Soft matrix and fixed point of Lennard-Jones potentials for different hard-clusters in size at glass transition

The existence of fixed point in self-similar Lennard-Jones (L-J) potentials has been proved based on the mosaic geometric structure theory of glass transition (GT) [J. L. Wu, Soft Nanoscience letters, 1, 3–86 (2011)]. A geometric local-global mode-coupling recursive equation, different from the current Mode-Coupling Theories, has been introduced to find out the non-integrable induced potential structure of boson peak at GT. The recursively defined variable in reduced recursive equation is the potential fluctuation of reduced L-J potentials associated with reduced geometric phase potentials. A series of results have been deduced directly at GT. (i) There are only 8 orders of molecule-clusters. (ii) Two orthogonally fast-slow reduced phase potentials, 3/8 and 5/8, are accompanied with density fluctuation and clusters hop-delocalization along 8 geodesics. (iii) The stability condition of potential fluctuation is the Lindemann ratio. (iv) A new reduced attractive potential of –17/16, lower than reduced potential well energy –1, occurs