Estimation of the parameters of q-Gaussian distributions in the standard map

We present a novel methodology for estimating the parameters of the q-Gaussian distribution within the framework of non-extensive statistical mechanics, applied to the standard map (Chirikov-Taylor map) across its chaotic, regular, and integrable regimes. Our approach combines a genetic algorithm with multi-objective optimization, simultaneously minimizing the discrepancy between the numerical probability density function (PDF) and the q-Gaussian while enforcing normalization to unit area. This framework yields highly accurate parameter estimates, revealing optimal q and b values that provide an exceptional fit to the numerical PDFs. Notably, the estimated q values differ from previous reports, highlighting the sensitivity of parameter inference to fitting methodology. For the integrable case (K=0), we uncover a striking asymptotic scaling law: q approaches its theoretical value of 2 extremely slowly with iteration number, following a power-law trend that implies an extraordinarily large number of iterations are required for convergence. Overall, our results demonstrate that the proposed method robustly captures the statistical properties of the standard map across different dynamical regimes and suggest its potential for broader applications to other deterministic dynamical systems, offering new insights into the emergence of non-extensive statistical behavior.