### 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, β. 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 β. 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.

Original language | English (US) |
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Pages (from-to) | 75-87 |

Number of pages | 13 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 506 |

DOIs | |

State | Published - Sep 15 2018 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics

### Cite this

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*Physica A: Statistical Mechanics and its Applications*, vol. 506, pp. 75-87. https://doi.org/10.1016/j.physa.2018.04.047

**On the Prony series representation of stretched exponential relaxation.** / Mauro, John; Mauro, Yihong Z.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the Prony series representation of stretched exponential relaxation

AU - Mauro, John

AU - Mauro, Yihong Z.

PY - 2018/9/15

Y1 - 2018/9/15

N2 - 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, β. 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 β. 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.

AB - 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, β. 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 β. 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.

UR - http://www.scopus.com/inward/record.url?scp=85045626160&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045626160&partnerID=8YFLogxK

U2 - 10.1016/j.physa.2018.04.047

DO - 10.1016/j.physa.2018.04.047

M3 - Article

AN - SCOPUS:85045626160

VL - 506

SP - 75

EP - 87

JO - Physica A: Statistical Mechanics and its Applications

JF - Physica A: Statistical Mechanics and its Applications

SN - 0378-4371

ER -