Generalized Radon--Nikodym Spectral Approach. Application to Relaxation Dynamics Study
The paper presents a new method for analyzing relaxation dynamics, but its impact is limited to the specific domain of relaxation processes and the results are demonstrated on a few examples without quantitative comparisons to existing methods.
The paper develops a Radon-Nikodym spectral approach for relaxation dynamics that avoids using a norm, instead building a virtual Hamiltonian whose eigenvalues represent dynamic characteristics. The method is applied to model and experimental time-series signals of degradation and relaxation processes, with a software implementation provided.
Radon--Nikodym approach to relaxation dynamics, where probability density is built first and then used to calculate observable dynamic characteristic is developed and applied to relaxation type signals study. In contrast with $L^2$ norm approaches, such as Fourier or least squares, this new approach does not use a norm, the problem is reduced to finding the spectrum of an operator (virtual Hamiltonian), which is built in a way that eigenvalues represent the dynamic characteristic of interest and eigenvectors represent probability density. The problems of interpolation (numerical estimation of Radon--Nikodym derivatives is developed) and obtaining the distribution of relaxation rates from sampled timeserie are considered. Application of the theory is demonstrated on a number of model and experimentally measured timeserie signals of degradation and relaxation processes. Software product, implementing the theory is developed.