Mesoscopic mean-field theory for supercooled liquids and the glass transition

The Weiss mean-field theory is applied to finite systems with unrestricted sizes, yielding a partition function for supercooled liquids. Finite-size effects broaden the transition and induce a Curie-Weiss-like energy reduction which provides an explanation for the Vogel-Tamman-Fulcher law. Because the energy reduction is intensive, the basic thermodynamic unit (aggregate) subdivides into smaller regions (clusters) which lowers the net internal energy. The distribution of aggregate sizes, combined with a relaxation rate that varies exponentially with inverse size, provides an explanation for the Kohlrausch-Williams-Watts law.