EXPLORANDO A ORDEM FRACIONÁRIA COMO MÉTRICA DE COMPLEXIDADE EM MATERIAIS IRRADIADOS – O CASO DO PVDC

Abstract

The interaction between radiation and matter involves phenomena ranging from atomic‑scale processes to macroscopic structural changes, requiring models capable of capturing memory, nonlocality, and anomalous behavior. Fractional calculus, a generalization of classical calculus, has emerged as a promising tool for describing complex systems, particularly in polymeric materials and disordered media. In this work, we investigate the application of the Grünwald–Letnikov (GL) fractional derivative to the logarithm of the linear attenuation coefficient (LAC) of the PVDC polymer, with the aim of evaluating the fractional order as a quantitative metric of radiological complexity. Using attenuation data obtained from Phy‑X, a fractional model of the form Log (LAC) = a.D^α (Log (LAC)) + b is fitted. The results show that the optimal fractional order α = 0.001 corresponds to a local behavior with no significant fractional memory. The model exhibits high accuracy, with R² = 0.999997, RMSE = 0.00528, and extremely low AIC and BIC values, reflecting strong parsimony. Although the residuals do not satisfy the assumptions of normality and independence, such behavior is characteristic of near‑perfect models applied to highly homogeneous materials. The analysis confirms that PVDC lies at the lower bound of a radiological complexity scale based on α and may therefore be treated as a reference material. Accordingly, a fractional‑based taxonomy for classifying irradiated materials is proposed, with potential applications to heavy metals, composites, and systems exhibiting hysteresis.