Another normality is possible. Distributive transformations and emergent Gaussianity

TL;DR

This paper introduces a novel distributional pathway to Gaussianity through Conservative Mixing Transformations, highlighting how ensemble statistics naturally tend toward Gaussian distributions under quadratic energy constraints, independent of additive mechanisms.

Contribution

It proposes a new route to Gaussianity based on ensemble transformations that preserve variance, differing from the classical Central Limit Theorem mechanism.

Findings

Gaussianity emerges as a supergeneric property under quadratic energy constraints.

The approach explains the occurrence of Gaussian distributions in kinetic variables.

The Juttner distribution is derived for non-quadratic energy cases.

Abstract

A distributional route to Gaussianity, associated with the concept of Conservative Mixing Transformations 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 stems from the low-velocity approximations of the physical conservation principles. Whenever, the energy constraint is not expressed in terms of quadratic functions (as in the relativistic case), the Juttner distribution is recovered from CMT.