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
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