Non-Debye relaxations: two types of memories and their Stieltjes character

TL;DR

This paper explores the mathematical properties of spectral functions in non-Debye relaxation models, revealing their connection to Stieltjes functions, and introduces a stochastic process framework to understand memory functions and their relationships.

Contribution

It demonstrates that spectral functions in non-Debye relaxation are Stieltjes functions, linking them to nonnegative response and relaxation functions, and introduces a stochastic process approach to model memory functions.

Findings

Spectral functions are Stieltjes functions supported on the positive semiaxis.

Response and relaxation functions are nonnegative and interconnected.

Memory functions can be identified with Laplace exponents of infinitely divisible stochastic processes.

Abstract

We show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semiaxis. Using only this property it can be shown that the response and relaxation functions are nonnegative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function $M(t)$ which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes based approach to the relaxation phenomena gives possibility to identify the memory function $M(t)$ with the Laplace (L\'evy) exponent of some infinitely divisible stochastic process and to introduce its partner memory $k(t)$. Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.