Hysteresis, avalanches, and disorder-induced critical scaling: A renormalization-group approach

Statistics of static avalanches in a random pinning landscape

We study the minimum-energy configuration of a d -dimensional elastic interface in a random potential tied to a harmonic spring. As a function of the spring position, the center of mass of the interface changes in discrete jumps, also called shocks or "static avalanches." We obtain analytically the distribution of avalanche sizes and its cumulants within an =4-d expansion from a tree and one-loop resummation using functional renormalization. This is compared with exact numerical minimizations of interface energies for random-field disorder in d=2,3 . Connections to dynamic avalanches are mentioned.

Size distributions of shocks and static avalanches from the functional renormalization group

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) approximately S;{-3/2}e;{-S4S_{m}} , where S_{m} is a large-scale cutoff, in some cases calculable. Resumming all one-loop contributions, we find P(S) approximately S;{-tau}exp(C(SS_{m});{1/2}-B/4(S/S_{m});{delta}) , where B , C , delta , and tau are obtained to first order in =4-d . Our result is consistent to O() with the relation tau=tau_{zeta}:=2-2/d+zeta , where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field, and random-periodic disorders. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/d+zeta to O(=2-d) . We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the relation tau=tau_{zeta} be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyperplane of codimension one is in mean field (valid close to and above d=4 ) given by P(S) approximately K_{13}(S)S , where K is the Bessel- K function, thus tau_{hyperplane}=4/3 .

Driven particle in a random landscape: disorder correlator, avalanche distribution, and extreme value statistics of records

We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .

Driving-induced crossover: from classical criticality to self-organized criticality

We propose a spin model with quenched disorder which exhibits in slow driving two drastically different types of critical nonequilibrium steady states. One of them corresponds to classical criticality requiring fine-tuning of the disorder. The other is a self-organized criticality which is insensitive to disorder. The crossover between the two types of criticality is determined by the mode of driving. As one moves from "soft" to "hard" driving the universality class of the critical point changes from a classical order-disorder to a quenched Edwards-Wilkinson universality class. The model is viewed as prototypical for a broad class of physical phenomena ranging from magnetism to earthquakes.

Critical fluctuations of time-dependent magnetization in a random-field Ising model

Cooperative behaviors near the disorder-induced critical point in a random-field Ising model are numerically investigated by analyzing time-dependent magnetization in ordering processes from a special initial condition. We find that the intensity of fluctuations of time-dependent magnetization, chi(t) , attains a maximum value at a time t=tau in a normal phase and that chi(tau) and tau exhibit divergences near the disorder-induced critical point. Furthermore, spin configurations around the time tau are characterized by a length scale, which also exhibits a divergence near the critical point. We estimate the critical exponents that characterize these power-law divergences by using a finite-size scaling method.

Loss separation for dynamic hysteresis in ferromagnetic thin films

We develop a theory for dynamic hysteresis in ferromagnetic thin films, on the basis of the phenomenological principle of loss separation. We observe that, remarkably, the theory of loss separation, originally derived for bulk metallic materials, is applicable to disordered magnetic systems under fairly general conditions regardless of the particular damping mechanism. We confirm our theory both by numerical simulations of a driven random-field Ising model, and by reexamining several experimental data reported in the literature on dynamic hysteresis in thin films. All the experiments examined and the simulations find a natural interpretation in terms of loss separation. The power losses' dependence on the driving field rate predicted by our theory fits satisfactorily all the data in the entire frequency range, thus reconciling the apparent lack of universality observed in different materials.

Gas adsorption and desorption in silica aerogels: a theoretical study of scattering properties

We present a numerical study of the structural correlations associated with gas adsorption and desorption in silica aerogels in order to provide a theoretical interpretation of scattering experiments. Following our earlier work, we use a coarse-grained lattice-gas description and determine the nonequilibrium behavior of the adsorbed gas within a local mean-field analysis. We focus on the differences between the adsorption and desorption mechanisms and their signature in the fluid-fluid and gel-fluid structure factors as a function of temperature. At low temperature, but still in the regime where the isotherms are continuous, we find that the adsorbed fluid density, during both filling and draining, is correlated over distances that may be much larger than the gel correlation length. In particular, extended fractal correlations may occur during desorption, indicating the existence of a ramified cluster of vapor filled cavities. This also induces an important increase of the scattering intensity at small wave vectors. The similarity and differences with the scattering of fluids in other porous solids such as Vycor are discussed.

Instanton analysis of hysteresis in the three-dimensional random-field Ising model

We study the magnetic hysteresis in the random-field Ising model in 3D. We discuss the disorder dependence of the coercive field Hc, and obtain an analytical description of the smooth part of the hysteresis below and above Hc, by identifying the disorder configurations (instantons) that are the most probable to trigger local avalanches. We estimate the critical disorder strength at which the hysteresis curve becomes continuous. From an instanton analysis at zero field we obtain a description of local two-level systems in the ferromagnetic phase.

Memory of the unjamming transition during cyclic tiltings of a granular pile

Discrete numerical simulations are performed to study the evolution of the microstructure and the response of a granular packing during successive loading-unloading cycles, consisting of quasistatic rotations in the gravity field between opposite inclination angles. We show that internal variables--e.g., stress and fabric of the pile--exhibit hysteresis during these cycles due to the exploration of different metastable configurations. Interestingly, the hysteretic behavior of the pile strongly depends on the maximal inclination of the cycles, giving evidence of the irreversible modifications of the pile state occurring close to the unjamming transition. More specifically, we show that for cycles with maximal inclination larger than the repose angle, the weak-contact network carries the memory of the unjamming transition. These results demonstrate the relevance of a two-phase description--strong- and weak-contact networks--for a granular system, as soon as it has approached the unjamming transition.

Helium condensation in aerogel: avalanches and disorder-induced phase transition

We present a detailed numerical study of the elementary condensation events (avalanches) associated to the adsorption of in silica aerogels. We use a coarse-grained lattice-gas description and determine the nonequilibrium behavior of the adsorbed gas within a local mean-field analysis, neglecting thermal fluctuations and activated processes. We investigate the statistical properties of the avalanches, such as their number, size and shape along the adsorption isotherms as a function of gel porosity, temperature, and chemical potential. Our calculations predict the existence of a line of critical points in the temperature-porosity diagram where the avalanche size distribution displays a power-law behavior and the adsorption isotherms have a universal scaling form. The estimated critical exponents seem compatible with those of the field-driven random field Ising model at zero temperature.

Friction versus texture at the approach of a granular avalanche

We perform an analysis of the granular texture of a granular bed close to stability limit. Our analysis is based on a unique criterion of friction mobilization in a simulated two-dimensional packing. In this way, we recover the bimodal character of granular texture and the coexistence of weak and strong phases in the sense of distinct contacts populations. Moreover, we show the existence of a well-defined subset of contacts within the weak contact network. These contacts are characterized by their important friction and form a highly coherent population in terms of fabric. They play an antagonistic role with respect to force chains. Thus, we are able to discriminate between incoherent contacts and coherent contacts in the weak phase and to specify the role that the latter plays in the destabilization process.

Breakdown of disordered media by surface loads

We model an interface layer connecting two parts of a solid body by N parallel elastic springs connecting two rigid blocks. We load the system by a shear force acting on the top side. The springs have equal stiffness but are ruptured randomly when the load reaches a critical value. For the considered system, we calculate the shear modulus G as a function of the order parameter phi describing the state of damage, and also the "spalled"; material (burst) size distribution. In particular, we evaluate the relation between the damage parameter and the applied force and explore the behavior in the vicinity of material breakdown. Using this simple model for material breakdown, we show that damage, caused by applied shear forces, is analogous to a first-order phase transition. The scaling behavior of G with phi is explored analytically and numerically, close to phi=0 and phi=1 and in the vicinity of phi(c), when the shear load is close to but below the threshold force that causes material breakdown. Our model calculation represents a first approximation of a system subject to wear induced loads.

Monte Carlo simulations of ferromagnetism in p-Cd1-xMnxTe quantum wells

Monte Carlo simulations, in which the Schrödinger equation is solved at each Monte Carlo sweep, are employed to assess the influence of magnetization fluctuations, short-range antiferromagnetic interactions, disorder, magnetic polaron formation, and spin-Peierls instability on the carrier-mediated Ising ferromagnetism in two-dimensional electronic systems. The determined critical temperature and hysteresis are affected in a nontrivial way by the antiferromagnetic interactions. The findings explain striking experimental results for modulation-doped p-Cd1-xMnxTe quantum wells.

Phase transitions in a disordered system in and out of equilibrium

The equilibrium and nonequilibrium disorder-induced phase transitions are compared in the random-field Ising model. We identify in the demagnetized state the correct nonequilibrium hysteretic counterpart of the T=0 ground state, and present evidence of universality. Numerical simulations in d=3 indicate that exponents and scaling functions coincide, while the location of the critical point differs, as corroborated by exact results for the Bethe lattice. These results are of relevance for optimization, and for the generic question of universality in the presence of disorder.

Local mean-field study of capillary condensation in silica aerogels

We apply local mean-field (i.e., density functional) theory to a lattice model of a fluid in contact with a dilute, disordered gel network. The gel structure is described by a diffusion-limited cluster aggregation model. We focus on the influence of porosity on both the hysteretic and the equilibrium behavior of the fluid as one varies the chemical potential at low temperature. We show that the shape of the hysteresis loop changes from smooth to rectangular as the porosity increases and that this change is associated with disorder-induced out-of-equilibrium phase transitions that differ in adsorption and in desorption. Our results provide insight in the behavior of 4He in silica aerogels.

Depinning with dynamic stress overshoots: a hybrid of critical and pseudohysteretic behavior

A model of an elastic manifold driven through a random medium by an applied force F is introduced and studied. The focus is on the effects of inertia and elastic waves, in particular stress overshoots in which motion of one segment of the manifold causes a temporary stress on its neighboring segments in addition to the static stress. Such stress overshoots decrease the critical force for depinning and make the depinning transition hysteretic with static and pinned configurations coexisting with the steadily moving phase for a range of F. We find that the steady-state velocity of the moving phase is, nevertheless, history independent and the critical behavior as the force is decreased is in the same universality class as in the absence of stress overshoots-the dissipative limit in which hysteresis cannot occur and theoretical analysis has been possible. To reach this conclusion, finite-size scaling analyses have been performed and a variety of quantities studied, including velocities, roughnesses, distributions of critical forces, and universal amplitude ratios. If the force is increased slowly from zero, the behavior is complicated with a spectrum of avalanche sizes occurring that seems to be quite different from the dissipative limit. Related behavior is seen as the force is increased back up again to restart the motion of samples that have been stopped from the moving phase. The restarting process itself involves both fractal-like and bubblelike nucleation. Hysteresis loops in small- and intermediate-size samples can be understood in terms of a depletion layer caused by the stress overshoots. Surprisingly, in the limit of very large samples the hysteresis loops vanish. Although complicated crossovers complicate the analysis, we argue that the underlying universality class governing this pseudohysteresis and avalanches is again that of the apparently very different dissipative limit. But there are history dependent amplitudes-associated with the depletion layer-that cause striking differences over wide ranges of length scales. Consequences of this picture for the statistics and dynamics of earthquakes on geological faults are briefly discussed.

Precursory dynamics in threshold systems

A precursory dynamics, motivated by the analysis of recent experiments on solid-on-solid friction, is introduced in a continuous cellular automaton that mimics the physics of earthquake source processes. The resulting system of equations for the interevent cycle can be decoupled and yields an analytical solution in the mean-field limit, exhibiting a smoothing effect of the dynamics on the stress field. Simulation results show the resulting departure from scaling at the large-event end of the frequency distribution, and support claims that the field leakage may parametrize the superposition of scaling and characteristic regimes observed in real earthquake faults.

Emergence of hysteresis in a network of nonhysteretic agents with continuous responses

We model the emergence of hysteresis as a collective behavior in a lattice network of individually nonhysteretic agents with continuous responses. The emergence, which depends on the agent's characteristic response to an external input, is optimized at a finite interaction size of the Moore neighborhood. The network also exhibits rich behavior including power-law variations, chaos, and system saturation. Our model is used to describe the hysteresis and high-frequency fluctuations in the oxygen isotope variation ( delta18O) data of ice (Greenland Ice Core Project) that exhibit the Younger Dryas cold event.

Depinning with dynamic stress overshoots: mean field theory

An infinite-range model of an elastic manifold pulled through a random potential by a force F is analyzed focusing on inertial effects. When the inertial parameter M is small, there is a continuous depinning transition from a small- F static phase to a large- F moving phase. When M is increased to M(c), a novel tricritical point occurs. For M>M(c), the depinning transition becomes discontinuous with hysteresis. Yet, the distribution of discrete "avalanche"-like events as the force is increased in the static phase for M>M(c) has an unusual mixture of first-order-like and critical features. The results may be relevant for the onset of crack propagation and for dynamics of geological faults.

Capillary condensation in disordered porous materials: hysteresis versus equilibrium behavior

We study the interplay between hysteresis and equilibrium behavior in capillary condensation of fluids in mesoporous disordered materials via a mean-field density functional theory of a disordered lattice-gas model. The approach reproduces all major features observed experimentally. We show that the simple van der Waals picture of metastability fails due to the appearance of a complex free-energy landscape with a large number of metastable states. In particular, hysteresis can occur both with and without an underlying equilibrium transition, and thermodynamic consistency is not satisfied along the hysteresis loop.

Crackling noise

Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behaviour over a huge range of sizes, their behaviour is likely to be independent of microscopic and macroscopic details, and progress can be made by the use of simple models. The fact that these models and real systems can share the same behaviour on many scales is called universality. We illustrate these ideas by using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses, and we highlight some continuing challenges in this still-evolving field.

Anomalous scaling in depinning transitions

It is demonstrated that the renormalization group (RG) flows of depinning transitions do not depend on whether the driving force or the system velocity is kept constant. This allows for a comparison between RG results and corresponding self-organized critical models. However, close to the critical point, scaling functions cross over to forms that can have singular behavior not seen in equilibrium thermal phase transitions. These can be different for the constant force and constant velocity driving modes, leading to different apparent critical exponents. This is illustrated by comparing extremal dynamics for interface depinning with RG results, deriving the change in apparent exponents. Thus, care has to be exercised in such comparisons.

Exact solution of return hysteresis loops in a one-dimensional random-field ising model at zero temperature

Minor hysteresis loops within the main loop are obtained exactly in the one-dimensional ferromagnetic random-field Ising model at zero temperature. Numerical simulations of the model show excellent agreement with the exact results.

Dynamics of a domain wall in soft-magnetic materials: barkhausen effect and relation with sandpile models

The CZDE model [P. Cizeau, S Zapperi, G. Durin, and H. E. Stanley, Phys. Rev. Lett. 79, 4669 (1997)] for the dynamics of a domain wall in soft-magnetic materials is investigated. The equation of motion for the domain wall is reduced to a dimensionless form where the control parameters are clearly identified. In this way we show that in soft-magnetic materials with low anisotropies the noise can be approximated by a columnar disorder, and perturbation theory gives a good estimate of the avalanche exponents. Moreover, the resulting exponents are found to be identical to those obtained for directed Abelian sandpile models. The analogies and differences with these models and the question of self-organized criticality in the Barkhausen effect are discussed.

Avalanches in breakdown and fracture processes

We investigate the breakdown of disordered networks under the action of an increasing external-mechanical or electrical-force. We perform a mean-field analysis and estimate scaling exponents for the approach to the instability. By simulating two-dimensional models of electric breakdown and fracture we observe that the breakdown is preceded by avalanche events. The avalanches can be described by scaling laws, and the estimated values of the exponents are consistent with those found in mean-field theory. The breakdown point is characterized by a discontinuity in the macroscopic properties of the material, such as conductivity or elasticity, indicative of a first-order transition. The scaling laws suggest an analogy with the behavior expected in spinodal nucleation.