A Generalized Coarse-Graining Procedure from Diffusive Master Equations to PDEs*

A key challenge in developing continuum models is to relate parameters to the underlying microscopic processes. Previous work [1] has shown that it is possible to relate phase field model parameters to the underlying hopping mechanisms of a microscopic master equation. However, one of the limitations is that established procedures for accomplishing this place restrictions on the admissible types of physics and/or dynamical processes. In this talk we will discuss how to use a symmetric and antisymmetric decomposition to extend this approach to a completely general diffusive microscopic process.

We show how the Langevin equation approximation of the diffusion master equation requires the introduction of a separability approximation for the antisymmetric component, which gives rise to the corresponding chemical potential. In addition, we demonstrate how the symmetric and antisymmetric decomposition implies that there is an intimate relationship between the coarse-grained kinetics and thermodynamics of stochastic processes. By studying this relationship, we discuss how certain popular choices of microscopic dynamics (e.g., Metropolis or Arrhenius dynamics) only correspond to a restricted class of admissible PDE models, and how such considerations can be used to improve model fitting/parameter estimation.

 

[1]    Q. Bronchart, Y. Le Bouar, and A. Finel, New Coarse-Grained Derivation of a Phase Field Model for Precipitation, Phys. Rev. Lett. 100, 015702 (2008).