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Inverse spectral problem for glassy state relaxation approximated by Prony series

Shuli Chen, Marrten V. de Hoop, Youjun Deng, Ching-Lung Lin, Gen Nakamura

TL;DR

This work addresses extracting relaxation parameters from glassy-state data by replacing the singular stretched exponential with a Prony-series approximation within the extended Burgers model (EBM). It develops an inverse spectral approach based on two clusters of eigenvalues of an augmented system to recover the Prony-series parameters ($D$, $r_i$, $b_i$). Numerical experiments demonstrate accurate reconstructions in noise-free scenarios and robustness to moderate noise, with improved recovery when the two clusters are well separated in frequency. The results suggest a practical, data-analytic tool for quantifying viscoelastic relaxation in glasses, potentially informing applications in DMA data analysis and beyond.

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.

Inverse spectral problem for glassy state relaxation approximated by Prony series

TL;DR

This work addresses extracting relaxation parameters from glassy-state data by replacing the singular stretched exponential with a Prony-series approximation within the extended Burgers model (EBM). It develops an inverse spectral approach based on two clusters of eigenvalues of an augmented system to recover the Prony-series parameters (, , ). Numerical experiments demonstrate accurate reconstructions in noise-free scenarios and robustness to moderate noise, with improved recovery when the two clusters are well separated in frequency. The results suggest a practical, data-analytic tool for quantifying viscoelastic relaxation in glasses, potentially informing applications in DMA data analysis and beyond.

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.

Paper Structure

This paper contains 4 sections, 2 theorems, 33 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Inverse clustered eigenvalue problem and its inversion method
  3. Numerical performance of the inversion method
  4. Conclusions and discussions

Key Result

Theorem 2.1

By knowing two clusters of eigenvalues associated with $k=k_1,\,k_2\in{\mathbb N}$, we can recover $D,\, b_i, r_i$ with $1\le i\le N$ of motion2.

Figures (2)

  • Figure 1: Comparisons between reconstructions $(r_j^{inv},\,b_j^{inv},\,D^{inv})$ and true values $(r_j^{true},\,b_j^{true},\,D^{true})$ for $N=5$ and noise level $\delta=0,\,0.05,\,0.1$. Here, each row represents the reconstructions for $D=0.5,\,1,\,5$, respectively. Each column represents the reconstructions for the pair of frequencies $(k_1,\,k_2)=(81,\,91),\,(81,\,501),\,(81,\,1001)$
  • Figure 2: Comparisons between reconstructions $(r_j^{inv},\,b_j^{inv},\,D^{inv})$ and true values $(r_j^{true},\,b_j^{true},\,D^{true})$ for $N=9$ and noise level $\delta=0,\,0.05,\,0.1$. Here, each row represents the reconstructions for $D=0.5,\,1,\,5$, respectively. Each column represents the reconstructions for the pair of frequencies $(k_1,\,k_2)=(81,\,91),\,(81,\,501),\,(81,\,1001)$

Theorems & Definitions (4)

  • Theorem 2.1
  • Lemma
  • Example 3.1
  • Example 3.2