Universality in self-organized criticality

Resumen

One of the most fascinating and important phenomena studied by Statistical Physics is the so-called Self-Organized Criticality (SOC). Since Bak, Tang and Wiesenfeld (BTW) coined this term in 1987, more than 30001 articles written in these last 21 years about the topic validate the statement which this thesis starts with. The concept of Self-Organized Criticality was introduced with the ambitious aim of being the explanation for the ubiquity of certain mathematical functions which describe some properties of real systems in Nature. Due to the initial impact of the work of BTW, SOC has been able to go beyond the frontiers, not only of its original discipline (Statistical Physics), but also of Physics itself, being a concept used on articles in Biology, Geology, Neuroscience, Ingeneering, Chemistry, Mathematics and even in Social Sciences like Psychology and Humanities. This burst of works and the broad range of fields in which SOC can be applied have made difficult to find one general and broadly accepted vision about what SOC really is; in fact, there is not a definition of SOC in literature, but (and depending on the discipline used to study it) there are many different definitions, sometimes mutually incompatible, of the term. The aim of this thesis is not to review all the work published about the topic up to date. This thesis tries to cover some general aspects of SOC from the perspective of phase transitions and their associated universal features, and this is what the reader should expect from the book. In chapter 1, the basic ideas about this topic are presented. After justifying its importance, the difference between self-organization, criticality and Self-Organized Criticality is clarified. It is pointed out under which conditions the latter is expected, and a preliminary definition for SOC, established. Also, it is introduced the paradigmatic example of SOC, the sandpile, with the original model of BTW as well as its stochastic version by Manna, and the more realistic Oslo ricepile model. These examples allow to understand how the features defined in the first part of the chapter as characteristic of SOC are applied to specific models. Meanwhile, the usual observables used in the study of these models are introduced, recalling the concept of critical exponent, which allows to talk about universality classes in SOC. The basic concepts related with phase transitions can also be found in appendix A. In chapter 2, SOC is treated from the point of view of a non-equilibrium phase transition with many absorbing states. This allows to use the typical observables defined in these systems to analyze SOC systems. Next, it is built a bridge connecting these magnitudes with the ones defined in the previous chapter, and conservation is used to define a universality class embracing both stochastic sandpiles and absorbing state systems: the Conserved Directed Percolation (C-DP) class. After that, similarities and differences between this universality class, which this thesis is focused on, and the absorbing-state-system paradigmatic class (Directed Percolation, DP) are pointed out. In the end, a continuous equation is deduced in order to describe at a mesoscopic level the behavior of the already defined C-DP universality class. Chapter 3 is devoted to compare the two above mentioned universality classes, as well as to justify the importance of the availability of tools able to distinguish clear and simply whether a system under suspect belongs to one or the other universality class. Two novel and conceptually simple criteria are presented for differentiation, and their performance is tested by using two models historically missclassified as representatives of DP among sandpiles. A different perspective from which SOC can be seen is by using elastic interfaces in random media. This is the scope of chapter 4. It is shown how to translate the SOC language into the interfaces one, and the opposite procedure. Next, the apparent incompatibilities between the observables measured in each description are explained. Thus, it is stated that certain types of SOC and interface models are just two different ways to observe the very same phenomenon. This last assertion is also justified by means of Renormalization Group (RG) arguments and RG functions measurements. Chapter 5 addresses the controversial topic of non-conservation in SOC, confirming its essential part in the existence of Self-Organized Criticality. Hence, an ultimate definition of SOC can be made. Also, examples in Nature, historically claimed to be SOC, are analyzed in this chapter, to determine whether they are critical or not. It is shown that a very concrete degree of dissipation can be present for a system to continue being critical, stating in this way up to which point the condition of conservation can be relaxed in SOC. The last chapter (chapter 6) is devoted to present some of the experiments showing the features of SOC theoretical models. Some of the earliest experiments, which did not observe the critical behavior of SOC, are studied. But also those which successfully exhibited real scale invariance, becoming the paradigmatic experimental realization of SOC. On the other hand, the circumstances and conditions under which SOC can be expected in a real system are studied. In the appendices, the above mentioned summary of some basic concepts about phase transitions and universality can be found (appendix A). But also the steps necessary to map a microscopic reaction-diffusion set of equations into a mesoscopic description of the same problem (appendix B), and the set of tables embracing the results exposed in this thesis (appendix C). This thesis is written in English, but a summary in Spanish is also included (appendix D) with the purpose to fulfill the requirements to obtain the degree of Philosophy Doctor in Physics with European level, as ruled by the regulations of the University of Granada. Also, a list of publications of the author is in the last pages of the thesis (see appendix E). The work content in this book is not, of course, all the made by its author. There is much work which has formed part of the learning necessary to obtain the results of some of the parts of this thesis, as well as other published work which, as it is not directly related to this topic, is not explicitely cited. Nonetheless, it can be found in the list of publications above mentioned.