  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.
