Functional-differential equations for $F_q$%-transforms of $q$-Gaussians

TL;DR

This paper investigates whether the q-Fourier transform of a q-Gaussian results in another q'-Gaussian, revealing that this holds for q > 1 except for two specific cases, with implications for statistical mechanics and PDEs.

Contribution

It establishes the conditions under which the q-Fourier transform of a q-Gaussian is another q'-Gaussian across the entire q range, clarifying its applicability.

Findings

The q-Fourier transform of a q-Gaussian is a q'-Gaussian if and only if q > 1.

Excluded cases are q=1/2 and q=2/3, which do not fit the general theory.

Applications to nonlinear PDEs like the porous medium equation are discussed.

Abstract

In the paper the question - Is the q-Fourier transform of a q-Gaussian a q'-Gaussian (with some q') up to a constant factor? - is studied for the whole range of $q\in (-\infty, 3)$. This question is connected with applicability of the q-Fourier transform in the study of limit processes in nonextensive statistical mechanics. We prove that the answer is affirmative if and only if q > 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, which are also out of the theory valid for q \ge 1. We also discuss some applications of the q-Fourier transform to nonlinear partial differential equations such as the porous medium equation.