Stochastic invertible mappings between power law and Gaussian probability distributions

TL;DR

This paper introduces stochastic, invertible mappings between power law and Gaussian distributions, enabling transformations useful in nonextensive thermodynamics and statistical physics.

Contribution

It constructs a novel class of invertible stochastic mappings between power law and Gaussian distributions, expanding tools for statistical physics applications.

Findings

Mappings are invertible via random scalar variables

Application to zero-th law in nonextensive thermodynamics

Provides a new perspective on distribution transformations

Abstract

We construct "stochastic mappings" between power law probability distributions (PD's) and Gaussian ones. To a given vector $N$, Gaussian distributed (respectively $Z$, exponentially distributed), one can associate a vector $X$, "power law distributed", by multiplying $X$ by a random scalar variable $a$, $N= a X$. This mapping is "invertible": one can go via multiplication by another random variable $b$ from $X$ to $N$ (resp. from $X$ to $Z$), i.e., $X=b N$ (resp. $X=b Z$). Note that all the above equalities mean "is distributed as". As an application of this stochastic mapping we revisit the so-called "zero-th law of thermodynamics problem" that bedevils the practitioners of nonextensive thermostatistics.