Distribution in TLM models for diffusion (Part II: multi-dimensional treatment)

The signals obtained at each time step of a transmission line matrix (TLM) simulation of Gaussian diffusion are analysed for two- and three-dimensional cases. A combinatorial formula is derived to provide the signal magnitude at any spatial position and any time step after a single-shot excitation in the two-dimensional link-line model. Formulae for the expectation and variance of the axial positions of a particle are determined for two- and three-dimensional link-line and link-resistor models. A generalization of these formulae is proposed for higher dimensions, and an entirely numerical proving scheme is devised. Finally, we briefly compare the resulting variances and that of the underlying diffusion process