Rigorous coarse-graining of defects at positive temperature

将一个大而复杂的系统近似为一个更简单、更低维的系统的过程称为粗粒化或模型简化，简化后的模型中的变量称为粗粒度或集合变量。晶体材料中含有各种缺陷，如空位、间隙、位错等。位错与其他缺陷之间的相互作用很大程度上决定了材料的宏观性质。由于缺陷尺度远小于宏观尺度，可靠的粗晶模型对于材料性能的控制和设计是必要的。最近，一般的(非平衡可逆和不可逆耦合的通用方程)框架被用来推导位错的集体动力学。然而，仍然缺乏对通用框架的数学严格验证；这引发了对该方法的可靠性的质疑，并限制了其应用。由于其非线性和奇异的性质，使这种形式的推导精确是具有挑战性的，并且目前是数学家和物理学家的基本兴趣。这个项目的目的是提供在正温度下缺陷粗晶化的严格推导。这将有助于对通用框架的重大理解，从而为其他概括打开大门。拟议的研究将以新颖的方式结合分析、统计力学和概率论的方法和技术。因此，预计该提案将加强和建立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，建议的结果将为严格推导粗粒度模式以及目前基本上缺乏坚实基础的正温度下的数值和多尺度方案提供分析基础。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.