Rounded stretched exponential for time relaxation functions

A rounded stretched exponential function is introduced, C(t)=exp{(τ0/τE)β[1−(1+(t/τ0)2)β/2]}, where t is time, and τ0 and τE are two relaxation times. This expression can be used to represent the relaxation function of many real dynamical processes, as at long times, t⪢τ0, the function converges to a stretched exponential with normalizing relaxation time, τE, yet its expansion is even or symmetric in time, which is a statistical mechanical requirement. This expression fits well the shear stress relaxation function for model soft soft-sphere fluids near coexistence, with τE⪡τ0. The function gives the correct limits at low and high frequency in Cole–Cole plots for dielectric and shear stress relaxation (both the modulus and viscosity forms). It is shown that both the dielectric spectra and dynamic shear modulus imaginary parts approach the real axis with a slope equal to 0 at high frequency, whereas the dynamic viscosity has an infinite slope in the same limit. This indicates that inertial effects at high f...