Mesoscale averaging of nucleation and growth models

The aim of this paper is to derive a general theory for the averaging of heterogeneous processes with stochastic nucleation and deterministic growth. We start by generalizing the classical Johnson--Mehl--Avrami--Kolmogorov theory based on the causal cone to heterogeneous growth situations. Moreover, we relate the computation of the causal cone to a Hopf--Lax formula for Hamilton--Jacobi equations describing the growth of grains. As an outcome of the approach we obtain formulae for the expected values of geometric densities describing the growth processes; in particular we generalize the standard Avrami--Kolmogorov relations for the degree of crystallinity. By relating the computation of expected values to mesoscale averaging, we obtain a suitable description of the process at the mesoscale. We show how the variance of these mesoscale averages can be estimated in terms of quotients of the typical length on the microscale and on the mesoscale. Moreover, we discuss the efficient computation of the mesoscale averages in the typical case when the nucleation and growth rates are obtained from mesoscopic fields (such as, e.g., temperature). Finally, we give a brief outlook to possible extensions such as polycrystalline growth, which turns out to be rather straightforward when starting from our general framework.