Dual Relativity and Lorentz Transformations for Liquid-Like Media and Statistics from Scaling Concepts for Geometry.

A dual class of Lorentz transformations (dual LT) for linear and Brownian motions in liquid-like media, is presented and discussed. It descends from two LT groups, for self-diffusion in simple liquids (BLT) and its geometrical analogy (GLT), which turned out to represent promising basic tools to deal with statistics at different length scales. Time dilation and length contraction of BLT occur upon ordering the originally indistinguishable molecular disorder in the liquid medium (i.e., a diffusivity lowering), giving rise to universality and scaling laws in polymer solutions. Dual LT exhibit a rich phenomenology, leading to a scale-dependent motion concept, where Brownian and quantum movements somewhat correspond. Density correlations in simple liquids, for instance, are suggested on this basis to behave like radial wave functions at the atomic scale. We also report some remarks on statistics in general, and its connection towards geometry. [ABSTRACT FROM AUTHOR]