Beyond pair entropy: orientational many-body correlations in supercooled glass-forming liquids from a four-point structural entropy.

Abstract

The conventional pair entropy, derived from the isotropic radial distribution function <i>g</i>(<i>r</i>), systematically underestimates structural ordering in supercooled glass-forming liquids. We argue that this failure arises because <i>S</i>₂, constructed from the isotropic <i>g</i>(<i>r</i>), is inherently insensitive to the many-body orientational correlations that become accessible only through a four-point conditional distribution evaluated in a local particle-centered frame. To recover this information, we introduce a three-dimensional four-point structural entropy <i>S</i>_3D, constructed from the four-point conditional distribution function <i>g</i>(<i>r</i>,<i>θ</i>,<i>ϕ</i>) evaluated in a local particle-centered reference frame, and derive its exact decomposition into a radial contribution <i>S</i>₂ and a weighted orientational entropy <i>S</i>_<i>Ω</i>. Applying this framework to the canonical KA binary mixture, we find that <i>S</i>_<i>Ω</i> accounts for a substantial fraction of <i>S</i>_3D across the full temperature range studied, reflecting genuine icosahedral and dodecahedral angular ordering present at all temperatures-as independently established by Zhang and Kob-rather than a numerical artifact. These results demonstrate that <i>g</i>(<i>r</i>,<i>θ</i>,<i>ϕ</i>) encodes angular structural information entirely invisible to the conventional <i>g</i>(<i>r</i>) and to the thermodynamic excess entropy <i>S</i>_ex, and that <i>S</i>_3D provides a tractable thermodynamic measure of packing-driven orientational ordering across the supercooled regime.