**5. Conclusions**

We deeply thank the enlightening research papers of D. Baleanu and A. Fernandez [4] and M.D. Ortigueira and J.A.T. Machado ([5–7]). These works are a source of inspiration in order to continue the studies about fractional calculus and its applications to real world phenomena. In the light of the results of these works, in this paper we present new scenarios of discussion that might complement the previous studies. The main novelties of this work, can be summarized in the following items:


Dirac obtained his famous equation by considering the square root of the Klein–Gordon equation. It is related to the basic idea of evolution depending only on the initial configuration of the system. At the same time, Dirac introduced the concept of internal degrees of freedom: the spin of a

particle. In this contribution, we apply the above idea of Dirac to the square root of the classical heat equation and we obtain a fractional diffusion equation with internal degrees of freedom.

We extend the idea of considering a general root equation of a given one, and we obtain a connection between the Silvester algebra and the fractional calculus.

• In Section 4, we show one example where a fractional diffusion equation does not satisfy the second law of Thermodynamics, and we consider the use of the fractional calculus to model the dust dynamics with the associated electromagnetic interaction in Earth and Mars atmospheres.

On the other hand, these developments set out interesting questions and discussions about the adjustment and the reliability of the mathematical models to the dynamic of the real processes; for instance, the objectivity in the descriptions of the models. In this sense, the following items are remarkable:


**Author Contributions:** Investigation, M.P.V., D.U., S.J., L.V., J.L.V.-P. and M.M.; Methodology, M.P.V., D.U., S.J., L.V., J.L.V.-P. and M.M.; Writing—original draft, M.P.V., D.U., S.J., L.V., J.L.V.-P. and M.M.; Writing—review and editing, M.P.V., D.U., S.J., L.V., J.L.V.-P. and M.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research has been carried out partially in the framework of the IN-TIME project, funded by the European Commission under the Horizon 2020 Marie Sklodowska-Curie actions Research and Innovation Staff Exchange (RISE) (Grant Agreement 823934). Futhermore, this research was funded by Ministerio de Economía, Industria y Competitividad of Spanish Government (ESP2016-79135-R).

**Acknowledgments:** Authors thank the collaboration of the research groups Finish Meteorological Institute (Ari-Matti Harri) and Space Research Institute of Russian Academy of Sciences (Oleg Korablev).

**Conflicts of Interest:** The authors declare no conflict of interest.
