β relaxation near glass transition singularities

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Abstract
Glass transition singularities as obtained by mode-coupling theory are classified within the framework of singularity theory for smooth mappings in parameter spaces. The general equations for the beta relaxation process are derived and characterised by sets of a few relevant parameters characteristic for every singularity. The simplest singularity, which is studied in detail, is specified by two relevant coordinates, the separation parameter and the exponent parameter. The scaling laws describing the dynamics and the leading corrections to scaling are discussed. It is shown that a strong asymmetry in the mode-coupling equations leads to a beta relaxation peak that can be described asymptotically by a Cole-Cole law. The corresponding dynamics is interpreted within the picture of renewal processes for motion in a high-dimensional potential landscape.