Another normality is possible. Distributive transformations and emergent Gaussianity

Abstract

A distributional route to Gaussianity, associated with the concept of Conservative Mixing Transformations (CMT) in ensembles of random vector-valued variables, is proposed. This route is completely different from the additive mechanism characterizing the application of Central Limit Theorem, as it is based on the iteration of a random transformation preserving the ensemble variance. Gaussianity emerges as a “supergeneric” property of ensemble statistics, in the case the energy constraint is quadratic in the norm of the variables. This result puts in a different light the occurrence of equilibrium Gaussian distributions in kinetic variables (velocity, momentum), as it shows mathematically that, in the absence of any other dynamic mechanisms, almost Gaussian distributions stem from the low-velocity approximations of the physical conservation principles. Whenever, the energy constraint is not expressed in terms of quadratic functions non-Gaussian distributions arise. This is case of relativistic collisional interactions where the Jüttner distribution is recovered from CMT.