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

TL;DR

This paper introduces a method to derive well-conditioned continuum reaction-drift-diffusion equations from master equations, capturing non-equilibrium drift effects in defect kinetics, applicable to atomic-scale Markov models.

Contribution

It presents a novel approach to derive continuum equations directly from master equations, including a previously overlooked non-equilibrium drift term, and employs spectral analysis for model reduction.

Findings

Identifies a non-equilibrium drift term in defect kinetics.

Develops a spectral method for efficient eigenvalue computation.

Produces well-conditioned continuum equations from discrete Markov models.

Abstract

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 (e.g., PCCA+) to define a proper reduced state space.