Robust determination of relaxation times spectra of long-time multirelaxation processes

Long-time relaxation processes occur in numerous physical systems. They are often regarded as multirelaxation processes, which are a superposition of exponential decays with a certain distribution of relaxation times. The relaxation times spectra often convey information about the underlying physics. Extracting the spectrum of relaxation times from experimental data is, however, difficult. This is partly due to the mathematical properties of the problem and partly due to experimental limitations. In this paper, we perform the inversion of time-series relaxation data into a relaxation spectrum using the singular value decomposition accompanied by the Akaike information criterion estimator. We show that this approach does not need any a priori information on the spectral shape and that it delivers a solution that consistently approximates the best one achievable for given experimental dataset. On the contrary, we show that the solution obtained imposing an optimal fit of experimental data is often far from reconstructing well the distribution of relaxation times.