Topological derivation of shape exponents for stretched exponential relaxation

In homogeneous (ideal) glasses, the important dimensionless stretched-exponential shape parameter beta is described by magic (not adjusted) simple fractions derived from fractal configuration spaces of effective dimension d* determined by different topological axioms (rules) in the presence (absence) of a forcing electric field. The rules are based on a new central principle for defining glassy states: equal a priori distributions of fractal residual configurational entropy. Our approach and its beta estimates are fully supported by the results of relaxation measurements involving many different glassy materials and probe methods. The present unique topological predictions for beta typically agree with observed values to approximately 1% and indicate that for field-forced conditions beta should be constant for appreciable ranges of such exogenous variables as temperature and ionic concentration, as indeed observed using appropriate frequency-domain data analysis. The present approach can also be inverted and used to test sample homogeneity and quality.