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EDWARD LORENZ’s singular contribution to science was his theory of
chaos.
He was not the first to discover chaotic
phenomena. That achievement belongs to Henri Poincaré who noticed
that the movement of three heavenly bodies around each other
followed the characteristics of a nonlinear system. Their motions
were so complex that they were almost impossible to calculate, much
less to predict.
Lorenz’s rediscovery was serendipitous. As
James Gleick tells it in his book, Chaos, Lorenz, a meteorologist,
ran in the winter of 1961 a weather simulation using a simple
computer program. A day or so later he repeated the simulation using
the same program and the same set of equations of the initial
conditions as in the first run. But instead of repeating the
simulation from the beginning, he started the second run in the
middle. What appeared on the screen were two divergent weather
trajectories.
Pondering this result, Lorenz concluded that the
smallest change could lead to radically different outcomes. He also
found that the differential equations that describe atmospheric
conditions were highly dependent on initial conditions. Therefore,
long-range and long-term weather forecasting was effectively
impossible. One must know conditions in all parts of the world at a
precise moment in time to produce a perfect prediction.
This phenomenon has since become known as the
butterfly effect. Small air movements, somewhere in the world,
could—in theory—stir a storm in a distant continent some weeks
later.
Lorenz published his findings in 1963 but it was
not until the late 70s, that other scientists took notice of them.
Edward Ott, a professor of physics at the University of Maryland,
told the New York Times: “When it finally penetrated the [science]
community that was what started people to really pay attention…and
[that] led to tremendous development.”
“Chaos,” according to David Millar of
Cambridge University, “[is] the irregular, unpredictable behavior
of deterministic, nonlinear dynamical systems.”
In less technical language, chaos is the
intermediate stage between highly ordered motion and fully random
movements. A slow moving river is “orderly”; as its velocity
increases, the flow becomes chaotic; in a flood, the flow becomes
turbulent or completely random. Turbulence is still an unsolved
problem in physics.
As a matter of incidental interest, the person
who can solve the problem will win US$1 million from the Clay
Mathematical Institute. The Navier-Stokes equations, first written
in the 1840s, hold the key to the mystery of both smooth and
turbulent flow.
Chaos is inherent in all natural systems. Even
the solar system, that model of Newtonian predictability, could over
time become chaotic. Stable motion is the exception rather than the
rule. Nonlinear dynamics and that branch of physics called
statistical mechanics in combination are today the tools of choice
in many fields of science.
Mitchell Feigenbaum using a technique called
renormalization showed that there are universal laws that govern the
transition from regular to chaotic behavior. These were confirmed in
fields as diverse as cardiology, chemistry, genetics and neurology.
They also helped explain the erratic reversals
of the Earth’s magnetic field throughout geological history.
The tumbling movement of the moon Hyperion
around Saturn was understood.
Even the fluctuations of wildlife populations
are now considered chaotic rather than deterministic.
The visual representation of chaos is the
mathematical object called a fractal, a simple shape that upon close
examination reveals an infinity of detail.
John Cage, in his musical compositions, allows
“any number of players, any sound or combinations of sounds
produced by any means, with or without other activities.”
Such a technique, according to Diana S. Dabby in
“Creating Musical Variations” (Science, April 4, 2008) uses a
“natural mechanism for variability found in the science of
chaos.” Cage’s chaotic trajectories are sensitive to initial
conditions. They are virtually infinite in number but close to the
original and achieve variations that are both extreme and original.
To illustrate chaotic dynamics, Lorenz invented
the Lorenz attractor, a three-dimensional curve that shows how the
motion of a system oscillates unpredictably between two directions,
never settling into a steady state.
Roger Penrose wondered in The Road to Reality if
“chaotic unpredictability” implies an evolution into the past
rather than into the future. But time-reversal is constrained by the
second law of thermodynamics that, in its simplest form, asserts
that heat flows from a hotter to a colder body. A reversal in time
“would be a practical impossibility to decide which body will get
hotter and which colder, how much, and when.” Dynamical
retrodiction like long-range weather prediction is impossible.
Let me end this tour with Steven Strogatz, the
author of Nonlinear Dynamics and Chaos: With Applications to
Physics, Biology, Chemistry and Engineering.
In 1942, Enrico Fermi, John Pasta, and Stanislaw
Ulam—all of them were part of the Manhattan Project that produced
the atom and thermonuclear bombs—decided to test the project’s
brand-new MANIAC, the world’s first supercomputer. The machine was
instructed to simulate the vibrations of an elastic chain of 32
particles. The mathematics for such computation was not available at
that time. They predicted that when the chain was disturbed from its
rest state, the system would degenerate into a state of randomness.
This was how the second law of thermodynamics would cause it to
behave. To their surprise, the particles returned to their starting
positions, the first evidence that nonlinearity could be a source of
order. Fermi, ecstatic, called it “a little discovery.” Strogatz,
impishly, intoned: “Nonlinearity giveth chaos, and nonlinearity
taketh it away.”
Edward Lorenz died of cancer on April 16, 2008.
He was 90.
opinion@manilatimes.net
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