Processos de ramificació, criticitat i autoorganització: aplicació als desastres naturals

Abstract

The statistics of natural catastrophes contains very counter-intuitive results. Using earthquakes as a working example, we show that the energy radiated by such events follows a power-law or Pareto distribution. This means, in theory, that the expected value of the energy is infinite, and in practice, that the mean of a finite set of data in not representative of the full population. Also, the distribution presents scale invariance, which implies that it is not possible to define a characteristic scale for the energy A simple model to account for this peculiar statistics is a branching process: the activation or slip of a fault segment can trigger other segments to slip, with a certain probability, and so on. Although not recognized initially by seismologists, this is a particular case of the stochastic process studied by Galton and Watson one hundred years in advance, in order to model the extinction of (prominent) families. Using the formalism of probability generating functions we will be able to derive the main properties of these models. Remarkably, a power-law distribution of energies is only recovered in a very special case, when the branching process is at the onset of attenuation and intensification, i.e., at criticality. In order to account for this fact, we introduce the self-organized critical models, in which, by means of some feedback mechanism, the critical state becomes an attractor in the evolution of such systems Along the text, analogies with basic statistical physics concepts are drawn. The bulk of the material presented here is self-contained, with the exception of some basic concepts in Theory of Probability.