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

了解大型相互作用粒子系统的定性性质在物理学和生物学的许多应用中都至关重要，例如分子动力学模拟、沉浸在流体中的粒子、器官发生模型、社会科学中用于优化或羊群的群体方法和意见形成模型，等等。这个项目将致力于进一步推进我们对粒子描述和连续体模型之间的联系的理解，通过通往热力学极限。我们的主要目标之一将是研究在崎岖不平的多尺度能量景观中相互作用粒子系统的热力学（平均场）极限，这种类型经常在生物物理学和化学物理学等应用中遇到。这种势能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。相变，即突然的变化的行为，由噪声驱动，将分析崎岖的能源景观。然后，我们将使用控制理论中的工具来开发稳定不稳定稳态的算法。此外，我们将开发求解非局部非线性平均场方程的有效数值技术，并将其应用于各种问题，如动态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极限，以开发基于共识的全局优化算法，该算法可应用于具有（无限）多个局部极小值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.