Motion of Free Particles in Fractal Space-time

Abstract

When considering time and space scales such that the description of the trajectories of these bodies is in the shape of non-differentiable curves, one gets fractal curves of fractal dimension 2. Continuity and non-differentiability yield a fractal space and a symmetry breaking of the differential time element which gives a doubling of the velocity fields. An application of these principles on the motion equation of free particles leads to the occurrence of a supplementary TISE (time independent, Schr dinger-type equation). When using a WKBJ method to solve this equation, one gets some interesting results. In other words, scale transformation laws produce, on the motion equation of free particles under some peculiar assumptions, effects which are analogous to those of a "macroscopic quantum mechanics".

Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

“…For D ¼_/2m and small distances of the order of Compton length, L¼_/mc (Nottale, 2004), an equation similar to (35) was solved using the WKBJ approximation method in Buzea et al (2009Buzea et al ( , 2010. We derived there, for different shapes of the potential, both asymptotic solutions and quantization conditions (Buzea et al, 2009(Buzea et al, , 2010)…”

Optimization and Numerical Anlysis of Mechanical Properties of Connecting Rod in the Internal Combustion Engine

“…Usage of the computer allows creating a real model, performing computer simulation, analysis of heat distribution, estimated deformation and aerodynamic researches [8][9][10][11][12][13]. Thanks to these above mention issue a company does not need to pay for useless costs of production, which in fact will not satisfy basic assumptions [14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Especially it is obvious in the process of car engines production, heavy industry, military, aerial and related industries, which needs many financial expenses for creating prototypes of machinery.…”

The Influence of Parameters of Heat Treatment on Thickens and Roughness of Oxide Layers on Titanium Alloy Ti6Al4V

“…The layers thus produced are characterized by up to six times greater durability in relation to the spontaneously formed layers [17][18][19][20][21][22][23].As part of this publication, the influence of time and temperature of annealing on the formation of oxide layers on Ti6Al4V titanium alloy is presented.A temperature range of 350 to 650 °C was selected for the tests however these layers also can be formed at higher temperatures. The selection of these temperatures is related to literature reports, where information appears that oxide layers produced at temperatures above 800 °C are very thin and characterized by high brittleness thus they burst during operation which is undesirable in particular in systems working under load [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].Research in the publication allowed to examine the relationship between roughness, thickness of the layer and time and temperature of annealing.…”

“…A temperature range of 350 to 650 °C was selected for the tests however these layers also can be formed at higher temperatures. The selection of these temperatures is related to literature reports, where information appears that oxide layers produced at temperatures above 800 °C are very thin and characterized by high brittleness thus they burst during operation which is undesirable in particular in systems working under load [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].…”