(XZ Plane Projection)
Construct a simple system: take a box, a simple solid rectangular solid.
Within this box, place a homogenous, preferably elemental, gaseous substance.
Heat the box, sit back, and observe.
What happens to the gas? It is, of course, common knowledge that warm gases
rise, while cooler gases sink; and initially, the portions of the gas closest
to the walls of the box (e.g. closest to the external heat source) will
become heated and rise. At certain temperatures, the gas will begin to form
cylindrical rolls spaced like jellyrolls lying lengthwise in the box. On one
side of the box, the gas rises, and on the other, it sinks; the rising gases
converge on one side and carry warmer gases up with them; as the gas cools,
it falls on the other side of the box.
With a regularly applied temperature, a smooth box interior, and a system
completely closed with respect to the gas itself, it might be expected that
the circular motion of the moving gas should be regular and predictable.
Nature, however, is neither regular nor predictable. It turns out that the
motion of the gaseous cylinders is chaotic. The rolls do not simply
roll around and around in one direction like a steam-roller; they roll for
a while in one direction, and then stop and reverse directions. Then,
seemingly at random, the gas reverses direction again; these fluctuations
continue at unpredictable times, at unpredictable speeds.
The Lorenzian Waterwheel
Most casual armchair scientists have no access to uniformly smooth boxes and
elemental gases, much less instruments to measure the rotational speed of
a moving cylinder of gas.
A metaphor for the gaseous system is found in the Lorenzian waterwheel.
This is a thought experiment. Imagine a waterwheel, with an arbitrary number
of buckets, usually more than seven, spaced equally around its rim. The
buckets are mounted on swivels, much like Ferris-wheel seats, so that the
buckets will always open upwards. At the bottom of each bucket is a small
hole. The entire waterwheel system is then mounted under a waterspout.
The scenario is set: now we commence the action.
Begin the flow of water from the waterspout. At low speeds, the water will
trickle into the top bucket, and immediately trickle out through the hole
in the bottom. Nothing happens. Real boring. Increase the flow just a bit,
however, and the waterwheel will begin to revolve as the buckets fill up
faster than they can empty. The heavier buckets containing more water let
water out as they descend, and when the water is gone, the now-light buckets
ascend on the other side, ultimately to be refilled. The system is in a
steady state; the wheel will, like a waterwheel mounted on a stream and
hooked to a grindstone, continue to spin at a fairly constant rate.
But even this simple system, sans boxes or heated gases, exhibits
chaotic motion. Increase the flow of water, and strange things will happen.
The waterwheel will revolve in one direction as before, and then suddenly
jerk about and revolve in the other direction. The conditions of the buckets
filling and emptying will no longer be so synchronous as to facilitate just
simple rotaton; chaos has taken over.
The explanation for the irregular movement of the gas lies at the molecular
level. While the box sides may seem smooth and thus the flow of the gas
should always be regular, at molecular levels the sides of the box are quite
irregular due to the motion of atoms and molecules. After all, in any solid
not at absolute zero, total entropy is positive and there must be some
irregularity in the molecular structure of the sides of the box.
Molecular interactions are tiny, however. How would such fluctations as small
as a slightly misplaced molecule affect the flow of the gas in such a
profound way as to cause seemingly random motion? The theory behind how small
deviations can lead to large deviations lies at the heart of chaos theory.
The explanation is simple and, in retrospect, obvious explanation commonly
known as sensitive dependence on initial conditions.
Edward Lorenz and Long-Term Meteorology
In the early sixties, a certain meteorologist named Edward Lorenz experimented
with computer simulations of weather on a relatively primitive "Royal McBee"
computer. His program used twelve recursive equations to simulate rudimentary
aspects of weather; he entered several variables into his program each time
he ran it, and watched to see what types of weather patterns such initial
conditions would generate. He could print out graphs of fluctuating
temperatures or other conditions, and his program captured the fancy of his
But mousie, thou art no thy lane
One day, Lorenz tried to recreate an interesting weather pattern, one he had
seen previously, by re-entering the values the computer had previously
calculated and reported. However, when he ran the program again, his results
were different from the initial run. Lorenz suspected a bug; blown diode?
burned-out vacuum tube? power surge? cosmic rays? After checking the
two plots, however, he realized his "error"; on his previous computer
printout, the one he had used to enter the initial conditions into the
computer for the second trial run, the figures were printed with three
significant digits. In the program, all values were calculated to six
significant digits. Lorenz had assumed that the difference, only one part in
a thousand, would be inconsequential; however, due to the recursive nature of
the equations, little errors would first cause tiny errors, which would then
affect the resulting next calculation a bit more, which would affect the
output of the next run even more. The final result of a long string of
recursive calculations would lead to a weather pattern totally different from
the expected values.
In proving that foresight may be in vain;
The best-laid plans of mice and men
Gang aft a'gley.
--Robert Burns, "To a Mouse"
The term "sensitive dependence on initial conditions" was coined to describe
the phenomenon that small changes in a recursive system can drastically
change the results of running that system. A term Lorenz coined to describe
sensitive dependence on initial conditions is the "butterfly effect." This
is another thought experiment which is hardly testable: imagine that there
exist two earths, so that an incorporeal observer could compare events on one
earth to another. Now imagine that both earths are identical except for one
fact; in one, a butterfly flaps its wings somewhere in South America, and in
the other, this butterfly remains still. One might think that such a small
discrepancy between the two earths would be inconsequential; after all,
nobody was there, nobody could even notice the butterfly's wings flapping,
and air currents would be affected only minorly by such a miniscule event.
After a period of time which is impossible to calculate, however, the weather
patterns of the two earths would be totally different. Why? Because of the
difference caused by the flap of one butterfly's wings! The miniscule event
affected air currents around that butterfly in a very miniscule fashion, true;
but those tiny air currents affected in turn slightly larger air currents,
which affected still larger air currents, and the small difference in air
flow between the two earths exponentially increases to become a large
difference. The wind patterns on the two earths, which started out otherwise
identical and had every reason to remain identical in a nice deterministic
manner, would now be different in every way. Eventually, as time multiplies
the differences between the two earths, completely different weather patterns
would emerge, all because of the fact that on one of the earths a butterfly
in South America decided to flap its wings, while on another of the earth,
the same butterfly did not.
The differences might not cause any major catastrophic events immediately;
the thought experiment does not suggest that murdering a butterfly could
cause a hurricane in a few minutes or a tornado in a few hours. However, the
air currents and wind patterns would be different.
It is thus completely impossible, even in theory, to perform long-term weather
prediction in any accurate manner. Unless a computer could be constructed which
could monitor each individual atom on earth, even the smallest undetected
anomaly could affect the weather in profound ways. Fascinated by this idea,
Edward Lorenz began drifting away from meteorology and began exploring the
realms of mathematics, looking for more unpredictable, nonlinear systems.
At one point, Edward Lorenz was looking for a way to model the action of the
chaotic behavior of the gaseous system first mentioned above. Lorenz took a
few "Navier-Stokes" equations, from the physics field of fluid dynamics.
He simplified them and got as a result the following three-dimensional
dx/dt = delta * (y - x)
Here delta represents the "Prandtl number." This number, which one absolutely
does not have to know the meaning of, is the ratio of the fluid viscosity of
a substance to its thermal conductivity (named after Ludwig Prandtl, a German
physicist). The value Lorenz used was 10.
dy/dt = r * x - y - x * z
dz/dt = x * y - b * z
The variable r represents the difference in temperature between the top and
bottom of the gaseous system. The value usually used in sample Lorenz
attractors such as the one displayed here is 28.
The variable b is the width to height ratio of the box which is being used
to hold the gas in the gaseous system. Lorenz happened to choose 8/3, which
is now the most common number used to draw the attractor.
The resultant x of the equation represents the rate of rotation of the
cylinder, y represents the difference in temperature at opposite sides of the
cylinder, and the variable z represents the deviation of the system from a
linear, vertical graphed line representing temperature.
Plotting the three differential equations requires the usage of a computer.
Plotted on a three-dimensional plane, a shape unlike any other forms. Instead
of a simple geometric structure or even a complex curve, the structure now
known as the Lorenz Attractor weaves in and out of itself. Projected on the
X-Z plane, the attractor looks like a butterfly; on the Y-Z plane, it
resembles an owl mask. The X-Y projection is useful mainly for glimpsing
the three-dimensionality of the attractor; it looks something like two
paper plates, on parallel but different planes, connected by a strand of
string. As the Lorenz Attractor is plotted, a strand will be drawn from one
point, and will start weaving the outline of the right butterfly wing. Then
it swirls over to the left wing and draws its center. The attractor will
continue weaving back and forth between the two wings, its motion seemingly
random, its very action mirroring the chaos which drives the process.
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