Crackling Noise

Abstracted from: "Crackling Noise", Nature 410, 242-250 (2001).

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 behavior over many decades of sizes, their behavior is likely independent of microscopic and macroscopic details, and progress can be made by the use of very simple models.

Figure 1: The Earth Crackles. (a) Time history of radiated energy from earthquakes throughout all of 1995. The earth responds to the slow strains imposed by continental drift through a series of earthquakes: impulsive events well separated in space and time. This time series, when sped up, sounds remarkably like the crackling noise of paper, magnets, and Rice Krispies©. (b) Histogram of number of earthquakes in 1995 as function of their magnitude (or, alternatively, their energy release). Earthquakes come in a wide range of sizes, from unnoticeable trembles to catastrophic events. The smaller earthquakes are much more common: the number of events of a given size forms a power law called the Gutenberg-Richter law. (Earthquake magnitude scales with the logarithm of the strength of the earthquake, e.g. its radiated energy. On a log-log plot of number vs. radiated energy, a power law is a straight line, as we observe in the plotted histogram.) One would hope that such a simple law should have an elegant explanation.

Many systems crackle; when pushed slowly, they respond with discrete events of a variety of sizes. The earth responds with violent and intermittent earthquakes as two tectonic plates rub past one another (see figure 1). A piece of paper (or a candy wrapper at the movies) emits intermittent, sharp noises as it is slowly crumpled or rumpled. (Try it: preferably not with this page.) A magnetic material in a changing external field magnetizes in a series of jumps (figure 2). These individual events span many orders of magnitude in size — indeed, the distribution of sizes forms a power law with no characteristic size scale. In the past few years, scientists have been making rapid progress in developing models and theories for understanding this sort of scale-invariant behavior in driven, nonlinear, dynamical systems.

Figure 2: One jump, or avalanche, in our model for crackling noise in magnets. The first spins to flip are colored blue, and the last pink. Notice the fractal structure: the avalanche is rough on all time scales.

Researchers have studied many systems that crackle. Simple models have been developed to study bubbles rearranging in foams as they are sheared, biological extinctions (where the models are controversial: of course we personally believe that the asteroid did in the dinosaurs), fluids invading porous materials and other problems involving invading fronts (where the model we describe was invented), the dynamics of superconductors and superfluids, sound emitted during martensitic phase transitions, fluctuations in the stock market, solar flares, cascading failures in power grids, failures in systems designed for optimal performance, group decision making, and fracture in disordered materials. A piece of iron will "crackle" as it enters a strong magnetic field, giving what is called Barkhausen noise. These models are driven systems with many degrees of freedom, which respond to the driving in a series of discrete avalanches spanning a broad range of scales — what we are calling crackling noise.

However, not all systems crackle. Some respond to external forces with lots of similar-sized, small events (popcorn popping as it is heated). Others give way in one single event (chalk snapping as it is stressed). Roughly speaking, crackling noise is in between these limits: when the connections between parts of the system are stronger than in popcorn but weaker than in the grains making up chalk, the yielding events can span many decades of sizes. Crackling forms the transition between snapping and popping.

There has been healthy skepticism by some established professionals in these fields to the sometimes grandiose claims by newcomers proselytizing for an overarching paradigm. But often confusion arises because of the unusual kind of predictions the new methods provide. If our models apply at all to a physical system, they should be able to predict all behavior on long length and time scales, independent of many microscopic details of the real world. This predictive capacity comes, however, at a price: our models typically don’t make clear predictions of how the real-world microscopic parameters affect the long-length-scale behavior.

Figure 3: Cross sections of the avalanches in our model of crackling noise in magnets. Each avalanche is drawn in a separate color. Left: a cube 100 on a side; right: a cube 1000 on a side (a billion domains); both systems are run at the border between popping and snapping where crackling noise occurs. The black background in each case represents a large avalanche that spans the system. Notice that the two pictures look similar to one another, if you blur your eyes to ignore the finer details on the right: the system is similar to itself on different scales. Our methods for studying crackling noise systematically use this self-similarity.

How do we derive the laws for crackling noise? There are two approaches. First, one can analytically calculate the behavior on long length and time scales by formally coarse-graining over the microscopic fluctuations. Second, one can make use of universality or the fact that simple models and real systems can share the same behavior on a wide range of scales (see figure 3). If the microscopic details don’t matter for the long length scale behavior, why not make up a simple model with the same behavior (in the same universality class, figure 4) and solve it?

Figure 4: Universality. Consider the system space of all possible models of crackling noise in magnets, plus all experimental magnetic materials exhibiting crackling noise. There is a separate dimension in system space for every possible parameter in a theoretical model or in an experiment (temperature, chemical composition, etc.) Only two dimensions are shown. To study crackling noise, we mathematically blur our eyes as in figure 3, removing some fraction of the microscopic local domains and finding new parameters for the model so that the remaining domains flip in the original patterns. Drawing an arrow from the original system to the blurred system gives us the flow on system space at left. The fixed point S* is self-similar, since it stays the same as you blur. The models on the surface C that flow into S* are self-similar except on the shortest, microscopic scales. Whether they are experimental systems or theoretical models, systems on C are self-similar on large scales in exactly the same way as S*. This universality is what makes our theories possible: study one model, get the rest for free! Models to the left of C transform in one large snap (a system-spanning avalanche): points on the right transform in many small pops. As our model crosses C, it goes through a phase transition from snap through crackle to pop.