# Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of SL(2R) Type

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## Abstract

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## 1. Introduction

## 2. Short Reminder on the motion fractal theories in the Form of Scale Relativity

## 3. Dynamics in a Complex System in the Form of Schrödinger Type Fractal “Regimes”

## 4. Dynamic of Complex Systems in the Form of Hydrodynamic Type Fractal “Regimes”

## 5. Conclusions

- (i)
- In the fractal theory of motion in the form of scale relativity, it is shown that in the case of the nonrotational movements of the entities of a complex system, its dynamics are associated with the geodesics on a fractal manifold in the form of Schrödinger equations at different scale resolutions. In such a context:
- (i1)
- Explaining the dynamics of complex system entities in the form of one-dimensional stationary Schrödinger equations at various scale resolutions involves joint invariant functions at the action of two isomorphic groups of type $SL\left(2R\right)$ (group of variables and parameters) as solutions of Stoka type equations. In such a context, special mechanisms for synchronizing the entities of a complex system become operable in the sense that not only the phases but the amplitudes of the complex system entities are homographically affected.
- (i2)
- The existence of a parallel transport of directions in the Levy–Civita sense, which specifies a certain amplitude-phase correlation for any entity of a complex system, explains a particular synchronization mechanism.
- (i3)
- (i4)
- A special synchronization through self-modulation in amplitudes between two entities of the same type of a complex systems is established. Such results are proved in the dynamic of laser ablation plasma (the synchronization on the oscillation modes between the Coulomb structure and thermal structure).

- (ii)
- In the fractal theory of motion in the form of scale relativity theory, it is shown that, in the general case, the dynamics in complex systems can be separated on the resolution scale (the differentiable and non-differentiable scales). Then, hydrodynamic type “regimes” to describe the dynamics become operational. In such a context:
- (ii1)
- In the static case of motions at a non-differentiable scale resolution, the velocity fields are given by means of nonlinear solutions of fractal soliton type and fractal soliton kink-type.
- (ii2)
- In the motion previous context, a fractal minimal vortex is generated. This case become the “source” of all turbulences in the dynamics of complex systems.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- (i)
- (ii)
- The dynamics of complex systems structural units are described through fractal variables. Then, two derivatives of the variable field $Q(t,dt)$ can be defined:$$\begin{array}{c}\hfill \frac{{d}_{+}Q(t,dt)}{dt}=\underset{\Delta t\to {0}_{+}}{lim}\frac{Q(t+\Delta t,\Delta t)-Q(t,\Delta t)}{\Delta t}\\ \hfill \frac{{d}_{-}Q(t,dt)}{dt}=\underset{\Delta t\to {0}_{-}}{lim}\frac{Q(t,\Delta t)-Q(t-\Delta t,\Delta t)}{\Delta t}\end{array}.$$

- (iii)
- The differential of the spatial coordinate field ${d}_{\pm}{X}^{i}(t,dt)$ is given by the expression:$${d}_{\pm}{X}^{i}(t,dt)={d}_{\pm}{x}^{i}\left(t\right)+{d}_{\pm}\xi (t,dt).$$

- (iv)

- (v)
- The differential time reflection invariance of any variable is recovered by means of the operator:$$\frac{\widehat{d}}{dt}=\frac{1}{2}\left(\frac{{d}_{+}+{d}_{-}}{dt}\right)-\frac{i}{2}\left(\frac{{d}_{+}-{d}_{-}}{dt}\right)$$

- (vi)
- Since the fractalization implies stochasticization [28], the whole statistic arsenal in the form of averages covariances, etc., becomes operational. Thus, let us choose for the average of ${d}_{\pm}{X}^{i}$ the following functionality:$$\u2329{d}_{\pm}{X}^{i}\u232a={d}_{\pm}{x}^{i}$$$$\u2329{d}_{\pm}{\xi}^{i}\u232a=0.$$

- (vii)
- The complex system dynamics can be described through the scale covariant derivative given by the operator:$$\frac{\widehat{d}}{dt}={\partial}_{t}+{\widehat{V}}^{i}{\partial}_{i}+\frac{1}{4}{\left(dt\right)}^{(2/{D}_{F})-1}{D}^{lk}{\partial}_{l}{\partial}_{k}$$$$\begin{array}{c}\hfill {D}^{lk}={d}^{lk}-i{\overline{d}}^{lk}\\ \hfill {d}^{lk}={\lambda}_{+}^{l}{\lambda}_{+}^{k}-{\lambda}_{-}^{l}{\lambda}_{-}^{k},\phantom{\rule{1.em}{0ex}}{\overline{d}}^{lk}={\lambda}_{+}^{l}{\lambda}_{+}^{k}+{\lambda}_{-}^{l}{\lambda}_{-}^{k}\end{array}.$$

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Agop, M.; Craus, M.
Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of *SL*(2*R*) Type. *Symmetry* **2020**, *12*, 226.
https://doi.org/10.3390/sym12020226

**AMA Style**

Agop M, Craus M.
Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of *SL*(2*R*) Type. *Symmetry*. 2020; 12(2):226.
https://doi.org/10.3390/sym12020226

**Chicago/Turabian Style**

Agop, Maricel, and Mitică Craus.
2020. "Towards Stochasticity through Joint Invariant Functions of Two Isomorphic Lie Algebras of *SL*(2*R*) Type" *Symmetry* 12, no. 2: 226.
https://doi.org/10.3390/sym12020226