Light PhD Seminar: Fractional Calculus and Material Clock Applied to Relaxation in Complex Systems

Title
Fractional Calculus and Material Clock Applied to Relaxation in Complex Systems

Abstract
The Debye model describes both relaxation and dispersion phenomena. However, this model, which is based on statistical mechanics, does not adequately describe experimental results, except for perfect liquids and crystals. The deviations from the Debye behaviour inspired a flowering of models called anomalous models, such as the Havriliak-Negami dispersion model in the frequency domain and the Kohlrausch-Williams-Watts relaxation in the time domain. These two models, which are obtained by inserting nonlinear variables into the Debye model, are not corresponding counterparts in the Fourier transform. In addition to this, there are other open questions, such as the physical origin of the Kohlrausch-Williams-Watts function with its singularity, the role of complete monotonicity, and the formulation of a general evolution equation for anomalous relaxation processes. My research results show that all these topics can be closely connected by introducing the concept of material clock. This strategic change of perspective make it possible to unravel the complexity that characterises anomalous relaxation processes and then formulate a model based on a second-order differential equation with time-dependent coefficients whose solution is the Kohlrausch-Williams-Watts function. The model, validated by experimental data, explains the purely mathematical meaning of the singularity and how the complete monotonicity or its absence affects the anomalous dynamics. The use of the material clock and the resulting game of frames point out that relaxation can be treated as an intensive property of the system and therefore the evolution equation takes the general form of a commutator based on the Reynolds-Leibniz theorem. Finally, a new integral transform bridges the Havriliak-Negami function and the generalised Kohlrausch-Williams-Watts function. Combining all the results, two different causes of the anomalous behaviour are found that explain the Cole-Cole plot. In conclusion it is possible to point out the conditions for the complete monotonicity of special functions useful in relaxation processes.

Link to the session: https://us06web.zoom.us/j/85610269039
Passcode: 774717