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MUSYX Report Summary

Project ID: 335120
Funded under: FP7-IDEAS-ERC
Country: United Kingdom

Mid-Term Report Summary - MUSYX (Multiscale Simulation of Crystal Defects)

The overarching purpose of this project is to develop a comprehensive numerical analysis theory for multi-scale coupling methods at the atomistic scale. More specifically the project focuses on defects in crystalline solids as an application area, and develops numerical algorithms, rigorous numerical analysis, and prototype software, for their efficient numerical simulation.
The project is split into four related themes, with the following main results achieved to date:

Theme A: Stability and bifurcation. The problem to be addressed in this theme are sharp stability estimates for atomistic/continuum (A/C) coupling methods and the consequence of such results for the a priori error analysis of A/C methods.
The biggest technical hurdle was to characterise for which A/C coupling methods stability of the underlying atomistic model implies stability of the A/C scheme; this has been achieved and work is now proceeding to develop and analyze models for which bifurcations can be studied.

Theme B: Transition rates. Theme B concerns the approximation error analysis of saddle points (as opposed to minima in the bulk of the A/C coupling literature).
Initial results establish (1) a thermodynamic limit convergence result for saddle points which forms the theoretical model of a potential energy saddle in an infinite crystalline medium and (2) a proof that the unstable mode (transition direction) is always exponentially localised. The latter in particularly ensures that approximation results for minimisers can be extended to approximation results for saddles.

Theme C: Temperature. The goal of this theme is to develop models and approximation results for defect formation *free* energy. In a a first paper a one-dimensional toy model and an extensive analysis have been developed that establishes the first steps towards an approximation error analysis for free energy with explicit convergence rates.

In addition, the project has established initial results on a decomposition of entropy into spatially localised contributions (in the low temperature regime) which will provide a new approach to finite-temperature coarse-graining and in particular also a mechanism to analyze the entropy contribution to the HTST rate in Theme B.

Theme D: MM/MM and QM/MM Coupling. The aim of Theme D is to explore QM/MM (quantum to classical) coupling methods. In initial work on this theme we have discovered a new decomposition of total potential energy for electronic structure models into spatially localised contributions. We then used this result to develop a new class of QM/MM coupling schemes with guaranteed and controllable approximation errors. Together these results kick off an extremely promising new direction for the investigation of interatomic potentials and QM/MM multi-scale models and related problems.

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United Kingdom
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