"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,
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
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.
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
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 dont make
clear predictions of how the real-world microscopic parameters affect
the long-length-scale behavior.
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 dont 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.