On the Prony Series Representation of Stretched Exponential Relaxation

TL;DR

This paper provides a detailed mathematical analysis of representing stretched exponential relaxation in glasses using Prony series, highlighting its accuracy, limitations, and physical implications for modeling glass relaxation.

Contribution

It offers optimized coefficients and analysis for Prony series approximation of stretched exponential functions, including their accuracy and limitations.

Findings

Prony series can accurately approximate stretched exponential decay with enough terms.

The approximation captures long-time behavior but not the initial divergence.

Frequency-domain analysis reveals physical insights into glass relaxation.

Abstract

Stretched exponential relaxation is a ubiquitous feature of homogeneous glasses. The stretched exponential decay function can be derived from the diffusion-trap model, which predicts certain critical values of the fractional stretching exponent. In practical implementations of glass relaxation models, it is computationally convenient to represent the stretched exponential function as a Prony series of simple exponentials. Here, we perform a comprehensive mathematical analysis of the Prony series approximation of the stretched exponential relaxation, including optimized coefficients for certain critical values of the exponent. The fitting quality of the Prony series is analyzed as a function of the number of terms in the series. With a sufficient number of terms, the Prony series can accurately capture the time evolution of the stretched exponential function, including its "fat tail" at long times. However, it is unable to capture the divergence of the first-derivative of the stretched exponential function in the limit of zero time. We also present a frequency-domain analysis of the Prony series representation of the stretched exponential function and discuss its physical implications for the modeling of glass relaxation behavior.