Properties of the glass instability treated within a mode coupling theory

Abstract

The anomalies at the liquid glass transition discussed recently by Bengtzelius et al. within a mode coupling theory are demonstrated to be due to an isolated eigenvalue of a certain stability matrix to approach unity at the critical point. Within this scenario it is shown how to derive the asymptotic results for the correlations functions analytically up to the determination of two eigenvectors and the evaluation of some wave vector integrals. As a result it is found that the Debye-Waller factor, the Lamb-Mossbauer factor, the localization length for a tagged particle, and the elastic moduli approach their asymptotic limit at the glass instability point with critical exponent one half. The critical dynamics for the coherent and incoherent scattering functions and for the transversal currents is given by a single wave vector independend scaling function. A formula for the critical exponent parameter is obtained and the scaling equation is shown to agree with the one discussed earlier for a schematic model.