From Self Organized Criticality to Patterns Formation. Theoretical Aspects, Occurence in Nature

Critical Phenomena play a fundamental role in the modern physics. Critical behavior is characterized by the absence of any characteristic scale for the physical quantities. Developing a tools describing these properties had represented a formidable challenge for the theoretical physicists. At the critical point the system fluctuates strongly at all wavelengths and no perturbation theory can be builted up. A more useful approach can be found in the K. G. Wilson Renormalization Group that bases its rules directly on scale invariance properties of a critical system. This new kind of approach to physical phenomena can be related to another conceptual revolution due to B. Mandelbrot (Mandelbrot (1982)). In his work he has pointed out that natural landscapes are by the most part composed by scale invariant (selfsimilar) objects and that in Nature the euclidean regularity is an exception. Furthermore i,vith the fractal geometry tools one is able to manage with very different phenomena in such a way to consider complexity as a self standing object. "Clouds are not spheres, mountains are not cones, coastlines are not circles, bark is not smooth nor does lightnings travel in a straight line ... " and from this point of view all these patterns show the same scale invariant properties of ordinary critical phenomena. Also in this case, in fact, one deals with systems characterized by an infinite correlation length. Despite of this fact a very deep difference exists between these two classes of physical processes: while the critical point in liquid-vapour transition is reached by a tuning of the system parameters (pressure and temperature), this seems not the case for the pattern formation in Nature. In this last case one deals with systems that are intrinsically irreversible and no hamiltonian treatment can be developed. The story of the whole process is necessary to describe the relative occurrence probability of a given configuration and therefore it cannot be ignored. As it will be pointed out in the following the scale invariance by itself does not imply a critical condition. The system may arrange in a fractal form and remain stable with respect to external perturbation. Sometimes one can observe that this widespread occurence is related to the fact that the critical state is an attractor for the dynamics of the system...