  Many technologically useful materials are polycrystals composed of small
monocrystalline grains that are separated by grain boundaries of crystallites
with different lattice orientations. The energetics and connectivities of the
grain boundaries play an essential role in defining the effective properties of
materials across multiple scales. In this paper we derive a Fokker-Planck model
for the evolution of the planar grain boundary network. The proposed model
considers anisotropic grain boundary energy which depends on lattice
misorientation and takes into account mobility of the triple junctions, as well
as independent dynamics of the misorientations. We establish long time
asymptotics of the Fokker-Planck solution, namely the joint probability density
function of misorientations and triple junctions, and closely related the
marginal probability density of misorientations. Moreover, for an equilibrium
configuration of a boundary network, we derive explicit local algebraic
relations, a generalized Herring Condition formula, as well as formula that
connects grain boundary energy density with the geometry of the grain
boundaries that share a triple junction. Although the stochastic model neglects
the explicit interactions and correlations among triple junctions, the
considered specific form of the noise, under the fluctuation-dissipation
assumption, provides partial information about evolution of a grain boundary
network, and is consistent with presented results of extensive grain growth
simulations.
