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Rigorous coarse-graining of defects at positive temperature

正温度下缺陷的严格粗晶化

基本信息

批准号：
EP/W008041/1
负责人：
Hong Duong
金额：
$ 4.93万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FW008041%2F1
关键词：
Rigorous coarse graining defects positive

项目摘要

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.

用一个更简单的低维系统来逼近一个大而复杂的系统的过程称为粗粒度化或模型约简，而约简模型中的变量称为粗粒度化或集合变量。晶体材料含有各种缺陷，如空位、间隙和位错。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，因此需要可靠的粗粒度模型来控制和设计材料性能。近年来，采用非平衡可逆与不可逆耦合的一般方程框架来推导位错的集体动力学。然而，仍然缺乏对GENERIC框架的数学上严格的验证；这引起了对该方法可靠性的质疑，并限制了其应用。由于其非线性和奇异性，使这种形式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一个严格的推导粗粒化缺陷在正温度。这将有助于对GENERIC框架的重要理解，从而为其他一般化打开大门。建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，预计该提案将加强并在这些数学领域之间建立新的联系。此外，通过发展定量的、误差可控的方法，该建议的结果将为粗粒度模型的严格推导以及目前在很大程度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，这个项目的成果将帮助我们更好地理解材料的变形行为。