Eur. Phys. J. D 49, 239-248 (2008)
https://doi.org/10.1140/epjd/e2008-00161-8

Fractal model of the atom and some properties of the matter through an extended model of scale relativity

¹ Department of Physics, University of Athens, 15771 Athens, Greece
² Department of Physics, Technical Gh. Asachi University, 700050 Iasi, Romania
³ Faculty of Applied Sciences, Politehnica University of Bucharest, Department of Physics I, 313 Splaiul Independentei street, 060042 Bucharest, Romania

Corresponding author: ^a m.agop@yahoo.com

Received: 3 April 2008
Revised: 16 June 2008
Published online: 15 August 2008

Abstract

The scale relativity model was extended for the motions on fractal curves of fractal dimension D_F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D_F=2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D_F=3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.