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

On the Prony Series Representation of Stretched Exponential Relaxation