Rigorous coarse-graining of defects at positive temperature

用一个更简单、更低维的系统来近似一个大而复杂的系统的过程被称为粗粒化或模型简化，简化模型中的变量被称为粗粒化或集合变量。晶体材料中含有各种缺陷，如空位、位错和位错。材料的宏观性能在很大程度上取决于位错和其他缺陷之间的相互作用。由于缺陷尺度远小于宏观尺度，因此可靠的粗粒度模型对于材料性能的控制和设计是必要的。最近，GENERIC（非平衡可逆和不可逆耦合的一般方程）框架已被用来推导位错的集体动力学。然而，数学上严格验证的GENERIC框架仍然缺乏，这引起了问题的可靠性的方法，并限制了其应用。由于其非线性和奇异性，使这种形式化的推导精确是具有挑战性的，目前是数学家和物理学家的根本利益。本项目的目的是提供一个严格的推导正温度下的缺陷粗粒化。这将有助于显着理解的通用框架，从而打开大门，其他generalisations.The拟议的研究将结合联合收割机的方法和技术，从分析，统计力学和概率论的新方法。因此，预计该提案将加强和创建这些数学领域之间的新联系。此外，通过发展定量的，误差可控的方法，该建议的结果将提供一个严格的推导粗粒度的模型，并在正温度下的数值和多尺度计划，目前在很大程度上缺乏坚实的基础的分析基础。从长远来看，该项目的成果将有助于我们更好地了解材料的变形行为。

The procedure of approximating a large and complex system by a simpler and lower dimensional one is referred to as coarse-graining or model-reduction, and the variables in the reduced model are called coarse-grained or collective variables. Crystalline materials contain a variety of defects such as vacancies, interstitials and dislocations. Macroscopic properties of materials are strongly determined by interactions between dislocations and other defects. Because the defect scale is vastly smaller than the macroscopic scale, trustworthy coarse-grained models are necessary for control and design of material properties. Recently, the GENERIC (General Equation for Non-Equilibrium Reversible and Irreversible Coupling) framework has been employed to derive the collective dynamics of dislocations. However, a mathematically rigorous validation of the GENERIC framework is still lacking; this raises questions about the reliability of the method and limits its application. Making this formal derivation precise is challenging because of its nonlinear and singular nature and is currently of fundamental interest to both mathematicians and physicists.The aim of this project is to provide a rigorous derivation of coarse-graining of defects at positive temperature. This will contribute to significant understanding of the GENERIC framework, thus opening the doors to other generalisations.The proposed research will combine methodology and techniques from analysis, statistical mechanics and probability theory in novel ways. Therefore, it is expected that the proposal will strengthen and create new connections between these areas of mathematics. Furthermore, by developing quantitative, error controllable methods, the outcome of the proposal will provide the analytical foundations for a rigorous derivation of coarse-grained models, and of numerical and multi-scale schemes at positive temperature which at present largely lack the solid foundations. In the long-term, the outcome of this project will help us to understand better the deformation behaviour of materials.