PHYSICAL REVIEW E 96, 042122 (2017) Analysis of crackling noise using the maximum-likelihood method: Power-law mixing and exponential damping Ekhard K. H. Salje, 1 Antoni Planes, 2 and Eduard Vives 2 1 Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, United Kingdom 2 Departament de Física de la Matèria Condensada, Facultat de Física, Universitat de Barcelona, Martí i Franquès,1, E-08028 Barcelona, Catalonia (Received 13 July 2017; published 11 October 2017) Crackling noise can be initiated by competing or coexisting mechanisms. These mechanisms can combine to generate an approximate scale invariant distribution that contains two or more contributions. The overall distribution function can be analyzed, to a good approximation, using maximum-likelihood methods and assuming that it follows a power law although with nonuniversal exponents depending on a varying lower cutoff. We propose that such distributions are rather common and originate from a simple superposition of crackling noise distributions or exponential damping. DOI: 10.1103/PhysRevE.96.042122 I. INTRODUCTION Universality allows to transfer physical observations from one scenario to another. A typical example is the observation of crackling noise in a wide variety of materials [1]. Crackling noise is the manifestation of avalanches which are commonly observed in ferroic materials under electric or magnetic fields, in porous materials under compression, and even in natural earthquakes [2–6], to name a few. Technological research has focused on domain switching in piezoelectric, ferroelectric, ferroelastic, and magnetic materials, which usually involves fine structures on a nanometer scale that, on aggregate, constitute the macroscopic switch [7]. A typical example is the lateral movement of a ferroelectric domain wall, which is composed of moving atom-sized kinks in the wall [8]. Most macroscopic observations will identify only the lateral domain wall movement as a time averaged process, while the elementary step of the kink movement is visible only in high time-resolution experiments [9]. The collective kink move- ments form avalanches, which constitute crackling noise and are expected to follow universal rules [10]. Experimentally, however, most facilities for the observation of ferroic crackling noise are limited in frequency to some 100 Hz (rather than THz) and amplitudes of acoustic emission (AE) of atomic kink movements are extremely weak [11]. Universality would allow us to compare ferroic crackling noise with that of porous materials during compression [5,12–18]. The collapse of porous materials produces excellent data sets of crackling noise. The reason is that compression in porous materials occurs via long sequences of failure events, which are well separated in time. Their distributions of size, energy, duration, and time intervals between events span many more decades than ever observed in ferroic materials. The processes appear to occur without specific length and time scales over long intervals of the event sequences, which suggests criticality of collapse avalanches. The vastly better resolved behavior of the porous collapse is then a consequence of dynamical constraints imposed by the intrinsically inhomogeneous nature of this class of systems and jamming effects [18]. Almost as complete data sets exist for magnetization processes [19,20], martensitic transitions [21–23], plastic deformation in solids [24–27], and geological earthquakes [28,29]. Power-law distributions of energies, aftershocks, and waiting times have been reported in many cases, with statistical laws borrowed in name from seismic studies such as the Gutenberg-Richter’s law, Omoris law, or the universal scaling law [5]. The transfer of results on porous collapse to ferroic materials depends on the validity of universality of avalanche dynamics and the equivalence of the relevant universality classes. While we expect to have very few universality classes, or even only one mean field critical point, experimental evidence appears to indicate the opposite. Even during porous collapse, critical parameters seem to vary greatly even in very similar materials and collapse mechanisms. The reasons behind the observed dispersion of critical exponents or lack of universality can be diverse: existence of experimental limitations associated with the detectors that deform the critical power-law distributions, limited sample size, intrin- sic damping factors, or mixture of several crackling noise mechanisms. In this work, we investigate how the maximum-likelihood (ML) fitting method, using a varying lower cutoff, might be used to investigate whether the recorded data set corresponds to a unique crackling noise source, whether it is distorted by exponential damping factors or whether it corresponds to a mixture of crackling noise mechanisms. We will focus in the analysis of the most important quantity which is the probability density g(E) that accounts for the distributions of energies of avalanche events. Similar analysis can be performed for distributions of other avalanche properties. At criticality it is expected to follow a power-law behavior g(E)dE = (ǫ − 1)  E E min  −ǫ dE E min , (1) where energy exponents ǫ vary between 1 and 2.5 (and even greater values have been reported occasionally). Note that the scale E min is required for the distribution to be normalized. Experimental observations include the collapse of berlinite where an overall exponent of 1.8 was reported [14], while collapse events near the final instability point show a well defined exponent of 1.4. The exponent was also slightly 2470-0045/2017/96(4)/042122(9) 042122-1 ©2017 American Physical Society