On the robustness of the q-Gaussian family

Abstract

We introduce three deformations, called α-, β-and γ-deformation respectively, of a N -body probabilistic model, first proposed by Rodríguez et al. (2008), having q-Gaussians as N → ∞ limiting probability distributions. The proposed α-and β-deformations are asymptotically scaleinvariant, whereas the γ-deformation is not. We prove that, for both α-and β-deformations, the resulting deformed triangles still have q-Gaussians as limiting distributions, with a value of q independent (dependent) on the deformation parameter in the α-case (β-case). In contrast, the γcase, where we have used the celebrated Q-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the q-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the q-Gaussian family.

Nonlinear inhomogeneous Fokker-Planck equations: Entropy and free-energy time evolution

“…On the other hand, the observation of a specific limit distribution does not allow us to infer the entropic form, unless a careful investigation of the structure of the effective potential is performed. This simple fact has been already pointed out in the study of elementary probabilistic toy models [42][43][44]. To further exemplify it and apply our formalism, let us consider, for example, the nonlinear inhomogeneous FPE for a fluid in a porous medium.…”

Nonextensive Itô-Langevin Dynamics

“…[38] would be warmly welcome). Furthermore, our contribution do not shed any light in explaining why 𝑞-Gaussians distributions appear quite often instead of any other possible deformation for the Gaussian distribution (the best know results in this directions are based on probabilistic scale invariant models [39][40][41]).…”

Nonlinear Fokker–Planck Equation Approach to Systems of Interacting Particles: Thermostatistical Features Related to the Range of the Interactions

“…where β is a real, positive parameter associated with the width of the q-Gaussian. These distributions appear in the study of diverse systems and processes in physics, biology, and other disciplines [1,2,[36][37][38][39]. They constitute solutions-both stationary and time-dependent-of some nonlinear evolution equations of mathematical physics.…”

“…In particular, we are going to show that this system admits exact, time-dependent solutions of the q-Gaussian form. The q-Gaussian densities are central to the S q -thermostatistics and to its diverse applications [1,2,[36][37][38][39]. The q-Gaussians are q-exponentials having an argument that is quadratic in the spatial or phase-space variables of the system under consideration.…”