On the Transport Phenomena in Composite Materials using the Fractal Space-Time Theory

Abstract

A new way to analyze the dynamics of the physical systems is to consider that the particle movements take place on continuous but non-differentiable curves, i.e. on fractals. Then, the complexity of these dynamics is substituted by fractality. There are some fundamental arguments which can justify such hypothesis: i) by interaction, the trajectory is no longer everywhere differentiable. The “uncertainty” in tracking the particle is eliminated by means of the fractal approximation of motion; ii) the complex dynamical systems, which display chaotic behavior, are recognized to acquire self-similarity and manifest strong fluctuations at all possible scales. Every type of “elementary” process of motion induces both spatiotemporal scales and the associated fractals. Moreover, the movement complexity is directly related to the fractal dimension: the fractal dimension increases as the movement becomes more complex. Different definitions were given for the fractal dimension (Kolmogorov dimension, Hausdorff dimension, etc.), but once we choose the fractal-type dimension in the study of motion we must work with it until the end. Therefore, considering that the complexity of the physical processes (from the system’s interactions) is replaced by fractality (situation in which the particle movements take place on fractal curves), it is no longer necessary to use notions as collision time, mean free path, etc., i.e., the whole classical “arsenal” of quantities from the dynamics of physical systems. Then, the physical systems will behave as a special interaction-less “fluid” by means of geodesics in a fractal space-time. The theory which treats the interactions in the previously mentioned manner is the Scale Relativity (SR). The SR is based on a generalization of Einstein’s principle of relativity to scale transformations. Namely, ”one redefines space-time resolutions as characterizing the state of reference systems scale, in the same way as speed characterizes their state of motion. Then one requires that the laws of physics apply whatever the state of the reference system, of motion (principle of motion-relativity) and of scale (principle of SR). The principle of SR is mathematically achieved by the principle of scale-covariance, requiring that the equations of physics keep their simplest form under transformations of resolution”. Another way of analyzing the system dynamics by means of the “fractals” is given by the Transfinite Physics (TP). The Transfinite Physics theory uses the Cantorian geometry as a working method. This geometry is a compromise between the discrete and the continuum. It is not simply discrete. It is transfinite discrete and has the cardinality of the continuum