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Entropy

Coherences in the Dynamics of Physical Systems from a Multifractal Perspective of Motion

Decebal Vasincu, Andreea Bianca Bruma, Oana Rusu, Cristina Marcela Rusu, Vlad Ghizdovăţ, Maricel Agop

Published Jul 30, 2023• 0 citations
Publisher

Abstract

Using an analogy between the multi-fractal Schrödinger equation and the dumped oscillator equation through a special ansatz, Stoler-type coherences in the dynamics of physical systems are highlighted. Such a result implies a Ricatti-type gauge, a process that can be considered a calibration of the difference between the kinetic and potential energy of a Lagrangian, specified as a perfect square in generic coordinates.

DOI
10.3390/e25081143

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Using an analogy between the multi-fractal Schrödinger equation and the dumped oscillator equation through a special ansatz, Stoler-type coherences in the dynamics of physical systems are highlighted. Such a result implies a Ricatti-type gauge, a process that can be considered a calibration of the difference between the kinetic and potential energy of a Lagrangian, specified as a perfect square in generic coordinates.
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.
The mathematical concept of minimal atomicity is extended to fractal minimal atomicity, based on the nondifferentiability of the motion curves of physical system entities on a fractal manifold. For this purpose, firstly, different results concerning minimal atomicity from the mathematical procedure of the Quantum Measure Theory and also several physical implications are obtained. Further, an inverse method with respect to the common developments concerning the minimal atomicity concept has been used, showing that Quantum Mechanics is identified as a particular case of Fractal Mechanics at a given scale resolution. More precisely, for fractality through Markov type stochastic processes, i.e., fractalization through stochasticization, the standard Schrödinger equation is identified with the geodesics of a fractal space for motions of the physical system entities on nondifferentiable curves on fractal dimension two at Compton scale resolution. In the one-dimensional stationary case of the fractal Schrödinger type geodesics, a special symmetry induced by the homographic group in Barbilian’s form “makes possible the synchronicity” of all entities of a given physical system. The integral and differential properties of this group under the restriction of defining a parallelism of directions in Levi-Civita’s sense impose correspondences with the “dynamics” of the hyperbolic plane so that harmonic mappings between the ordinary flat space and the hyperbolic one generate (by means of a variational principle) a priori probabilities in Jaynes’ sense. The explicitation of such situation specifies the fact that the hydrodynamical variant of a Fractal Mechanics is more easily approached and, from this, the fact that Quantum Measure Theory can be a particular case of a possible Fractal Measure Theory.
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.
Inconsistencies of some standard quantum mechanical models (Madelung’s, de Broglie’s models) are eliminated as- suming the micro particle movements on continuous, but non-differentiable curves (fractal curves). This hypothesis, named by us the fractal approximation of motion, will allow an unitary approach of the phenomena in quantum me-chanics (separation of the physical motion of objects in wave and particle components depending on the scale of resolution, correlated motions of the wave and particle, i.e. wave-particle duality, the mechanisms of duality, by means of both phase wave-particle coherence and wave-particle incoherence, the particle as a clock, particle incorporation into the wave and the implications of such a process). Moreover, correspondences with standard gravitational models (Einstein’s model, string theory) can be also distinguished.