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, baptis
ed 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.
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.
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 m
anifestation 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.
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. The
reafter, 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.
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.
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 t
agging 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.
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 irregul
ar 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.
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 u
sers 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, academ
ic 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.
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 coast
lines, 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.
