Asymptotic Relaxation Laws
WOLFGANG GÖTZE · 2008
Abstract In order to identify the essential items of the mode-coupling-theory scenarios, this chapter describes asymptotic expansions of the correlation functions for small frequencies and small separations of the coupling constants from their critical values. For generic glass transitions the leading-order asymptotic contributions are specified by two scaling laws. The slowing down of the dynamics is governed by two time scales, which exhibit power-law dependencies on the separation coordinates. Stretching is caused by the interplay of a critical power-law relaxation and a von-Schweidler-power-law decay. Logarithmic relaxation is the leading-order behaviour near higher-order singularities. The leading-order asymptotic corrections determine the range of validity of the leading-order formulas. Strong-coupling effects are shown to cause Cole–Cole relaxation processes for the critical decay near generic transitions and sublinear power-law variations of the mean-squared displacements near higher-order ones.