  The transport properties of disordered systems are known to depend critically
on dimensionality. We study the diffusion coefficient of a quantum particle
confined to a lattice on the surface of a tube, where it scales between the 1D
and 2D limits. It is found that the scaling relation is universal and
independent of the disorder and noise parameters, and the essential order
parameter is the ratio between the localization length in 2D and the
circumference of the tube. Phenomenological and quantitative expressions for
transport properties as functions of disorder and noise are obtained and
applied to real systems: In the natural chlorosomes found in light-harvesting
bacteria the exciton transfer dynamics is predicted to be in the 2D limit,
whereas a family of synthetic molecular aggregates is found to be in the
homogeneous limit and is independent of dimensionality.
