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The traction evolution is a fundamental ingredient to model the dynamics of an earthquake rupture which ultimately controls, during the coseismic phase, the energy release, the stress redistribution and the consequent excitation of seismic waves. In the present paper we explore the use of the friction behavior derived from laboratory shear experiments performed on granular materials at low normal stress. We find that the rheological properties emerging from these laboratory experiments can not be described in terms of preexisting governing models already presented in literature; our results indicate that neither rate-and state-dependent friction laws nor nonlinear slip-dependent models, commonly adopted for modeling earthquake ruptures, are able to capture all the features of the experimental data. Then, by exploiting a novel numerical approach, we directly incorporate the laboratory data into a code to simulate the fully dynamic propagation of a 3-D slip failure. We demonstrate that the rheology of the granular material, imposed as fault boundary condition, is dynamically consistent. Indeed, it is able to reproduce the basic features of a crustal earthquake, spontaneously accelerating up to some terminal rupture speed, both sub- and supershear.

Data from experiments and theoretical arguments are the two pillars sustaining the job of modelling physical systems through inference. In order to solve the inference problem, the data should satisfy certain conditions that depend also upon the particular questions addressed in a research. Here we focus on the characterization of systems in terms of a distance from equilibrium, typically the entropy production (time-reversal asymmetry) or the violation of the Kubo fluctuation-dissipation relation. We show how general, counter-intuitive and negative for inference, is the problem of the impossibility to estimate the distance from equilibrium using a series of scalar data which have a Gaussian statistics. This impossibility occurs also when the data are correlated in time, and that is the most interesting case because it usually stems from a multi-dimensional linear Markovian system where there are many time-scales associated to different variables and, possibly, thermal baths. Observing a single variable (or a linear combination of variables) results in a one-dimensional process which is always indistinguishable from an equilibrium one (unless a perturbation-response experiment is available). In a setting where only data analysis (and not new experiments) is allowed, we propose - as a way out - the combined use of different series of data acquired with different parameters. This strategy works when there is a sufficient knowledge of the connection between experimental parameters and model parameters. We also briefly discuss how such results emerge, similarly, in the context of Markov chains within certain coarse-graining schemes. Our conclusion is that the distance from equilibrium is related to quite a fine knowledge of the full phase space, and therefore typically hard to approximate in real experiments., Comment: 23 pages - 8 figures

Several years ago, in the context of the physics of hysteresis in magnetic materials, a simple stochastic model has been introduced: the ABBM model. Later, the ABBM model has been advocated as a paradigm for a broad class of diverse phenomena, baptised "crackling noise phenomena". The model reproduces many statistical features of such intermittent signals, as the statistics of burst (or avalanche) durations and sizes, with their power law exponents that would characterise the dynamics as critical. Beyond such "critical exponents", the measure of the average shape of the avalanche has also been proposed. Here, the exact calculation of average and fluctuations of the avalanche shape for the ABBM model is presented, showing that its normalised shape is independent from the external drive. Moreover, average and fluctuations of the multi-avalanche shape, that is a sequence of avalanches of fixed total duration, is also computed. Surprisingly, the two quantities (avalanche and multi-avalanche normalised shapes) are identical. This result is obtained using the exact solution of the ABBM model, obtained leveraging the equivalence with the Cox-Ingersoll-Ross process (CIR), through an exact "time change". A presentation of this and other known exact results is provided: notably the correspondence of the ABBM/CIR model with the generalised Bessel process, describing the dynamics of the modulus of the multi dimensional Ornstein-Uhlenbeck process. As a consequence, the correspondence between the excursion (avalanche) and bridge (multi-avalanche) shape distributions, turns to apply to all the aforementioned stochastic processes. In simple words: considering the distance from the origin of such diffusive particles, the (normalised) average shape (and fluctuations) of its trajectory until a return in a time T is the same, whether it has returned before T or not., Comment: 30 pages, 7 figures. Revised version to appear in Jstat

In the context of stochastic thermodynamics, a minimal model for non equilibrium steady states has been recently proposed: the Brownian Gyrator (BG). It describes the stochastic overdamped motion of a particle in a two dimensional harmonic potential, as in the classic Ornstein-Uhlenbeck process, but considering the simultaneous presence of two independent thermal baths. When the two baths have different temperatures, the steady BG exhibits a rotating current, a clear signature of non equilibrium dynamics. Here, we consider a time-dependent potential, and we apply a reverse-engineering approach to derive exactly the required protocol to switch from an initial steady state to a final steady state in a finite time $\tau$. The protocol can be built by first choosing an arbitrary quasi-static counterpart - with few constraints - and then adding a finite-time contribution which only depends upon the chosen quasi-static form and which is of order $1/\tau$. We also get a condition for transformations which - in finite time - conserve internal energy, useful for applications such as the design of microscopic thermal engines. Our study extends finite-time stochastic thermodynamics to transformations connecting non-equilibrium steady states., Comment: 5 pages, 1 figure plus supplementary material 10 pages, 2 figures. To appear in PRE Rapid communications

The way granular materials response to an applied shear stress is of the utmost relevance to both human activities and natural environment. One of the their most intriguing and less understood behavior, is the stick-instability, whose most dramatic manifestation are earthquakes, ultimately governed by the dynamics of rocks and debris jammed within the fault gauge. Many of the features of earthquakes, i.e. intermittency, broad times and energy scale involved, are mimicked by a very simple experimental set-up, where small beads of glass under load are slowly sheared by an elastic medium. Analyzing data from long lasting experiments, we identify a critical dynamical regime, that can be related to known theoretical models used for "crackling-noise" phenomena. In particular, we focus on the average shape of the slip velocity, observing a "breakdown of scaling": while small slips show a self-similar shape, large does not, in a way that suggests the presence of subtle inertial effects within the granular system. In order to characterise the crossover between the two regimes, we investigate the frictional response of the system, which we trat as a stochastic quantity. Computing different averages, we evidence a weakening effect, whose Stribeck threshold velocity can be related to the aforementioned breaking of scaling., Comment: 11 pages, 14 figures

We review a few representative examples of granular experiments or models where phase separation, accompanied by domain coarsening, is a relevant phenomenon. We first elucidate the intrinsic non-equilibrium, or athermal, nature of granular media. Thereafter, dilute systems, the so-called "granular gases" are discussed: idealized kinetic models, such as the gas of inelastic hard spheres in the cooling regime, are the optimal playground to study the slow growth of correlated structures, e.g. shear patterns, vortices and clusters. In fluidized experiments, liquid-gas or solid-gas separations have been observed. In the case of monolayers of particles, phase coexistence and coarsening appear in several different setups, with mechanical or electrostatic energy input. Phenomenological models describe, even quantitatively, several experimental measures, both for the coarsening dynamics and for the dynamic transition between different granular phases. The origin of the underlying bistability is in general related to negative compressibility from granular hydrodynamics computations, even if the understanding of the mechanism is far from complete. A relevant problem, with important industrial applications, is related to the demixing or segregation of mixtures, for instance in rotating tumblers or on horizontally vibrated plates. Finally, the problem of compaction of highly dense granular materials, which has many important applications, is usually described in terms of coarsening dynamics: there, bubbles of mis-aligned grains evaporate, allowing the coalescence of optimally arranged islands and a progressive reduction of total occupied volume., Comment: 12 pages, 10 figures, to appear in "Dynamics of coarsening" Comptes Rendus Physique special issue, https://sites.google.com/site/ppoliti/crp-special-issue

The size of large cliff failures may be described in several ways, for instance considering the horizontal eroded area at the cliff top and the maximum local retreat of the coastline. Field studies suggest that, for large failures, the frequencies of these two quantities decrease as power laws of the respective magnitudes, defining two different decay exponents. Moreover, the horizontal area increases as a power law of the maximum local retreat, identifying a third exponent. Such observation suggests that the geometry of cliff failures are statistically similar for different magnitudes. Power laws are familiar in the physics of critical systems. The corresponding exponents satisfy precise relations and are proven to be universal features, common to very different systems. Following the approach typical of statistical physics, we propose a "scaling hypothesis" resulting in a relation between the three above exponents: there is a precise, mathematical relation between the distributions of magnitudes of erosion events and their geometry. Beyond its theoretical value, such relation could be useful for the validation of field catalogs analysis. Pushing the statistical physics approach further, we develop a numerical model of marine erosion that reproduces the observed failure statistics. Despite the minimality of the model, the exponents resulting from extensive numerical simulations fairly agree with those measured on the field. These results suggest that the mathematical theory of percolation, which lies behind our simple model, can possibly be used as a guide to decipher the physics of rocky coast erosion and could provide precise predictions to the statistics of cliff collapses., Comment: 20 pages, 13 figures, 1 table. To appear in Earth Surface Processes and Lanforms (Rocky Coast special issue)

Folksonomies provide a rich source of data to study social patterns taking place on the World Wide Web. Here we study the temporal patterns of users' tagging activity. We show that the statistical properties of inter-arrival times between subsequent tagging events cannot be explained without taking into account correlation in users' behaviors. This shows that social interaction in collaborative tagging communities shapes the evolution of folksonomies. A consensus formation process involving the usage of a small number of tags for a given resources is observed through a numerical and analytical analysis of some well-known folksonomy datasets., Comment: 6 pages, 10 figures; Proceedings of the 20th ACM conference on Hypertext and hypermedia, 2009

We discuss various situations where the formation of rocky coast morphology can be attributed to the retro-action of the coast morphology itself on the erosive power of the sea. Destroying the weaker elements of the coast, erosion can creates irregular seashores. In turn, the geometrical irregularity participates in the damping of sea-waves, decreasing their erosive power. There may then exist a mutual self-stabilization of the wave amplitude together with the irregular morphology of the coast. A simple model of this type of stabilization is discussed. The resulting coastline morphologies are diverse, depending mainly on the morphology/damping coupling. In the limit case of weak coupling, the process spontaneously builds fractal morphologies with a dimension close to 4/3. This provides a direct connection between the coastal erosion problem and the theory of percolation. For strong coupling, rugged but non-fractal coasts may emerge during the erosion process, and we investigate a geometrical characterization in these cases. The model is minimal, but can be extended to take into account heterogeneity in the rock lithology and various initial conditions. This allows to mimic coastline complexity, well beyond simple fractality. Our results suggest that the irregular morphology of coastlines as well as the stochastic nature of erosion are deeply connected with the critical aspects of percolation phenomena., Comment: 34 pages, 15 figures. arXiv admin note: substantial text overlap with arXiv:cond-mat/0311509

The enormous increase of popularity and use of the WWW has led in the recent years to important changes in the ways people communicate. An interesting example of this fact is provided by the now very popular social annotation systems, through which users annotate resources (such as web pages or digital photographs) with text keywords dubbed tags. Understanding the rich emerging structures resulting from the uncoordinated actions of users calls for an interdisciplinary effort. In particular concepts borrowed from statistical physics, such as random walks, and the complex networks framework, can effectively contribute to the mathematical modeling of social annotation systems. Here we show that the process of social annotation can be seen as a collective but uncoordinated exploration of an underlying semantic space, pictured as a graph, through a series of random walks. This modeling framework reproduces several aspects, so far unexplained, of social annotation, among which the peculiar growth of the size of the vocabulary used by the community and its complex network structure that represents an externalization of semantic structures grounded in cognition and typically hard to access.

A distributed classification paradigm known as collaborative tagging has been widely adopted in new Web applications designed to manage and share online resources. Users of these applications organize resources (Web pages, digital photographs, academic papers) by associating with them freely chosen text labels, or tags. Here we leverage the social aspects of collaborative tagging and introduce a notion of resource distance based on the collective tagging activity of users. We collect data from a popular system and perform experiments showing that our definition of distance can be used to build a weighted network of resources with a detectable community structure. We show that this community structure clearly exposes the semantic relations among resources. The communities of resources that we observe are a genuinely emergent feature, resulting from the uncoordinated activity of a large number of users, and their detection paves the way for mapping emergent semantics in social tagging systems., Comment: 14 pages, 8 figures

Geometrical properties of landscapes result from the geological processes that have acted through time. The quantitative analysis of natural relief represents an objective form of aiding in the visual interpretation of landscapes, as studies on coastlines, river networks, and global topography, have shown. Still, an open question is whether a clear relationship between the quantitative properties of landscapes and the dominant geomorphologic processes that originate them can be established. In this contribution, we show that the geometry of topographic isolines is an appropriate observable to help disentangle such a relationship. A fractal analysis of terrestrial isolines yields a clear identification of trenches and abyssal plains, differentiates oceanic ridges from continental slopes and platforms, localizes coastlines and river systems, and isolates areas at high elevation (or latitude) subjected to the erosive action of ice. The study of the geometrical properties of the lunar landscape supports the existence of a correspondence between principal geomorphic processes and landforms. Our analysis can be easily applied to other planetary bodies., Comment: 21 pages, 7 figures

We experimentally investigate the response of a sheared granular medium in a Couette geometry. The apparatus exhibits the expected stick-slip motion and we probe it in the very intermittent regime resulting from low driving. Statistical analysis of the dynamic fluctuations reveals notable regularities. We observe a possible stability property for the torque distribution, reminiscent of the stability of Gaussian independent variables. In this case, however, the variables are correlated and the distribution is skewed. Moreover, the whole dynamical intermittent regime can be described with a simple stochastic model, finding good quantitative agreement with the experimental data. Interestingly, a similar model has been previously introduced in the study of magnetic domain wall motion, a source of Barkhausen noise. Our study suggests interesting connections between different complex phenomena and reveals some unexpected features that remain to be explained., Comment: 10 pages, 7 figures, statphys23 conference proceeding

We analyze a large-scale snapshot of del.icio.us and investigate how the number of different tags in the system grows as a function of a suitably defined notion of time. We study the temporal evolution of the global vocabulary size, i.e. the number of distinct tags in the entire system, as well as the evolution of local vocabularies, that is the growth of the number of distinct tags used in the context of a given resource or user. In both cases, we find power-law behaviors with exponents smaller than one. Surprisingly, the observed growth behaviors are remarkably regular throughout the entire history of the system and across very different resources being bookmarked. Similar sub-linear laws of growth have been observed in written text, and this qualitative universality calls for an explanation and points in the direction of non-trivial cognitive processes in the complex interaction patterns characterizing collaborative tagging., Comment: 6 pages, 7 figures

We present a two-dimensional theoretical model for the slow chemical corrosion of a thin film of a disordered solid by suitable etching solutions. This model explain different experimental results showing that the corrosion stops spontaneously in a situation in which the concentration of the etchant is still finite while the corrosion surface develops clear fractal features. We show that these properties are strictly related to the percolation theory, and in particular to its behavior around the critical point. This task is accomplished both by a direct analysis in terms of a self-organized version of the Gradient Percolation model and by field theoretical arguments., Comment: 7 pages, 3 figures

We present results from a series of experiments on a granular medium sheared in a Couette geometry and show that their statistical properties can be computed in a quantitative way from the assumption that the resultant from the set of forces acting in the system performs a Brownian motion. The same assumption has been utilised, with success, to describe other phenomena, such as the Barkhausen effect in ferromagnets, and so the scheme suggests itself as a more general description of a wider class of driven instabilities., Comment: 4 pages, 5 figures and 1 table

We compute the average shape of trajectories of some one--dimensional stochastic processes x(t) in the (t,x) plane during an excursion, i.e. between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independently from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces non trivial asymmetric shapes, which are studied numerically., Comment: 16 pages, 16 figures; accepted for publication in Phys. Rev. E

We study the formation and the dynamics of correlations in the velocity field for 1D and 2D cooling granular gases with the assumption of negligible density fluctuations (``Homogeneous Velocity-correlated Cooling State'', HVCS). It is shown that the predictions of mean field models fail when velocity fluctuations become important. The study of correlations is done by means of molecular dynamics and introducing an Inelastic Lattice Maxwell Models. This lattice model is able to reproduce all the properties of the Homogeneous Cooling State and several features of the HVCS. Moreover it allows very precise measurements of structure functions and other crucial statistical indicators. The study suggests that both the 1D and the 2D dynamics of the velocity field are compatible with a diffusive dynamics at large scale with a more complex behavior at small scale. In 2D the issue of scale separation, which is of interest in the context of kinetic theories, is addressed., Comment: 24 pages, 16 figures, conference proceeding

Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the dimension of the so-called "hull" or external perimeter of the incipient percolation cluster, controls the width and length of gradient percolation frontiers whatever the gradient magnitude. This behavior is extended to previous model studies of etching by a finite volume of etching solution in contact with a disordered solid. In such a model, the dynamics stop spontaneously on an equilibrium self-similar surface similar to the fractal frontier of gradient percolation. It is shown that the power laws describing the system geometry involves also the fractal dimension of the percolation hull, whatever the value of the dynamically generated gradient, i.e. even for a non-fractal frontier. The comparison between numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law valid from the smallest possible scale up to infinity. These results suggest the possible existence of an underlying conservation law, relating the length and the statistical width of percolation gradient frontiers., Comment: 20 pages, 14 figures, to appear on "Proceedings of Symposia in Pure Mathematics" by "American Mathematical Society"

We study the average shape of a fluctuation of a time series x(t), that is the average value _T before x(t) first returns, at time T, to its initial value x(0). For large classes of stochastic processes we find that a scaling law of the form _T = T^\alpha f(t/T) is obeyed. The scaling function f(s) is to a large extent independent of the details of the single increment distribution, while it encodes relevant statistical information on the presence and nature of temporal correlations in the process. We discuss the relevance of these results for Barkhausen noise in magnetic systems., Comment: 5 pages, 5 figures, accepted for publication in Phys. Rev. Lett

We present a general scheme to calculate within the independent interval approximation generalized (level-dependent) persistence properties for processes having a finite density of zero-crossings. Our results are especially relevant for the diffusion equation evolving from random initial conditions, one of the simplest coarsening systems. Exact results are obtained in certain limits, and rely on a new method to deal with constrained multiplicative processes. An excellent agreement of our analytical predictions with direct numerical simulations of the diffusion equation is found., Comment: 21 pages, 4 figures, to appear in Journal of Physics A

We present a numerical study of the non-equilibrium dynamics of the Sherrington-Kirkpatrick model. We analize the overlap distribution between the configurations visited at the time t and in particular its scaling behaviour with the size of the system. We find two different non-equilibrium dynamical regimes. The first is a proper Out of Equilibrium Regime, that is the relevant regime for the dynamics of an infinite system. The second is an Intermediate Regime that separates the Out of Equilibrium Regime from equilibrium. After studying the crossover beetween the two regimes, we focus on the Out of Equilibrium Regime and we reveal some of the geometrical features of phase space. According to recent analitical work, we find that the asymptotic out of equilibrium energy density is the equilibrium one. The same happens to the staggered-magnetisation, suggesting there is a deep geometrical similarity between equilibrium and out of equilibrium configurations. A procedure, that we call clonation, shows that the dynamics follows sort of canyons, as opposed to a system of independent traps., Comment: 20 pages, LaTeX2e, 11 figures, epsf.sty. Also available at ftp://hpteo.roma1.infn.it/pub/preprints/andreab/sk-offeq.tar.gz