  Results are presented from numerical experiments aiming at the computation of
stochastic phase-field models for phase transformations by coarse-graining
molecular dynamics. The studied phase transformations occur between a solid
crystal and a liquid. Nucleation and growth, sometimes dendritic, of crystal
grains in a sub-cooled liquid is determined by diffusion and convection of
heat, on the macroscopic level, and by interface effects, where the width of
the solid-liquid interface is on an atomic length-scale. Phase-field methods
are widely used in the study of the continuum level time evolution of the phase
transformations; they introduce an order parameter to distinguish between the
phases. The dynamics of the order parameter is modelled by an Allen--Cahn
equation and coupled to an energy equation, where the latent heat at the phase
transition enters as a source term. Stochastic fluctuations are sometimes added
in the coupled system of partial differential equations to introduce nucleation
and to get qualitatively correct behaviour of dendritic side-branching. In this
report the possibility of computing some of the Allen-Cahn model functions from
a microscale model is investigated. The microscopic model description of the
material by stochastic, Smoluchowski, dynamics is considered given. A local
average of contributions to the potential energy in the micro model is used to
determine the local phase, and a stochastic phase-field model is computed by
coarse-graining the molecular dynamics. Molecular dynamics simulations on a two
phase system at the melting point are used to compute a double-well reaction
term in the Allen--Cahn equation and a diffusion matrix describing the noise in
the coarse-grained phase-field.
