Integral equation approach to condensed matter relaxation

A model of relaxation in supercooled and entangled polymer liquids is devel oped starting from an integral equation describing relaxation in liquids ne ar thermal equilibrium and probabilistic modelling of the dynamic heterogen eity presumed to occur in these complex fluids. The treatment of stress rel axation considers two types of dynamic heterogeneity-temporal heterogeneity reflecting the intermittency of particle motion in cooled liquids and spat ial heterogeneity or particle clustering governed by Boltzmann's law. Exact solution of the model relaxation integral equation by fractional calculus methods leads to a two parameter family of relaxation functions for which t he memory indices (beta, phi) provide measures of the influence of the temp oral and spatial heterogeneity on the relaxation process. The exponent beta is related to the geometrical form of the spatial heterogeneity. Relaxatio n function classes are identified according to the asymptotics of the psi(t ; beta, phi) functions at long and short times and their integrability prop erties. The integral equation model for relaxation provides a framework for understanding the existence of 'universality' in condensed matter relaxati on under restricted circumstances.