Stretched Exponential Relaxation Arising from a Continuous Sum of Exponential Decays

TL;DR

This paper investigates the origin of stretched exponential relaxation by analyzing the distribution of relaxation rates in systems with independent exponential relaxations, providing insights into the physical meaning of key parameters.

Contribution

It introduces a detailed analysis of the probability distribution of relaxation rates that produce stretched exponential behavior, linking mathematical properties to physical interpretations.

Findings

Derived properties of the relaxation rate distribution P(lambda/lambda*,beta)

Connected the parameters lambda* and beta to physical relaxation processes

Provided a framework to understand stretched exponential relaxation in various systems

Abstract

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 the stretched exponential relaxation arises from the 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.