Reaction-drift-diffusion models from master equations: application to material defects

We present a general method to produce well-conditioned continuum reaction-drift-diffusion equations directly from master equations on a discrete, periodic state space. We assume the underlying data to be kinetic Monte Carlo models (i.e. continuous-time Markov chains) produced from atomic sampling of point defects in locally periodic environments, such as perfect lattices, ordered surface structures or dislocation cores, possibly under the influence of a slowly varying external field. Our approach also applies to any discrete, periodic Markov chain. The analysis identifies a previously omitted non-equilibrium drift term, present even in the absence of external forces, which can compete in magnitude with the reaction rates, thus being essential to correctly capture the kinetics. To remove fast modes which hinder time integration, we use a generalized Bloch relation to efficiently calculate the eigenspectrum of the master equation. A well conditioned continuum equation then emerges by searching for spectral gaps in the long wavelength limit, using an established kinetic clustering algorithm to define a proper reduced, Markovian state space.

Automated summary

The study presents a method to derive well-conditioned continuum reaction-drift-diffusion equations from master equations on a discrete, periodic state space. This method is applicable to kinetic Monte Carlo models and discrete, periodic Markov chains, and involves modifying the drift term to accurately capture the kinetics. A generalized Bloch relation is employed to calculate the eigenspectrum of the master equation and a kinetic clustering algorithm is used to define a reduced, Markovian state space, eliminating fast modes that hinder time integration.