Order parameters in the diffusion of rods through two- and three-dimensional fixed scatterers

Characteristic diffusion in a Lorentz gas system and its extensions are typically described through a scale-free order parameter. The relative validity of possible order parameters for the diffusion of a thick tracer rod through spherical scatterers in two and three dimensions as a function of the rod length L and scatterer density rho has been examined by simulation. We find that the often-used two-dimensional order parameter, rho L-2, is accurate. However, its naive generalization to three dimensions, changing the exponent according to dimensionality, does not fit the observed results. A revised order parameter can be generated through a theory based on the behavior of the mean-free path of the tracer. We find that this generalized order parameter captures the multidimensional diffusive behavior.