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Buletinul Institutului Politehnic din Iaşi. Secţia, Matematică. Mecanică teoretică. Fizică

Motion Dynamic Scenarios in the Scale Relativity Theory

Monica Molcăluţ, Cristina Marcela Rusu

Published Dec 1, 2024• 0 citations
Publisher

Abstract

Abstract Assuming that any complex system, both structural and functional, can be assimilated to a mathematical object of multifractal type, it is shown that its dynamics can be described by multifractal curves. In such a framework, based on the Multifractal Theory of Motion, the Schrödinger multifractal and the Madelung multifractal scenarios become not only compatible, but also complementary in the dynamic descriptions.

DOI
10.2478/bipmf-2024-0007

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Abstract Assuming that any complex system, both structural and functional, can be assimilated to a mathematical object of multifractal type, it is shown that its dynamics can be described by multifractal curves. In such a framework, based on the Multifractal Theory of Motion, the Schrödinger multifractal and the Madelung multifractal scenarios become not only compatible, but also complementary in the dynamic descriptions.
By assimilating any complex system into a multifractal, a new approach for describing the dynamics of such systems is proposed by means of the Multifractal Theory of Motion. In such context, the description of these dynamics is accomplished through continuous and non-differentiable curves (multifractal curves), giving rise to two scenarios. The first scenario is a Schrödinger-type multifractal scenario, a situation in which the motion laws can be related to the SL(2R) algebra invariant functions. The second scenario is a Madelung-type multifractal scenario, a situation in which if the differentiable and non-differentiable components of the velocity field satisfy a particular restriction, an SL(2R) symmetry can also be highlighted. Moreover, correlative dynamics in either of the two scenarios, based on the same SL(2R) symmetry, can be obtained by Riccati-type gauges, which imply Stoler coherent states. Several cases induced by the SL(2R) symmetry are also analyzed.
We perform a critical analysis of some quantum mechanical models such as the hydrodynamic model (Madelung's model), de Broglie's theory of double solution etc., specifying both mathematical and physical inconsistencies that occur in their construction.These inconsistencies are eliminated by means of the fractal approximation of motion (physical objects moving on continuous and non-differentiable curves, i.e. fractal curves) developed in the framework of Scale Relativity (SR) (Nottalle, L., 1993;Chaline, J. et al, 2009;
In this paper the fractal model of the atom, using the hydrodynamic approach of the scale relativity theory, is obtained. Thus, assuming that the electron motion around the nucleus takes place on fractal curves of fractal dimension DF (continuous but non-differentiable curves), it is shown that its dynamics, in the second order approximation of the equation of motion, is described in complex speed field by a generalized Navier-Stokes type equation with imaginary viscosity coefficient. Applying this model to study the atom, it resulted that the real part of the complex velocity field describes the electron averaged movement. The electron moves on stationary orbits according to a quantification condition and the imaginary part of the complex velocity describes the fractality through a fractal potential. In the DF=2 fractal dimension and for the D=h / 2m viscosity coefficient, the classical results of quantum mechanics for the hydrogen atom are obtained.
Considering that the chaotic effect of associated wave packet on the particle itself results in movements on the fractal (continuous and non-differentiable) curves of fractal dimension DF, wave–particle duality through an extension of the scale relativity theory is given. It results through an equation of motion for the complex speed field, that in a fractal fluid, the convection, dissipation and dispersion are reciprocally compensating at any scale (differentiable or non-differentiable). From here, for an irrotational movement, a generalized Schrodinger equation is obtained. The absence of dispersion implies a generalized Navier–Stokes type equation, whereas, for the irrotational movement and the fractal dimension, DF = 2, the usual Schrodinger equation results. The absence of dissipation implies a generalized Korteweg–de Vries type equation. In such conjecture, at the differentiable scale, the duality is achieved through the flowing regimes of the fractal fluid, i.e. the wave character by means of the non-quasi-autonomous flowing regime and the particle character by means of the quasi-autonomous flowing regime. These flowing regimes are separated by '0.7 structure'. At the non-differentiable scale, a fractal potential acts as an energy accumulator and controls through the coherence the duality. The correspondence between the differentiable and non-differentiable scales implies a Cantor space-time. Moreover, the wave–particle duality implies at any scale a fractal.