On inversion of the Williams–Watts function for large relaxation times

For glassy materials relaxation as a function of time is generally well fitted with the Williams–Watts function exp[−(t/τ0)β] with β≊1/3. One frequently interprets such nonexponential behavior in terms of a superposition of exponential relaxation processes with density ρβ(τ). We have determined ρβ(τ), corresponding to the Williams–Watts function, as an asymptotic (divergent) series. The leading behavior is ρβ(τ)∼τ−1+1/2β/(1−β) exp[−(1−β)(βτ/τ0)β/(1−β)]. This compliments the discussion of ρβ(τ) given by Lindsey and Patterson in terms of a convergent series, but one numerically difficult to evaluate for large τ. The density ρβ(τ) for β=1/3, 1/2, and 2/3 is expressed in closed form in terms of known functions. Suggestions for the extraction of the spectral density for long relaxation times for experimental data are given, applicable to relaxation processes slower than exponential.