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

非局部偏微分方程：熵、梯度流、相变和应用

基本信息

批准号：
EP/P031587/1
负责人：
Grigorios Pavliotis
金额：
$ 57.24万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2017
资助国家：
英国
起止时间：
2017 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP031587%2F1
关键词：
Nonlocal Partial Differential Equations entropies

项目摘要

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.

了解相互作用粒子的大系统的定性性质在物理学和生物学的许多应用中至关重要，例如分子动力学模拟，浸入流体中的粒子，器官形成建模，社会科学中的优化或羊群的群集方法和意见形成模型，仅举几例。这个项目将致力于进一步推进我们对粒子描述和连续模型之间通过热力学极限的联系的理解。我们的主要目标之一将是研究热力学（平均场）的限制，在崎岖的，多尺度的能量景观，人们经常遇到的应用，如生物物理学和化学物理学的类型的相互作用的粒子系统。在这样一个潜在的动力学表现出亚稳态行为在所有尺度。特别是，我们想了解的多尺度结构的存在性和稳定性的平均场动力学的定态的效果。相变，即，将针对崎岖的能源景观分析由噪声驱动的行为突变。然后，我们将采用控制理论的工具，以开发稳定不稳定稳态的算法。此外，我们将开发有效的数值技术来解决非局部，非线性平均场方程，我们将把它们应用到不同的问题，如动态密度泛函理论和数学模型，从社会科学，包括意见形成模型。此外，我们将使用适当的系统的相互作用的粒子和它们相应的平均场的限制，以开发基于共识的全局优化算法，可以应用于潜在的多尺度结构的特点是（无限）许多局部极小值。