Non-Debye relaxations: The characteristic exponent in the excess wings model
arXiv:2109.07415 · doi:10.1016/j.cnsns.2021.106006
Abstract
The characteristic (Laplace or Lévy) exponents uniquely characterize infinitely divisible probability distributions. Although of purely mathematical origin they appear to be uniquely associated with the memory functions present in evolution equations which govern the course of such physical phenomena like non-Debye relaxations or anomalous diffusion. Commonly accepted procedure to mimic memory effects is to make basic equations time smeared, i.e., nonlocal in time. This is modeled either through the convolution of memory functions with those describing relaxation/diffusion or, alternatively, through the time smearing of time derivatives. Intuitive expectations say that such introduced time smearings should be physically equivalent. This leads to the conclusion that both kinds of so far introduced memory functions form a "twin" structure familiar to mathematicians for a long time and known as the Sonine pair. As an illustration of the proposed scheme we consider the excess wings model of non-Debye relaxations, determine its evolution equations and discuss properties of the solutions.
References in corpus (16)
- Langevin equations for continuous time Lévy flights
- A practical guide to Prabhakar fractional calculus
- Models of dielectric relaxation based on completely monotone functions
- Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics
- A comment on a controversial issue: a Generalized Fractional Derivative cannot have a regular kernel
- On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics
- Analytical representations for relaxation functions of glasses
- The excess wing in the dielectric loss of glass-forming ethanol: A relaxation process
- General fractional calculus and Prabhakar's theory
- On relation between generalized diffusion equations and subordination schemes
- On the complete monotonicity of the three parameter generalized Mittag-Leffler function
- Stochastic Tools Hidden Behind the Empirical Dielectric Relaxation Laws
- Non-Debye relaxations: smeared time evolution, memory effects, and the Laplace exponents
- The Volterra type equations related to the non-Debye relaxation
- A note on paper "Anomalous relaxation model based on the fractional derivative with a Prabhakarlike kernel" [Z. Angew. Math. Phys. (2019) 70:42]
- Non-Debye relaxations: two types of memories and their Stieltjes character