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On the Prony series representation of stretched exponential relaxation

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  • Mauro, John C.
  • Mauro, Yihong Z.

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.

Suggested Citation

  • Mauro, John C. & Mauro, Yihong Z., 2018. "On the Prony series representation of stretched exponential relaxation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 75-87.
    • Handle: RePEc:eee:phsmap:v:506:y:2018:i:c:p:75-87
    DOI: 10.1016/j.physa.2018.04.047
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    References listed on IDEAS

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    1. Mauro, John C. & Smedskjaer, Morten M., 2012. "Minimalist landscape model of glass relaxation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(12), pages 3446-3459.
    2. Alvarez-Martinez, R. & Martinez-Mekler, G. & Cocho, G., 2011. "Order–disorder transition in conflicting dynamics leading to rank–frequency generalized beta distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(1), pages 120-130.
    3. Naumis, G.G. & Phillips, J.C., 2012. "Diffusion of knowledge and globalization in the web of twentieth century science," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3995-4003.
    4. Mauro, John C. & Smedskjaer, Morten M., 2012. "Unified physics of stretched exponential relaxation and Weibull fracture statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(23), pages 6121-6127.
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