A Model for Approximately Stretched-Exponential Relaxation with Continuously Varying Stretching Exponents

Relaxation in glasses is typically approximated by a stretched-exponential form: $f\left(t\right)=A\exp [-(t/\tau {)}^{\beta }]$. Here, we show that the relaxation in a model of sheared non-Brownian suspensions developed by Corté et al. (Nat Phys 4:420–424, 2008) can be well approximated by a stretched exponential with an exponent $\beta$ that depends on the strain amplitude: 0.25 < $\beta$ < 1. In a one-dimensional version of the model, we show how the relaxation originates from density fluctuations in the initial particle configurations. Our analysis is in good agreement with numerical simulations and reveals a functional form for the relaxation that is distinct from, but well approximated by, a stretched-exponential function.