Static and free time-dependent fractal systems through an extended hydrodynamic model of the scale relativity theory

Abstract

Considering that the motions of the particles take place on ‘arbitrary’ fractals, an extended hydrodynamic model of the scale relativity is built. In this approach, static (particle in a box) and time-dependent (free particle) systems are analyzed. The particle in a box can be associated with a fractal fluid: the zero value of the real (differentiable) part of the complex speed field specifies the coherence, while the non-zero value of the imaginary (non-differentiable or fractal) part implies, through a quantization relation, a Reynolds criterion. For a minimal value of the Reynolds number, a Heisenberg's ‘egalitarian’ relationship results, whereas for big Reynolds numbers, the flow regime of the fractal fluid becomes turbulent. In such a context, the microscopic–macroscopic scale transition could be associated with an evolution scenario towards chaos. The free time-dependent particle can be associated with an incoherent fractal fluid: the differentiable and fractal components of the complex speed field are inhomogeneous in fractal coordinates due to the action of a fractal potential. There exists a momentum transfer on both speed components and the ‘observable’ in the form of a uniform motion is generated through a specific mechanism of ‘vacuum’ polarization induced by the same fractal potential.