Rigorous coarse-graining of defects at positive temperature

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.