Stretched exponential relaxation arising from a continuous sum of exponential decays

Stretched exponential relaxation of a quantity n versus time t according to n=n₀0.2em0exexp[-(λ*t)^β] is ubiquitous in many research fields, where λ* is a characteristic relaxation rate and the stretching exponent β is in the range 0<β<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(λ∕λ*,β) of relaxation rates λ. We study the properties of P(λ∕λ*,β) and their dependence on β. Physical interpretations of λ* and β, derived from consideration of P(λ∕λ*,β) and its moments, are discussed.