On one hand it decreases because the number of primes at stake decreases with frequency. Therefore the configurationl entropy goes down. On the other hand, because the only entropy that may be involved in the process, the 2d-entropy, is lower than the statistical entropy that would result from backward (Kan extension) and forward approximation of the „Zeit Objekt‟ characterized by Z ( 0) , a Kan entropy appears. It must lead holistic characteristics. In

other words, the lowering of the dissipation within the competition between two types of metric (Euclidean and 2d-hyperbolic) contributes to the building of an auto organization (and/or Holistic rules) during the frequency deployment of the dissipative process. This would be a general property of all dissipative processes except for the diffusive one. The emergence is clearly a constructive-dissipation. Up to now this issue was hidden by implicit hypothesis of limitation of dissipative processes by diffusive transportation.

Conclusion

Far from inconsistency, the fractional diff-integral operator enlarges the epistemological perspectives, open by integer analysis. Due to the fact that the set of rational numbers is larger than the set of integers and according to the couple of Godel‟s theorems about incompleteness, we cannot reduce the fractional analysis to the paradigmatic point of view of integer dynamics. In particular, the energy is no more an invariant of space time dynamics. It has to take into account the behaviour at infinity, that is to say the structure of the environment. This assertion is clearly a key point of what is often called „inconsistency‟ of fractional analysis. This apparent „inconsistency‟ is due to the fact that energy stays the main factor of experimental performance as well as the application of the theory of fractal distribution requires regularization via a test function. The paradoxes are due to the incompleteness of the paradigmatic analysis with respect to the fractional one. This incompleteness of the standard paradigms, explains the divergence characterizing the behaviour of complex systems analysis with respect to the system able to accept a reverse engineering. Far from the very nice artefact of renormalization technics, how to overcome in physics the divergence between Euclidean and hyperbolic systems? The physical systems do that naturally through what we call „emergence‟, a response of the system to bypass the paradox led by non notherian internal structure. The theory of categories highlights this question through the concepts of Kan extension and the related virtual sets which close precisely the systems upon fractality as shown by Bagshi and recently Riot.

This analysis shows why the expression of a dissipative process by means of a derivative operator of non-integer order is relevant within the meaning of fractality (recursive structures), but also why it requires a tricky physical analysis involving the introduction of Kan entropy. In fact, the energy being not a natural invariant of the fractional operator, the practical use of this operator for the representation of physical systems and its extensions in engineering (for example as in the CRONE control devices) requires to understand a subtle problem of closure at infinity. This question can be expressed in various ways. For his part, the author chooses to express this problem in Fourier space by considering the link between the dynamic transfer functions, the hyperbolic spaces with angle at infinity (or equivalently the Gromov hyperbolic groups) and finally the Riemann zeta properties in theory of numbers. P. Riot chooses another way. He has shown recently that incompleteness can be analysed using the category theory. In all cases, the complex systems require, for being analysed, an

Complimentary Contributor Copy

embeddement, that is to say at least two points of view. The paradox of compatibility between the experience that requires the use of energy and the theory ends into the emergence of self similarity and vice versa. New holistic hierarchies then appear and contribute to the emergence of new properties or geometrical features such as aggregation, dendritic growth, the geographical contours, etc... The model TEISI contributes for his part to show why the concept of complex time is the hearth of the problematic.

Alongside the traditional time (seen as inverse transform of the Fourier frequency) the concept of complex time contributes to the emergence of a singular time which is precisely the time of the holistic emergence, self-organization and innovation. The emergence is expressed by splitting between internal and external referential during the frequencial deployment of the process.

We have seen that all the considerations that are needed to understand the physical meaning of the fractional differential equations naturally open upon heuristics issues. Recall that these opportunities were very early noticed. It concerned in particular (i) the physical sense of the wave function in quantum mechanics (d = 2) now illuminated by zeta functions and Kan extension, (ii) the topological superconductivity; (iii) the question of the topological gravity by inversion of space-time variable. All these questions and associated opportunities remain yet open upon future developments.

Acknowledgment

The author thanks D. Tayurski and J. P. Badiali for deep exchanges in quantum and complex systems physics, as well as P. Riot and I. Polubny for mathematical partnership.

References

(Angot, 1972) Angot A., Compléments de mathématiques à l’usage des ingénieurs de l’électrotechnique et des télécommunications, Masson, Paris, 1972.

(Badiali, 2013) Badiali J.P., Toward a new fundations of statistical mechanics, arXiv: 1311.4995, V1, cond-math.stat-mech, 20 nov., 2013.

(Bagshi, 1987) Bagshi B., Recurence into topological dynamics and Riemann Hypothesis, Acta Math Hungar, 50, p. 227-240, 1987.

(Cole, 1942) Cole K.S. and Cole R.H., Dispersion and absorption in dielectric. Direct current characteristics, J. Phys. Chem 10, p. 98-105, 1942.

(Connes, 1994) Connes A. Non Commutative Geometry, Acad. Press. San Diego C.A. 1994. (Derrida, 1973) Derrida Jacques, ‘Différance’ in speech and phenomena, Northwest

University Press Ilinois, 1973.

(Gemant, 1935) Gemant A. The conception of concept viscosity and application to dielectric,

Transaction of the faraday society, V.31, pp. 1582-1590, 1935.

(Ghys, 1990) Ghys E and de La Harpe P : sur les groupes hyperboliques d‟après Gromov, (Bern 1988) 1-25, Prog. Math., 83, Birkhäuser Boston, MA, 1990.

(Holland, 1998) Holland J. H., Emergence: from chaos to order, Helix books AddisonWesley MA, 1998.

Complimentary Contributor Copy

(Le Méhauté, 1974) Le Méhauté A, Application à la hiérarchie départementale française d‟un modèle d‟adaptation des individus à une structure de désutilité en arbre, Annales du Centre de Recherche de l’urbanisme Presses du CRU, Paris, 1974.

(Le Méhauté, 1977) Le Méhauté A. and Apelby J., A parametric analysis of the structure of international energy consumption, Energy, 2, p 105-114, 1977.

(Le Méhauté, 1980) Le Méhauté A., de Guibert A., Delaye M. et Filippi C., Phénoménologie du transfert de matière et d‟énergie en milieu présentant une similitude interne. Rapport de Recherche, Laboratoire de Marcoussis, Division Electrochimie, 458, Décembre 1980.

(Le Méhauté, 1982a) Le Méhauté A., de Guibert A., Delaye M. et Filippi C., Note d‟introduction à la cinétique des échanges d‟énergies et de matières sur des interfaces fractales, C.R. Acad. Sci., Paris, t. 294, Paris, pp.835-837, 1982.

(Le Méhauté, 1982b) Le Méhauté A. et Crepy G., Sur quelques propriétés de transferts électrochimiques en géométrie fractales, C.R. Acad. Sci., T. 294, Paris, pp.685-688, 1982.

(Le Méhauté, 1983) Le Méhauté A., Dugast A., Introduction of scaling properties in electrochemistry: from TEISI model to lithium layered structure, Solid State Ionics, 9/10, P 17-30, 1983.

(Le Méhauté, 1984) Le Méhauté A, Transfer process in fractal media, J. Stat. Phys. Vol. 36 N°5/6 N.Y. 665-676, 1984.

(Le Méhauté, 1986) Le Méhauté A, Frushter L., Crépry G., le Méhauté A., Batteries, Identified Fractal objects, J. power Sources, 18, 51-62, 1986.

(Le Méhauté, 1992) Le Méhauté A., Heliodore F, Cottevieille D., Propagation des ondes en milieux hétérogènes, Revue scientifique et technique de la défense, p 22-33, 1992.

(Le Méhauté, 2010) Le Méhauté A., El Kaabouchi A., Nivanen N., Riemann‟s conjecture and fractal derivative, Computer and mathematics with application Chaos Solitons and Fractals, Vol 59 (6), pp. 1610-1613, 2010.

(Mandelbrot, 1975) Mandelbrot B., Les géométries fractales, Flammarion Paris 1975. (Mandelbrot, 1982) Mandelbrot B., The fractal geometry of nature, Freeman, San Francisco,

1982.

(Nigmatullin, 1975) Nigmatullin R. Sh., Kader B.A., Krylov V.S., Sokolov L.A., Electrochemical methods of transfer processes research in liquids. //Success in Chemistry Uspekhi Khimii, v. XLIV, issue.11, pp.2008-2034, 1975.

(Nivanen, 2005) Nivanen L, Wang Q.A., le Méhauté A. „from complexe systems to generalized statistics and algebra In fractional differentiation and application Eds A. le Méhauté, J.A. Tenreiro-Machado, J.C. Trigeassou, and J. Sabatier U-Book, 2005

(Oldham, 1974) Oldham K.B. and Spanier J., The fractional calculus, Academic Press NY, 1974.

(Oustaloup, 1983) Oustaloup A., Systèmes asservis linéaires d‟ordre fractionnaire: théorie et pratique. Masson, Paris, 1983.

(Peukert, 1897) Peukert W., Über die Abhängigkeit der Kapazität von der Entladestromstärke bei Bleiakkumulatoren, Electrotech. Z. V.18, pp.287-292, 1897.

(Podlubny, 1999)Podlubny I, Fractional differential Equations, Academic press NY, 1999. (Riot, 2013) Riot P. http://repmus.ircam.fr/mamux/saisons/saison13-2013-2014/2013-12-06 (Rovelli, 2004) Rovelli C., Quantum Gravity, Cambridge University Press, 2004.

(Rovelli, 2006) Rovelli C., Qu’est ce que le temps? Qu’est ce que l’espace?, Bernard Gilson, 2006.

(Schwartz, 1950) Schwartz L., Théorie des distributions, Herman, Paris, 1950.

Complimentary Contributor Copy

(Tenenbaum, 2011) Tenenbaum G. et Mendes France M., Les nombres premiers entre ordre et chaos, Dunod, Paris, 2011.

(Tricot, 1999) Tricot Claude, Courbes et dimensions fractales, Springer Verlag, 1999. (Voronin, 1975) Voronin S., Theorem of the universality of Riemann zeta function, Izv. Akad.

Nauk. SSSR Ser. Matem., Vol. 39, p. 476-486, 1975, Reprinted: Math SSSR. Izv, Vol.9, pp.443-445, 1975.

Complimentary Contributor Copy