Inhomogeneities as a result of non-exponential relaxation in disordered systems

In many disordered systems such as supercooled liquids the relaxation function does not decay exponentially with time. In this paper, it is shown from a consideration of the initial conditions that a system of identical molecules, each of which has the same non-exponential relaxation function, must be inhomogeneous. As a result, the relaxation function for the system is not the same as that of an individual molecule, contrary to what is often assumed. As an example, the general formulae are applied to a simple model system in which the dielectric relaxation function of the individual molecules is of the stretched exponential form. It is found that while the dielectric relaxation of the system can be well approximated by a function of this form, its stretching parameter and relaxation time differ from those for the individual molecules.