  We solve numerically the Boltzmann equation describing the evolution of a
cosmic string network which contains only loops. In Minkowski space time the
equilibrium solution predicted by statistical mechanics is recovered, and we
prove that this solution is stable to non-linear perturbations provided that
their energy does not exceed the critical energy for the Hagedorn transition.
In expanding Einstein - de Sitter Universes we probe the distribution of loops
with length much smaller than the horizon. For these loops we discover stable
scaling solutions both in the radiation and matter dominated epochs. The shape
of these solutions is very different in the two eras, with much higher energy
density in the radiation epoch, and a larger average loop length in the matter
epoch. These results suggest that if the conditions for formation of loop
networks are indeed satisfied, these could in principle be good candidates for
structure formation.
