Buckingham’s approximants to physical laws

The typical problem is addressed of system and process modelling depending on a large number of variables. Demonstration is given that the number of independent variables can be dramatically reduced, by a modified application of Buckingham's theorem of dimensional analysis, resulting in a simplified formulation of the problem in terms of a limited number of dimensionless arguments. This simplified formulation eventually leads, by interpolation of numerical data, to the derivation of practical approximants to the physical laws governing the system or process. This approach is demonstrated, in case of electron scattering, through a general layer in the elastic regime, as modelled by Monte Carlo methods. In particular, a single dimensionless quality factor is introduced, allowing remarkable simplification both in forward and backward scattering analysis. Hence, Buckingam approximants are derived, effectively describing the scattering by universal laws, applicable to all elements of the periodic table and all variations of electron energy and layer thickness, in the ranges 5 to 150 keV and 1 to 3000 nm. The proposed method may be considered as a physical approximation, compared to purely mathematical methods, such as for example Padé approximants. As a consequence, the method is understandably highly effective, and may be extensively applied to all systems and processes susceptible to being described by mathematical modelling.