  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. 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 the exponent. 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.
