Hyperbolic distance u / v 1 / d

Figure 1. Typical Cole-Cole transfer function for a canonical fractional differential equation (Le Méhauté, 1982).

To give a content to this remarks it is possible to refer to the impedance related to diffusive TEISI degeneration when fractal interface fulfils the 2D space.

5. Diffusion Under Field

The diffusive model can be written in the frame of the TEISI model as . Let us compare this equation with the fractional expression of the diffusion equation that uses non integer differential equations (Oldham, 1974) (Podlubny, 1999):

The second factor of that equation is a non-causal operator, whose physical meaning, to our knowledge, has never been considered. Even hidden, this factor plays a role which is crucial especially if some characteristics can fit the complementary set of

in the complex plan, that is to say, the role played by the co-dimension in the general TEISI analysis. In order to fit this meaning, we have to observe that the thermodynamic „gradient‟ must be written

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It is expressed by the means of a pivot which is an energy in the process used for measurement. The harmonic ratio

is then precisely the impedance. As shown by J. P. Badiali in the frame of statistical thermodynamics when based on path integrals (Badiali, 2013), this energy is used as a reference point for energy balance. This pivot gives a generalization of the physical concept of equilibrium and a reference for causality principle. The main question is the following: when such a reference is used, is the entropy increasing or decreasing with respect to the fractal correlations along the process? Indeed, the conditions of the existence of a flow of „negentropy‟ remain to be clarified to address the issue of non-causal factor. The diffusive example can serve as a guide to address the issue of the absence of energetical dimensional fitting because this absence is clearly related to this non-causal factor.

45

t

0

Forward Backward

Figure 2. Analysis of diffusive process with field using the merging of forward and backward fractional process in the Poincaré fundamental domain (Nivanen, 2005).

As evoked above, due to the use of energy as the essential experimental factor for defining the equilibrium, the representation of empirical data by the means of a derivative operator of non-integer order, over fit like the ball of measurement the fractal interface properties. So the description of the fractal reality with non-integer operator is almost surely incomplete with respect to the physical experiment. Nevertheless this assertion is paradoxical if coupled to the Godel incompleteness theorems: the set of rational numbers is obviously larger than the set of natural numbers. To precise the content of this paradox we have to analyse the response given when half diff-integral is considered in the frame of non-integer operator application? Unlike transport models and the use of temporal operators without precaution, the Fourier space used to express the TEISI model opens into the origin of a natural incompleteness. We have already shown that the model is related to the chaotic dynamics upon punctuated torus characterized by an angle at boundary (Nivanen, 2005). In the diffusive frame, the angle at infinity is/ 2 / 2 . This angle is critical: the canonical diffusive process (d=2) under field may be

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considered as the merging schematized in the diagram given below. A couple of causal (forward) and a-causal (backward) tiles of a generalized Poincaré fundamental domain mapping (Figures 2 and 3) enters in confusion if 1/ 2 .

Because for d=2, the diagram merges a couple of components for stating a new meaning of the resistance R, that is to say the role of local energy within the whole dissipative process. The merging renders obvious the role played by the referential of the representation. The passing from the non-integer order differential equation (irrelevance of the energy) to its integer counterpart (relevance of the energy) requires a rotation of the external referential with respect to the angle. The referential attached to the fractal interface (external with regard to energy) is opposed to the overall-referential (qualified as internal). In the first referential (limited to the harmonic analysis) the operational time is associated with the inverse Fourier transform of omega. The process is linearly approximated. In the frame of the second referential, the relevance of the time requires to take into account a singularity at boundary. In the diffusive traditional equation, an initialization of a process in the Atlantic east coast of United States, involved an immediate modification of the state of the sea in the west coast of Britany. This non-physical singularity is fortunately a forgotten non-physical characteristic of integer representation of a diffusive process. This paradox refers to internal equation referential. In the first referential, nonobstant the singularity at infinity, the process states a pseudo-equilibrium, as „Zeit Objekt‟. This object is given under the sole linear constraint. It is qualified as a pseudo-equilibrium because it satisfies independence from the clock-time variable, seen as an inverse transform of Fourier frequency. However this time is not the relevant variable with respect to the energy balance. This balance requires a meaning for the local concept of differentiation. It forgets the singularity on the time variable mainly ignores on the way of fractional representation when taking into account the sole fractality. Somehow paradoxically, the singularity involved with the complex time, restores the Noetherianhomogeneity of lost space-time via the sole fractality.

We shall show that this singularity contributes to the emergence of a negentropique factor. This factor balances the disequilibrium of the „energy‟ between the fractal interface characterized by the forward process, using „something happening‟ in the co-fractal environment. The theory of the categories may explain this emergence as a Kan extension of the fractal space-time (Riot, 2013). At this stage the controversy opened in 1982 reaches its heuristic pungency. The will stays focused on the TEISI transfer process supported by a huge undeniable set of experimental data that ignores transportation. The angle at infinity appears naturally as a key factor of the paradoxes concerning the energy. Since the incompleteness of non-integer operators is pointed out, it also asks for understanding the precise choice of the space time referential. From that point of view, the problem looks like the issue of the Coriolis momentum within a rotational referential. The analogy is peculiarly relevant because fractal geometry is strongly linked to the hyperbolic geometry. In this frame the fractal dimension d is related to a curvature of the geodesics, i.e., their common acceleration. This acceleration is characterized by the singularity of the dynamics that characterizes an extended „time‟ at infinity. Then, the paradox of the energy, led by the role of the resistance R, that is to say, the term of dissipation of energy, is related to the geometrical phase angle. According to the canonical model TEISI this angle carries a factor related to the internal correlations upon recursion which states, as for Coriolis momentum, the duality of the referential system when

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fractal space is considered from inside or from outside. This is clearly unveiled when we relate both Natural and Fourier spaces.

For example, the inverse Fourier transform of gives a measure of local Dirac (Angot, 1972), but also gives meaning to the ball for measurement upon the fractal object. Conversely, the measurement of the gauge , does not have any analytic expression in Fourier space. The gauge is the only variable which may simply be expressed in physical space via except if we forget the Fourier Transform limitations

After analysis, the splitting between natural and Fourier space is normal, but may be surprising especially since the changes associated with d = 2, lead the exact identity of the diffusion operators involving a simple square roots:

This simplicity is not general if. Therefore we cannot consider the diffusive process (respectively the deterministic case) as a mere extension of the general fractal process. As already pointed by Peano and Cantor for recursive geometries and in another area by Galois and Abel with non-solvable groups, the diffusive process leads to face a degenerate form which cannot easily be derived from the general fractal process. This explains the reported link within others notes between our analysis and Gromov‟s hyperbolic groups: due to the absence of sharpness on branching (small-triangle at infinity), the scaling process cannot be completely defined from the hyperbolic tree seen as a set of branches. There is an uncertainty at branching. The difficulty lies precisely in the degenerate nature of this issue if d = 2 (respectively if d = 1). In the case d = 1 the semi-circle fully explains the local exchange of energy. In the case d = 2, we can consider the isometry between the arc and the euclidean process by passing the nonlinear scaling. As shown in the scheme below, due to equivalence of metric between hyperbolic and Euclidean (Nash theorem), we can use the fundamental Poincaré‟s group to lead an isometry represented as a pseudo-linear application.

The illustration of the isomorphism gives a simple obvious form to the geometric content of the Laplace operator d=2. This isomorphism, well understood from external algebra, manages the relationship between Euclidean space, associated to the local exchange of energy (gradient) and the hyperbolic geometry of the whole set of exchanges upon the interface characterized by a Peano structure (divergence of the gradient vs. TEISI). Insofar d = 2, the interface environment is none other than the interface itself. It prohibits any leakage of energy out of the local entropy production zone.

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Figure 3. Illustration of the Nash isomorphism in the case of diffusive process with field (d=2).

Although Nash‟s theorem ensures that, such an isometry exists almost surely, even for a more complex hyperbolic structure than a Peano one, that is to say , intuition is much less immediate and the associated application is in this case, no longer linear as shown in the diagram below. In this diagram the tilling of the automorphism (related to the dynamics with ) is very distinct of the Poincaré‟s mapping.

0

Z1 / d ( )

0

Z ( )

Figure 4. Representation in the frame of TEISI model of Nash‟s isomorphism in the general case.

The difficulties already pointed out can then be summarized as follows: merged in the same „object‟, the sums of building (constructivist view and co-product) and of the products of processes (leading to partitions or reverse engineering), the extension of the concept of exponential in Fourier space, offers almost all the good properties of a generic functions. These functions are strongly connected with fractal metrics. Its representation in the complex plan is indeed a piece of hyperbolic geodesic. Nevertheless, this piece cannot possess one of the crucial properties required to perform a rigorous reverse engineering of the process: the distributivity. A „difference‟ remains after any proceeding along a loop of building and deconstruction. This „difference‟ is expressed by a phase factor (time irreversibility). Once again, for understanding this subtle issue we have to refer to the theory of categories. It explains the paradox of the energy, by suggesting the Kan extension as an operation to be considered for understanding the storage of information within the phase factor.

The use of any types of generalized exponential must be done with caution. It requires a deep understanding of physical phenomena in its own and external space time structure. A geometric understanding must guide the use of related fractional operators for the physical

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