The fourfold way to Gaussianity. Physical Interactions, distributional models and monadic transformations

The Central Limit Theorem stands as a milestone in probability theory and statistical physics, as the privileged, if not the unique, universal route to normal distributions. This article addresses and describes several other alternative routes to Gaussianity, stemming from physical interactions, related to particle-particle and radiative particle–photon elementary processes. The concept of conservative mixing transformations of random ensembles is addressed, as it represents the other main universal distributional route to Gaussianity in classical low-energy physics. Monadic ensemble transformations are introduced, accounting for radiative particle–photon interactions, and are intimately connected with the theory of random Iterated Function Systems. For Monadic transformations, possessing a thermodynamic constraint, Gaussianity represents the equilibrium condition in two limiting cases: in the low radiative-friction limit in any space dimension, and in the high radiative-friction limit, when the dimension of the physical space tends to infinity.