Probabilistic Models with Nonlocal Correlations: Numerical Evidence of q-Large Deviation Theory

TL;DR

This paper introduces a family of correlated probabilistic models that generalize previous nonextensive statistical mechanics models, providing numerical evidence supporting a $q$-generalized Large Deviation Theory and deepening understanding of complex systems with global correlations.

Contribution

It generalizes a previous model to produce a broader family of $Q$-Gaussian attractors and supports a $q$-generalized Large Deviation Theory through numerical analysis.

Findings

Family of models yields $Q_c$-Gaussians in the large N limit.

Numerical evidence supports $q$-generalized Large Deviation Theory.

Models are isomorphic via a monotonic transformation Q_c(Q).

Abstract

The correlated probabilistic model introduced and analytically discussed in Hanel et al (2009) is based on a self-dual transformation of the index $q$ which characterizes a current generalization of Boltzmann-Gibbs statistical mechanics, namely nonextensive statistical mechanics, and yields, in the $N\to\infty$ limit, a $Q$-Gaussian distribution for any chosen value of $Q \in [1,3)$. We show here that, by properly generalizing that self-dual transformation, it is possible to obtain an entire family of such probabilistic models, all of them yielding $Q_c$-Gaussians ($Q_c \ge 1$) in the $N\to\infty$ limit. This family turns out to be isomorphic to the Hanel et al model through a specific monotonic transformation $Q_c(Q)$. Then, by following along the lines of Tirnakli et al (2022), we numerically show that this family of correlated probabilistic models provides further evidence towards a $q$-generalized Large Deviation Theory (LDT), consistently with the Legendre structure of thermodynamics. The present analysis deepens our understanding of complex systems (with global correlations among their elements), supporting the conjecture that generic models whose attractors under summation of $N$ strongly-correlated random variables are $Q$-Gaussians, might always be concomitantly associated with $q$-exponentials in the LDT sense.