Weak self-similarity of the Mittag–Leffler relaxation function

Abstract

The Mittag–Leffler (ML) relaxation function, E α (−t α ) (0 < α ⩽ 1), describes multiscale relaxation processes with a broad range of relaxation rates, where α = 1 corresponds to exponential relaxation. For 0 < α < 1 it decays asymptotically ∼t −α and is thus asymptotically self-similar, i.e. form invariant under a scale transform t → μt. In the language of asymptotic analysis, such functions are referred to as regularly varying. Based on this observation we derive a refined, ‘weakly self-similar’ asymptotic form by applying a theorem due to J Karamata. Reasoning along the same lines, we derive also a corresponding weakly self-similar form for the time derivatives of the ML relaxation function in the short time limit. In both cases the respective asymptotic power law forms are approached by slowly varying functions in the sense of asymptotic analysis and we show that the range of validity of the respective approximations increases strongly with the decrease of α.