Beyond Gaussian Statistics in Polymer Melts: Statistical Masking of Persistent Local Constraints
Short polymer chains exhibit clear deviations from Gaussian end-to-end distance statistics, yet the molecular mechanism by which Gaussian behavior is recovered in long chains remains unestablished. Atomistic molecular dynamics simulations of polyethylene melts reveal that conformational heterogeneity persists at the Kuhn scale across all chain lengths, consisting of a mosaic of slow-relaxing, extended aligned chain segments (ACS) and coiled segments -- random conformational sequences (RCS) and chain ends (CE). We show that the end-to-end distance distributions for both unentangled and entangled chains are accurately described by a $q$-Gaussian function, with the entropic index $q$ increasing systematically from $0.67$ (C50) to $0.99$ (C500). This evolution tracks the emergence and accumulation of RCS segments, which are absent in short chains, establishing $q$ as a quantitative ``heterogeneity index''. The $q < 1$ values are a signature of non-extensive statistics, with the ratio of Tsallis to Boltzmann-Gibbs entropy ($S_q/S_1$), computed directly from simulation data without fitting, decreasing from $1.80$ (C50) to $1.03$ (C500). Crucially, we demonstrate that Gaussian recovery does not result from the erasure of Kuhn-scale heterogeneities, as ACS domains persist in all chain lengths above the critical mass ($\approx 35\%$). Instead, the transition to Gaussian statistics is a statistical masking effect, where the accumulation of independent RCS segments progressively obscures the non-Gaussian signatures of the persistent ACS domains.