Avalanches in Out of Equilibrium Systems: Statistical Analysis of Experiments and Simulations

Abstract

Instead of a linear and smooth evolution, many physical system react to external stimuli in avalanche dynamics. When an out of equilibrium system governed by disorder is externally driven the evolution of internal variables is local and non-homogeneous. This process is a collective behaviour adiabatically quick known as avalanches. Avalanche dynamics are associated to the transformation of spatial domains in different scales: from microscopic, to large catastrophic events such as earthquakes or solar flares. Avalanche dynamics is also involved in interdisiplinar topics such as the return prices of stock markets, the signalling in neuron networks or the biological evolution.

Many avalanche dynamics are characterised by scale invariance, trademark of criticality. The physics in a so-called critical point are the same in all observational scales. Some avalanche dynamics share empirical laws and can define Universality Classes, reducing the complexity of systems to simpler mathematical models.

Scale invariance causes avalanche distributions to be power-laws, determined by critical exponents. It is difficult to predict the behaviour of critical avalanche dynamics. Maximum Likelihood techniques are the standard method to determine critical exponents. Whether by physical reasons or instrumental resolutions, scale invariance can only be found within a certain magnitude range. The Maximum Likelihood Exponent Maps (MLEM) is a methodology developed to determine both the critical exponent and the boundaries of the scale invariance regime.

Hazard assessment related to avalanche dynamics rely on the event probabilities within temporal intervals. We adopted some analytical techniques to study the avalanches as events in a point process. The intensity of the point process can be defined by exogenous effects or the history of the process itself (endogenous effects). The distribution of waiting times had shown the existence of a new scale invariance in the case of earthquakes, the so called Unified Scaling Law (USL). We have found similar scaling behaviours in other experimental systems. We used the so-called Bi test to identify endogenous effects in a given point process. Such effects can sometimes be adjusted to the so-called Epidemic Type Aftershock Sequence (ETAS) model by the method of the Mean Aftershock Sequence (MAS). All these techniques have been used to analyse different systems exhibiting avalanche dynamics.

The mechanical failure under uniaxial compression of heterogeneous materials is caused by a sequence of fractures and displacements that can be detected by Acoustic Emission techniques. We performed compression experiments over samples of porous materials while registering the Acoustic Emission. This work discusses the results obtained with a mesoporous silica glass (SiO_2), natural goethite (FeO(OH)) ore and artificially sintered alumina (Al2O3). Scale invariance and endogenous effects were identified in such samples. Furthermore, the silica glass rendered strong analogies with the statistical laws of earthquakes.

Martensitic transformations are a sort of structural phase transitions found in several materials sometimes referred as Shape Memory Alloys. Such transformations take place in an avalanche process close to criticality that can be detected by high sensitivity calorimetry, acoustic emission or magnetic (Barkhausen) noise in the case of magneto-structural transitions. Our statistical analysis rendered scale invariance and endogenous effects.

Finally, the same methodology was applied to the numerical results of metastable loops in the Random Field Ising Model (RFIM). A classification method of spanning avalanches was verified by means of MLEM. The RFIM doesn't manifest endogenous production of avalanches. Hence, criticality in avalanche dynamics may not be a sufficient condition to find this phenomenology.

En comptes de mostrar una evolució lineal i suau, molts sistemes físics reaccionen als estímuls externs en forma de dinàmica d'allaus. Al conduir externament un sistema dominat pel desordre fora de l'equilibri, l'evolució de les variables internes es produeix de forma local i no homogènia en processos col·lectius i instantanis que anomenem allaus.

Les dinàmiques d'allaus presenten sovint invariància d'escala associada a un fenomen de criticalitat. Les distribucions d'allaus crítiques són del tipus lleis de potència i estan determinades per exponents crítics, avaluats amb tècniques de Màxima Versemblança. Per poder determinar els extrems de la invariància d'escala, hem desenvolupat els anomenats Mapes d'Exponent per Màxima Versemblança.

L'estudi de les allaus com a esdeveniments en 'point processes' i l'avaluació d'efectes exògens i endògens és fonamental per entendre les dinàmiques d'allaus i estimar riscos. Hem corroborat la Llei d'Escala Unificada de temps d'espera en sistemes experimentals. El test de Bi ens permet determinar si un 'point process' conté efectes endògens i els podem ajustar un model de rèpliques si hi identifiquem certs patrons a través del mètode de la Seqüència de Repliques Mitjana. Amb l'ajut d'aquestes tècniques hem analitzat les propietats de les allaus generades en: - L'emissió acústica en processos de fallida mecànica sota compressió uniaxial de diversos materials porosos: vidre porós Vycor® (SiO2); mineral de goethita (FeO(OH)) i alúmina sinteritzada artificialment (Al2O3). - Les senyals calorimètriques l'emissió acústica i el soroll Barkhausen associat a les transformacions martensítiques de diferents aliatges (FePd, CuZnAl i NiMnGa). - El creixement de dominis en les simulacions numèriques de dinàmica metastable en el Random Field Ising Model (RFIM). Tots els processos experimentals exhibeixen estadístiques d'invariància d'escala en un cert rang de magnituds, i presenten, sovint, producció endògena d'allaus. En el cas de la compressió del Vycor®, la similitud amb les lleis estadístiques dels terratrèmols és notable. Els estudis sobre el RFIM han permès verificar un sistema de classificació d'allaus. No hem observat que el model sigui capaç de generar producció endògena d'allaus, posant en dubte que la criticalitat sigui condició suficient per trobar aquesta fenomenologia.