Stretched exponential relaxation arising from a continuous sum of exponential decays

Stretched exponential relaxation of a quantity n versus time t according to n=n(0) exp[-(lambda(*)t)(beta)] is ubiquitous in many research fields, where lambda(*) is a characteristic relaxation rate and the stretching exponent beta is in the range 0 <beta < 1. Here we consider systems in which stretched exponential relaxation arises from global relaxation of a system containing independently exponentially relaxing species with a probability distribution P(lambda/lambda(*),beta) of relaxation rates lambda. We study the properties of P(lambda/lambda(*),beta) and their dependence on beta. Physical interpretations of lambda(*) and beta, derived from consideration of P(lambda/lambda(*),beta) and its moments, are discussed.