibit metastable behavior at all scales. In particular, we want to understand the effect of the multiscale structure on the existence and stability of stationary states of the mean field dynamics. Phase transitions, i.e., abrupt changes of behavior, driven by noise will be analyzed for rugged energy landscapes. We will then employ tools from control theory in order to develop algorithms for stabilizing unstable steady states. In addition, we will develop efficient numerical techniques for solving nonlocal, nonlinear mean field equations and we will apply them to diverse problems such as dynamical density functional theory and mathematical models from the social sciences, including models for opinion formation. Furthermore, we will use appropriate systems of interacting particles and their corresponding mean field limit in order to develop consensus-based global optimization algorithms that can be applied to potentials with a multiscale structure characterized by (infinitely) many local minima.