Random walk in a quasicontinuum

A continuous-time random walk on a spatial lattice described by the Kramers-Moyal expansion has a continuum limit described by a Fokker-Planck equation. It is often desirable to know corrections to quantities computed in the continuum limit, but truncation of the Kramers-Moyal expansion at any level other than the Fokker-Planck either breaks down or yields unphysical results. Here we introduce an alternative approximation to the Kramers-Moyal expansion which circumvents the problems of a naive truncation and correctly incorporates the first-order corrections due to the discrete lattice.