# Informational Non-Differentiable Entropy and Uncertainty Relations in Complex Systems

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## Abstract

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## 1. Introduction

## 2. Hallmarks of Non-Differentiability

**X**, we can write [6,7]:

**v**

_{±}the forward and backward mean speeds,

**ξ**

_{±}a measure of non-differentiability (a fluctuation induced by the fractal properties of trajectory) having the average:

**v**

_{+}and

**v**

_{−}), instead of one. These “two-values” of the speed vector represent a specific consequence of non-differentiability that has no standard counterpart (according to differential physics).

**v**

_{+}as compared to

**v**

_{−}. The only solution is to consider both the forward (dt > 0) and backward (dt < 0) processes. Then, it is necessary to introduce the complex speed [6,7]:

**V**

_{D}is differentiable and resolution scale (dt) speed independent, then

**V**

_{F}is non-differentiable and resolution scale (dt) speed dependent.

**x**

_{±}= d

_{±}

**x**, Equation (6) becomes:

**X**of components X

^{i}(i = 1, 2, 3) is the position vector of a point on the curve. Let us also consider a function f(

**X**, t) and the following series expansion up to the second order:

_{±}X

^{i}and dt are independent. Therefore, the averages of their products coincide with the product of averages. Thus, Equation (10) becomes:

_{±}describes the fractal properties of the trajectory with the fractal dimension D

_{F}[23], it is natural to impose that ${(d{\xi}_{\pm})}^{{D}_{F}}$ is proportional with resolution scale dt [6,7],

^{i}and dξ

^{l}. Therefore, using Equation (13), we can write:

^{i}and dt are standard infinitesimals of order one, while dξ

^{i}is an infinitesimal of order 1/D

_{F}, the terms dx

^{i}dx

^{l}/dt, dt

^{2}/dt, dx

^{i}dt/dt are infinitesimals of order one and are null; the last term is finite by means of Relation (14)).

## 3. Geodesics Equation

_{F}= 2, Equation (24) takes the Nottale’s form [6,7]. Moreover, for motions of complex system entities on Peano’s curves at the Compton scale, $D=\frac{\overline{)h}}{2{m}_{0}}$ (for details, see [6,7]), with ħ the reduced Planck constant and m

_{0}the rest mass of the complex system entities, Relation (24) becomes the standard Schrödinger equation.

- Any entity of the complex system is in permanent interaction with the fractal medium through a specific fractal potential.
- The fractal medium is identified with a non-relativistic fractal fluid described by the specific momentum and state density conservation laws (probability density conservation law [6,7]). For motions of complex system entities on Peano’s curves at the Compton scale, the fractal medium is identified with Bohm’s “subquantum level” [7].
- Any interpretation of Q should take cognizance of the “self” or the internal nature of the specific momentum transfer. While the energy is stored in the form of mass motion and potential energy (as it actually is), some is available elsewhere, and only the total one is conserved. It is the conservation of energy and specific momentum that ensures the reversibility and existence of eigenstates, but denies a Brownian motion-type form of interaction with an external medium.
- The specific fractal potential (27) generates the viscosity stress tensor [8,13]:$${\widehat{\sigma}}_{il}={D}^{2}{(dt)}^{\frac{4}{{D}_{F}}-2}\left({\nabla}_{i}{\nabla}_{l\rho}-\frac{{\nabla}_{i\rho}{\nabla}_{l\rho}}{\rho}\right)=\eta \left(\frac{\partial {V}_{Fi}}{\partial {x}_{l}}+\frac{\partial {V}_{Fl}}{\partial {x}_{i}}\right)$$$${\nabla}_{i}{\widehat{\sigma}}_{il}=-\rho {\nabla}_{l}Q$$
- For motions of complex system entities on Peano’s curves, at spatial scales higher than the mean free path and at temporal scales higher than the oscillation periods of the pulsating velocities, which overlaps the average velocity of the complex system motion, FHM reduces to the standard hydrodynamics model [24].
- Since the position vector of the complex system entity is assimilated to a Wiener-type stochastic process [6,7,23], ψ is not only the scalar potential of complex velocity (through ln ψ) in the fractal hydrodynamics, but also the density of probability (through |ψ|
^{2}) in the Schrödinger-type theory. Then, the equivalence between the fractal hydrodynamics formalism and the Schrödinger one results. Moreover, chaoticity, either through turbulence in the fractal hydrodynamics approach [24] or by means of stochasticization in the Schrödinger-type approach, is exclusively generated by the non-differentiability of the movement trajectories in a fractal space.

## 4. Non-Differentiable Entropy, Uncertainty Relations

_{0}is the rest mass of the complex system entity. Accordingly, non-differentiable stresses are, in their possible effects, potentially equivalent to momentum stresses p

_{i}p

_{j}= −m

_{0}

**V**

_{Di}m

_{0}

**V**

_{Dj}imparted to the fractal hydrodynamic fluid associated with the entity:

_{i}p

_{j}〉 represent the observable momentum stresses of the complex system entity. According to Equation (35),

_{i}p

_{j}, Equation (35), are generated by unobservable (first term) and observable (second term) stresses. The observable momentum stresses are given by the dyad:

_{ij}of the conjugated components of the position tensor

**rr**of the complex system entity. Thus, one finds from Equation (38) the relation:

_{ij}(s) is a function of the set of quantum numbers specifying the state of the complex system, as we shall establish in the following.

**r**, t) = ρ

_{1}(x

_{1}, t)ρ

_{2}(x

_{2}, t)ρ

_{3}(x

_{3}, t), the non-diagonal variances vanish: ∆x

_{ij}= 0 for i ≠ j. In this case, Equations (39)–(41) give:

## 5. Informational Non-Differentiable Entropy

_{0}= const. In a fractal space, substituting this value in the expression −∇Q, with Q given by (27), the force is found:

_{Q}(p, q) with the constraints:

## 6. Informational Non-Differentiable Energy and Uncertainty Relations

- The informational non-differentiable energy is an indication of the dispersion distribution (56), since the quantity:$$A=\frac{2\pi}{\sqrt{ac-{b}^{2}}}$$
_{Q}= const. Then, the normalized Gaussian becomes even more clustered, so that their informational non-differentiable energy will be higher. - The class of statistical hypotheses is specific to the Gaussians having the same mean given by the constant value of the informational non-differentiable energy.
- If the informational non-differentiable energy is constant, then Relations (57) and (58) give the egalitarian uncertainty relation:$${(\delta p)}^{2}{(\delta q)}^{2}=\frac{1}{4{\pi}^{2}{E}^{2}(a,b,c)}+co{v}^{2}(p,q)$$$$\delta p\delta q\ge \frac{1}{2\pi E(a,b,c)}$$

- The informational non-differentiable energy is quantified:$$E(a,b,c)=\frac{1}{nh}$$
- (63) implies the uncertainty relation:$$\delta p\delta q\ge n\u0127$$$$\delta p\delta q\ge \u0127$$

## 7. Conclusions

- Any complex structure implies test particles, field sources, etc., correlated with various types of forces, together with the non-differentiable medium in which they evolve. The non-differentiable (fractal) medium is assimilated to a fractal fluid, whose particles are moving on continuous, but non-differentiable, curves. Moreover, the non-differentiable medium that cannot be separated from test particles and field sources is described either by a Schrödinger-type equation or by non-differentiable hydrodynamics with non-differentiable potential, which works simultaneously with standard potentials. The non-differentiable potential is induced by the non-differentiability of the movement curves of fractal fluid entities.
- The dynamics of a complex system is described by motion equations for a complex speed field and exhibit rheological properties (memory).
- Separation movements on the interaction scales imply non-differentiable hydrodynamics, which, at the differentiable scale, contains the law of momentum conservation and, at the non-differentiable scale, the law of probability density (states density) conservation.
- The correlation fractal potential-non-differentiable entropy provides uncertainty relations in the fractal hydrodynamic approach. These relations are explained for the case of a test particle motion in spherically symmetric Coulomb or Newton fields.
- The correlation informational non-differentiable entropy-informational non-differentiable energy provides specific uncertainty relations through a maximization principle of the informational non-differentiable entropy and for a constant value of the informational non-differentiable energy. For a linear harmonic oscillator, the constant value of the informational non-differentiable energy is equivalent to a quantification condition.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Agop, M.; Gavriluț, A.; Crumpei, G.; Doroftei, B.
Informational Non-Differentiable Entropy and Uncertainty Relations in Complex Systems. *Entropy* **2014**, *16*, 6042-6058.
https://doi.org/10.3390/e16116042

**AMA Style**

Agop M, Gavriluț A, Crumpei G, Doroftei B.
Informational Non-Differentiable Entropy and Uncertainty Relations in Complex Systems. *Entropy*. 2014; 16(11):6042-6058.
https://doi.org/10.3390/e16116042

**Chicago/Turabian Style**

Agop, Maricel, Alina Gavriluț, Gabriel Crumpei, and Bogdan Doroftei.
2014. "Informational Non-Differentiable Entropy and Uncertainty Relations in Complex Systems" *Entropy* 16, no. 11: 6042-6058.
https://doi.org/10.3390/e16116042