  Metastable condensed matter typically fluctuates about local energy minima at
the femtosecond time scale before transitioning between local minima after
nanoseconds or microseconds. This vast scale separation limits the
applicability of classical molecular dynamics methods and has spurned the
development of a host of approximate algorithms. One recently proposed method
is diffusive molecular dynamics which aims to integrate a system of ordinary
differential equations describing the likelihood of occupancy by one of two
species, in the case of a binary alloy, while quasistatically evolving the
locations of the atoms. While diffusive molecular dynamics has shown to be
efficient and provide agreement with observations, it is fundamentally a model,
with unclear connections to classical molecular dynamics.
  In this work, we formulate a spin-diffusion stochastic process and show how
it can be connected to diffusive molecular dynamics. The spin-diffusion model
couples a classical overdamped Langevin equation to a kinetic Monte Carlo model
for exchange amongst the species of a binary alloy. Under suitable assumptions
and approximations, spin-diffusion can be shown to lead to diffusive molecular
molecular dynamics type models. The key assumptions and approximations include
a well defined time scale separation, a choice of spin exchange rates, a low
temperature approximation, and a mean field type approximation. We derive
several models from different assumptions and show their relationship to
diffusive molecular dynamics. Differences and similarities amongst the models
are explored in a simple test problem.
