  An open question in studying normal grain growth concerns the asymptotic
state to which microstructures converge. In particular, the distribution of
grain topologies is unknown. We introduce a thermodynamic-like theory to
explain these distributions in two- and three-dimensional systems. In
particular, a bending-like energy $E_i$ is associated to each grain topology
$t_i$, and the probability of observing that particular topology is
proportional to $\frac{1}{s(t_i)}e^{-\beta E_i}$, where $s(t_i)$ is the order
of an associated symmetry group and $\beta$ is a thermodynamic-like constant.
We explain the physical origins of this approach, and provide numerical
evidence in support.
