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Probability and
Statistical Physics Connecting Microscopic
and Macroscopic
I
n 1827, a botanist named Robert Brown noticed that grains of pollen suspended in
water did a strange sort of dance when examined under a microscope. At first he
thought the pollen was alive. But in 1905, Albert Einstein explained the real cause of
“Brownian motion,” which had nothing to do with biology. The grains are constantly
buffeted by collisions with water molecules, which cause them to jiggle in random
directions.
It is surprisingly common for random microscopic events to produce predictable
effects at a macroscopic level. The Brownian motion of any single particle of smoke
is highly unpredictable, yet smoke spreads in a room at a predictable rate. Iron
atoms make innumerable random choices on which way to spin, but at a predictable
temperature their spins spontaneously line up and the iron becomes magnetized.
The voids in a porous material may be distributed randomly, but at a certain density,
which is predictable, they connect up and the material becomes permeable. There
are three kinds of “random path” here: the meanderings of a smoke particle, the
boundary between spin-up and spin-down atoms in a magnet, and the path of water
through a rock. But remarkably, all these disparate phenomena (or at least a simplified
mathematical model of each one) can be brought under one roof.
In 2000, a universal mechanism was discovered that shows how microscopic disorder
can lead to macroscopic order for two-dimensional systems. This discovery is now
called Schramm-Loewner evolution, which allows precise calculations of macroscale
phenomena that could until then only be predicted nonrigorously. Not only that, the
mechanism applies to the random processes mentioned above as well as to others. A
single parameter, k (the Greek letter kappa), distinguishes Brownian motion (k = 8)
in the 21st Century
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from measures of the boundaries between small regions of
materials (“magnetic grains”) that are aligned magnetically
(k = 3) or paths for water percolation (k = 6) (Figure
25). Schramm-Loewner evolution is a wonderful unified
description of all these separate phenomena, illuminating
how disorder can create order.
The most essential feature of Schramm-Loewner
evolution is a symmetry property called conformal
invariance. Conformal invariance has two components:
scale invariance and rotation invariance. The first means
that a Brownian trajectory will look just the same at any
level of magnification. If you blow it up by a factor of
10, it will look just as jiggly as before. The formerly small
bounces will become big bounces—but you will see new,
even smaller bounces that you couldn’t make out before.
Rotation invariance means that Brownian motion has no
preferred direction. For example, in a closed room, smoke
will go everywhere.
Consider a crack that grows randomly inward from
the edge of an infinite pane of glass. According to the
idea on which Schramm-Loewner evolution is based,
the pane of glass can always be “healed” by a conformal
transformation, deforming the glass in a way that pushes
the crack back out to the boundary. If the crack grows,
the glass can be healed again. In this never-ending
process of cracking and healing, the attachment point of
the crack moves around. In fact, it jiggles very erratically
along the edge of the glass. Does this sound familiar? The
attachment point is actually undergoing one-dimensional
Brownian motion. The intensity of the jiggling is described
by the parameter k; larger values of k correspond to more
intense jiggling and to more jagged cracks. The basic
idea of Schramm-Loewner evolution converts any two-
dimensional, conformally invariant random process into
25 / Schramm-Loewner evolution for different values
a one-dimensional Brownian motion. Many questions
of k. For example, critical percolation (top) gives
trajectories with k = 6. The middle image shows become simpler after they are restated as a Schramm-
k = 0.5, and the bottom image k = 8/3. Reprinted
Loewner evolution. What is the probability that the
with permission from Scott Sheffield, Massachusetts
trajectories of two pollen grains in a petri dish will intersect
Institute of Technology. /
before the grains reach the edge of the dish? What is the
probability that water will percolate from one side of a
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26 / Critical percolation model. When each cell
has an equal probability of being red or white,
the red and white cells connect up into very long
networks, and the interface between them is a
curve described by Schramm-Loewner evolution.
Reprinted with permission from Michael Kozdron,
University of Regina, Saskatchewan. /
rectangle to the other before it escapes out the top or bottom? What is the fractional
dimension (or roughness) of the outside of a cloud of smoke? All of these questions have
precise answers now.
Schramm-Loewner evolution is a purely mathematical construct. There is no known
physical mechanism that can duplicate the cracking-and-healing process described
above. It’s “mathemagic” in the best sense.
Conformally invariant processes are particularly relevant to the physics of phase
transitions, such as the freezing of water or the magnetization of iron. (Just as there is a
freezing point for water, above a certain temperature, iron will not magnetize; below that
So 21st century
temperature, it will.) These processes are scale-invariant because it is precisely at a phase
mathematical
transition that, when the temperature is dropping, small-scale, local correlations become
scientists still have
large-scale as the ice crystals or iron crystals lock into place.
Though Schramm-Loewner evolution is a key to understanding such random
their work cut out
processes in two dimensions, it has two caveats. First, it is anything but routine to prove
for them as they try
that a given random process corresponds to a particular value of k. There are some
to explain phase
processes, such as the growth of polymers (which are like self-avoiding random walks),
where the appropriate value of k is strongly suspected but not rigorously established. transitions in our
Second, Schramm-Loewner evolution is—unfortunately—limited to two dimensions.
three-dimensional
It seems very likely that three-dimensional random processes cannot be classified by
world.
a single parameter such as k. It is likely also that the critical exponents describing the
correlation of nearby molecules are not as simple as they are in the two-dimensional
case. So 21st century mathematical scientists still have their work cut out for them as
they try to explain phase transitions in our three-dimensional world.
However, Schramm-Loewner evolution provides a model for how a theory of phase
transitions might look. Research based on Schramm-Loewner evolution has twice won
the Fields Medal, one of the highest honors in the mathematical sciences. An award
like this shows the remarkable amount of esteem among mathematical scientists for a
discovery that is scarcely a decade old.
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Mathematical Sciences Inside...
Remember how magical it seemed when CD players started to appear in automobiles? How
could a precision instrument, which has to detect pits less than a micron across, possibly
function in an environment where it is routinely jolted distances that are tens of thousands
of times larger than that? The magic is not in the shock absorbers, it’s in the mathematical
sciences. A method called maximum likelihood sequence estimation, based on the statisti-
cal technique called maximum likelihood, works out the most likely sequence of 1s and 0s
recorded on the disk and compensates for the noise and errors created by the bumpy car ride.
Many other technologies that we now take for granted are based on mathematical ideas.
Other inventions that seem visionary today but might be commonplace 20 years from now,
likewise depend on math. To illustrate that point, the table below lists 10 inventions whose
patents cite a method from the mathematical sciences. The data are from the Google patent
database at www.google.com/patents, accessed on August 3, 2011. The terms are ordered
roughly by their frequency of occurrence in the Google database.
Fast Fourier Transform (FFT)
The industry standard way of decomposing
an electronic signal into its constituent
frequencies, based on samples taken Correlation coefficient
at regular time intervals.
This patent is for a “monolithic” A basic statistical method for determining how
silicon chip that can compute closely related two paired sets of data are.
FFT’s; such chips are used An optical scanner locates a “bull’s-eye” on
in digital image processing, a label by finding the correlation between
speech recognition, and scanned pixel sequences and the expected
transmission as in cell phones. sequence for a cross section of the bull’s-eye.
Patent No. 4547862 (1985) TRW, Inc. Patent No. 6122310 (2000) United Parcel Service
Viterbi algorithm
Algorithm used in cell phones and CD and DVD players to decode noisy
signals. Its key idea is to use “soft,” probabilistic decision procedures.
This patent is one of hundreds that tweaks the original version, here
by speeding up the “traceback” part of the algorithm.
Patent No. 6904105 (2005) Intel
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Inventions Complex number
Numbers of the form a + bi (where i = −1).
Applications such as FFT require integrated
circuits capable of adding and multiplying
complex numbers, as described in this patent.
Elliptic curve
Patent No. 4858164 (1989) United Technologies
Algebraic structure used in public-key
cryptography—for example, to authenticate the
user of a smart card. Minimal surface
In this patent, users can pick their own elliptic
curve instead of selecting one from a centrally Surfaces, such as soap films,
managed registry. that have the least area
spanning a given boundary.
Patent No . 6446205 (2002) Citibank
This patent proposes the
Schwarz triply periodic
minimal surface as a scaffold
for regenerating human bone
and organ tissue.
Patent No. 7718109 (2010) Mayo
Foundation
Support vector machine (SVM)
Recently discovered (1995) method for
partitioning data into classes.
SVMs are used in an implantable “brain
pacemaker” for Parkinson’s disease patients, to
determine when the patient is having a seizure
or movement disorder.
B-spline
Patent No. 12/694035 (applied 2010) Medtronic, Inc.
The industry standard method of representing smooth surfaces,
used in computer-aided design and manufacturing. Quaternion
B-splines have become popular in recent years with video game
manufacturers; in this patent, they are used to generate smooth Hypercomplex numbers used
motions of three-dimensional figures under user control. primarily for composing spatial
rotations.
Patent No. 5982389 (1999) Microsoft
This patent is for a toothbrush
that will automatically track its
Conjugate gradient method location relative to the user’s
teeth. Quaternions are used to
compensate for motion of the
An iterative method for solving linear equations (Ax = b) or
user’s head.
energy minimization problems involving many variables.
Used in this patent to compute the electronic structure Patent No. 12/866,381 (applied 2010) Philips
of simple molecules like glass. The energy depends on
thousands of variables, each representing a possible
electron orbit.
Patent No. 6106562 (2000) Corning 57
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