Inverse spectral problem for glassy state relaxation approximated by Prony series

TL;DR

This paper develops and tests an inversion method based on Prony series approximation of the stretched exponential function to analyze glassy state relaxation data, demonstrating its effectiveness and robustness.

Contribution

It introduces a numerical inversion method for the relaxation tensor using Prony series approximation, extending previous work on viscoelasticity models.

Findings

The inversion method accurately identifies the relaxation tensor.

The method effectively analyzes glassy state relaxation data.

Numerical experiments confirm its robustness and efficiency.

Abstract

The stretched exponential relaxation function is used to analyze the relaxation of the glassy state data. Due to the singularity of this function at the origin, this function is inconvenient for data analysis. Concerning this, a Prony series approximation of the stretched exponential relaxation function (J. Mauro, Y. Mauro, 2018), which is the extended Burgers model (abbreviated by EBM) known for viscoelasticity equations, was introduced. In our previous paper [arXiv:2509.16714], we gave an inversion method to identify the relaxation tensor of the EBM using clustered eigenvalues of the quasi-static EBM. As a next important research subject of this study, we numerically examine the performance of the inversion method. The performance reveals that it is a powerful method of data analysis, analyzing the relaxation of the glassy state data.