A relaxation function encompassing the stretched exponential and the compressed hyperbola

TL;DR

This paper introduces a unified relaxation function that smoothly transitions between the stretched exponential and the compressed hyperbola, capturing diverse relaxation behaviors with a simple mathematical form.

Contribution

A new simple relaxation function is proposed that unifies the stretched exponential and compressed hyperbola, with detailed analysis of its properties and parameter effects.

Findings

The function reduces to exponential, stretched exponential, or hyperbola under specific parameter conditions.

It accurately models relaxation phenomena across different regimes.

Parameters alpha and beta control the transition between different relaxation behaviors.

Abstract

A simple relaxation function I(t/tauzero; alpha, beta) unifying the stretched exponential with the compressed hyperbola is obtained, and its properties studied. The scaling parameter tauzero has dimensions of time, whereas the shape-determining parameters alpha and beta are dimensionless, both taking values between 0 and 1. For short times, the relaxation function is always exponential, with time constant tauzero. For small values of alpha, the function is close to exponential for all times, irrespective of beta. The function is also close to an exponential when beta is near unity, irrespective of alpha. For large values of alpha and long times, the function is close to a stretched exponential, provided that beta>0. The compressed hyperbola is recovered for beta=0.