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

Maricel Agop,Mitică Craus

doi:10.3390/sym12020226

Abstract

In the motion fractal theory, the scale relativity dynamics of any complex system are described through various Schrödinger or hydrodynamic type fractal “regimes”. In the one dimensional stationary case of Schrödinger type fractal “regimes”, synchronizations of complex system entities implies a joint invariant function with the simultaneous action of two isomorphic groups of the S L ( 2 R ) type as solutions of Stoka type equations. Among these joint invariant functions, Gaussians become in the Jeans’s sense, probability density (i.e., stochasticity) whenever the information on the complex system analyzed is fragmentary. In the two-dimensional case of hydrodynamic type fractal “regimes” at a non-differentiable scale, the soliton and soliton-kink of fractal type of the velocity field generate the minimal vortex of fractal type that becomes the source of all turbulences in the complex systems dynamics. Some correlations of our model to experimental data were also achieved.

Highlights

The theory of measurable Lie groups [1,2,3] is constituted as a fundamental procedure for integral geometry, necessary in generating the so-called geometric probabilities
As long as we limit ourselves to thermodynamic constraints, the quantity that we are interested in is the energy of the system in relation to which the distribution density typical of the system is an exponential function
When wanting a connection of the thermodynamics with the mechanics by which we study, for example, the harmonic oscillator, the distribution density becomes, automatically, a Gaussian bivariate in impulse-coordinate [14]

Summary

Introduction

The theory of measurable Lie groups [1,2,3] is constituted as a fundamental procedure for integral geometry, necessary in generating the so-called geometric probabilities. The notion of geometric probability has a long history [4,5], except that recently [6,7,8,9,10,11] it has been explicitly realized that, by the theory of measurable groups, it could have applicability in physics or more generally, in fields involving statistical inference The basis of this applicability lies in Jaynes’ observation given in [12,13]. The various achievements of this group will be considered in more detail in this paper, insofar as they can be related to statistical physics In this conjecture, we will show that the density of such a distribution can be obtained as joint invariant function of two SL(2R) isomorphic Lie algebras, operating in a motion fractal theory [10,11,15]. Dynamics of complex systems at non-differentiable scale resolutions using various hydrodynamic type fractal “regimes” are analyzed

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

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

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

Conclusions