Mathematical Underpinnings of Diffusive Molecular Dynamics

Gideon Simpson
Drexel University

Metastable condensed matter typically fluctuates about local energy minima at the femtosecond time scale before transitioning between local minima after nanoseconds or microseconds. This vast scale separation limits the applicability of classical molecular dynamics methods and has spurred the development of a host of approximate algorithms. One recently proposed method is diffusive molecular dynamics which aims to integrate a system of ordinary differential equations describing the likelihood of occupancy by one of two species, in the case of a binary alloy, while quasistatically evolving the locations of the atoms. While diffusive molecular dynamics has shown to be efficient and provide agreement with observations, its relationship to classical molecular dynamics and other methods remains unclear.

In this work, we formulate a spin-diffusion stochastic process and show how it can be connected to diffusive molecular dynamics. The spin-diffusion model couples a classical overdamped Langevin equation to a kinetic Monte Carlo model for exchange between the species of a binary alloy. Under suitable assumptions and approximations, spin-diffusion can be shown to lead to diffusive molecular molecular dynamics type models. The key assumptions and approximations include a well defined time scale separation, a choice of spin exchange rates, a low temperature approximation, and a mean field type approximation. We derive several models from different assumptions and show their relationship to diffusive molecular dynamics. Differences and similarities among the models are explored in a simple test problem.

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