  This paper is concerned with a singular limit of the Kobayashi-Warren-Carter
system, a phase field system modelling the evolutions of structures of grains.
Under a suitable scaling, the limit system is formally derived when the
interface thickness parameter tends to zero. Different from many other
problems, it turns out that the limit system is a system involving fractional
time derivatives, although the original system is a simple gradient flow. A
rigorous derivation is given when the problem is reduced to a gradient flow of
a single-well Modica-Mortola functional in a one-dimensional setting.
