  We study the evolution, the transverse spreading and the subsequent
thermalization of string states in the Weyl static axisymmetric spacetime. This
possesses a singular event horizon on the symmetry axis and a naked singularity
along the other directions. The branching diffusion process of string bits
approaching the singular event horizon provides the notion of temperature that
is calculated for this process. We find that the solution of the Fokker-Planck
equation in the phase space of the transverse variables of the string, can be
factored as a product of two thermal distributions, provided that the classical
conjugate variables satisfy the uncertainty principle. We comment on the
possible physical significance of this result.
