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ity of a given algorithm designed to solve a problem, or the complexity of a
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machine that responds to feedback, or the complexity of a static image or
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design.
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For the moment, let’s take a “we know it when we see it” attitude toward
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complexity, and be prepared to develop more formal definitions when cir-
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cumstances require.
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It’s not just cups of coffee in which complexity grows and then fades
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as entropy increases: the universe as a whole does exactly the same thing.
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At early times, near the Big Bang, the entropy is very low. The state is
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also extremely simple: it’s hot, dense, smooth, and rapidly expanding.
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That’s a complete description of what is going on; there’s no real differ-
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ence in conditions in the universe from place to place. In the far future the
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entropy will be very high, but conditions will once again be simple. If
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we wait long enough, the universe will appear cold and empty, and will
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have regained its smoothness. All of the matter and radiation we currently
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see will have left our observable horizon, diluted away by the expansion
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of space.
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It is today, in between the far past and the far future, when the universe
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is medium- entropy but highly complex. The initially smooth configuration
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has become increasingly lumpy over the last several billion years as tiny
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perturbations in the density of matter have grown into planets, stars, and
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galaxies. They won’t last forever; as we saw in chapter 6, eventually all the
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stars will burn out, black holes will swallow them up, and then even the
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black holes will evaporate away. The era of complex behavior that our uni-
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verse is currently enjoying is, alas, a temporary one.
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Entropy
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Complexity
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Time
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The evolution of entropy and complexity in a closed system over time.
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This similarity between the development of complexity in coffee cups and
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the universe, even as entropy is constantly increasing, is provocative. Is it possi-
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ble that there is a new law of nature yet to be found, analogous to the second law
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of thermodynamics, that summarizes the evolution of complexity over time?
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The short answer is “We don’t know.” The somewhat longer answer is
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“We don’t know, but maybe, and if so, there’s good reason to believe it will
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be— appropriately enough— complicated.”
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•
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I have been working on precisely this issue in my own research, with col-
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laborators Scott Aaronson, Varun Mohan, Lauren Ouellette, and Brent
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Werness. It all started on a ship sailing through the North Sea. This was
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part of an unusual interdisciplinary conference devoted to the nature of
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time, which was literally international in scope: it began in Bergen, Nor-
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way, continued during the ship voyage, and finished up in Copenhagen,
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Denmark. I gave the opening lecture, and Scott was in the audience. I
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talked a bit about how complexity seems to come and go as closed systems
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evolve, using coffee and the universe as my examples.
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Scott is one of the world’s experts on “computational complexity,” which
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organizes different kinds of questions into categories based on how hard they
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are to solve. He was intrigued enough to think about making the question
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more precise. He recruited Lauren, an undergraduate at MIT at the time, to
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write a simple computer code representing an automaton that would simulate
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cream and coffee mixing into each other. After we wrote a first draft of a paper
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and put it on the Internet, Brent wrote to us to point out a flaw in our results—
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not one that undermined the basic idea, but one that indicated the specific
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example we were looking at wasn’t appropriate. In the spirit of moving science
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forward, rather than blackballing Brent and trying to destroy his scientific
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career as punishment for his impertinence, we recognized that he was right
08
and brought him on as a collaborator. Scott recruited Varun, another MIT
09
undergraduate, to update the code and perform more simulations, until we
10
finally fixed our problems. Such is the majestic progress of science.
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•
12
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For our investigation, we were specifically interested in what we called the
14
apparent complexity of the cup of coffee. It’s related to what computer sci-
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entists call the “algorithmic” or “Kolmogorov” complexity of a string of
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bits. (Any image can be represented as a string of bits, for example, in a data
17
file.) The idea is to pick some computer language that has the ability to
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output such strings, such as 01001011011101. The algorithmic complexity of
19
a string is simply the length of the shortest program that, when run, outputs
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that string. Simple patterns have low complexity, while completely random
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strings have high complexity— the only way to output them is simply to
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have a “Print” statement that includes an explicit copy of the string.
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For our purposes of characterizing images of cream mixing with coffee,
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random noise would count as “simple,” not complex. So, following
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Boltzmann’s treatment of entropy, we defined “apparent complexity” by
26
coarse- graining. Rather than observing the position of every single particle
27
in our simulation, we looked at the average number in a small region of
28
space. The apparent complexity is then the
algorithmic complexity of the
29
coarse- grained distribution of cream and coffee. It’s a nice way to formalize
30
our intuitive notion of “how complex an image appears to be.” High appar-
31
ent complexity corresponds to a coarse- grained (smeared) image that
32
contains a lot of interesting structure.
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Unfortunately, there is no way to directly calculate the apparent com-
34
plexity of an image. But there is a very good approximation: just stick the
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image into a file- compression algorithm. Everyone’s computer has programs
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that do that, so we were off and running.
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At the beginning of the simulation, the apparent complexity is low: a
01
complete description is just “cream on top, coffee below.” At the end, the
02
apparent complexity is low once again: all we need to say is that there are
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equal amounts of cream and coffee at every point. In between, when the
04
mixing is occurring, is where things become interesting. What we found
05
was that complexity doesn’t necessarily develop— whether it does or not de-
06
pends on how the cream and coffee interact with each other.
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Roughly speaking, if the cream and coffee molecules interact only with
08
other nearby molecules, you don’t see much development of complexity.
09
Everything just smoothly blends together rather than forming a jagged pat-
10
tern of tendrils.
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If we introduce long- range effects— analogous to the spoon stirring the
12
coffee— that’s when things get interesting. Rather than just blurring to-
13
gether, the boundary between the cream and coffee takes on a fractal as-
14
pect. The resulting images have a high degree of apparent complexity; in
15
order to describe them accurately, you would have to specify the intricate
16
shape of the cream- coffee boundary, which would require a relatively large
17
amount of information.
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A simple computer simulation of cream and coffee mixing together. The configuration
starts out simply and grows increasingly complex; further evolution would show it becom-S35
ing simple once again, as the black and white became completely mixed.
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The relationship between “fractal” and “complex” is more than just a
02
cosmetic one. A fractal is a geometric figure that looks basically the same at
03
any magnification. In the cream and coffee, we see roughly fractal patterns
04
appear in the configuration of the molecules before they eventually fade
05
away in equilibrium. This is a hallmark of complexity; interesting things are
06
going on when we look at the system up close, with just a few moving parts,
07
and also when we look at it all at once.
08
In both physics and biology, complexity often emerges in a hierarchical
09
fashion: small pieces conglomerate into larger units, which then conglom-
10
erate into even larger ones, and so on. Smaller units maintain their integrity
11
while interacting together within the whole. In this way, networks are built
12
up that exhibit complex overall behavior emerging from simple underlying
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rules. The coffee- cup automaton is too simple a system to model this pro-
14
cess faithfully, but the appearance of a fractal shape is a reminder of how
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robust and natural complexity can be.
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Keep going, and the apparent complexity disappears. All of the cream
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and coffee is simply mixed together. If we wait long enough, any isolated
18
system reaches equilibrium, where nothing interesting happens.
19
•
20
21
There is no law of nature, therefore, that says complexity necessarily devel-
22
ops as systems evolve from low entropy to high entropy. But it can develop—
23
whether it does or not depends on the details of the system you are thinking
24
of. On the strength of one simple computer simulation, it seems that a key
25
issue is the existence of effects that stretch over long distances, rather than
26
only involving particles right next to each other.
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The real world features interactions both on short ranges, when particles
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bump into each other, and on ones that stretch over longer ranges, like the
29
influence of gravity or electromagnetism. When we see complex structures
30
arise as the universe expands and cools, what we’re seeing is an interplay
31
between competing influences. The expansion of the universe draws things
32
apart; mutual gravitational forces pull them together; magnetic fields push
33
them sideways; collisions between atoms shove matter around and allow it
34
to cool down. If interesting complex structures can arise in a computer
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simulation with nothing more than white dots and black dots, it’s not
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surprising that they arise in something as multifaceted as the expanding
01
universe.
02
The appearance of complexity isn’t just compatible with increasing
03
entropy; it relies on it. Imagine a system that didn’t have any Past Hypoth-
04
esis, and was simply in a high- entropy equilibrium state right from the
05
start. Complexity would never develop; the whole system would remain
06
featureless and uninteresting (apart from rare random fluctuations) for all
07
time. The only reason complex structures form at all is because the universe
08
is undergoing a gradual evolution from very low entropy to very high en-
09
t
ropy. “Disorder” is growing, and that’s precisely what permits complexity
10
to appear and endure for a long time.
11
The microscopic laws of physics don’t distinguish between past and fu-
12
ture. So any tendency of things to behave differently in one direction in
13
time as opposed to the other— whether it’s birth and death, biological evo-
14
lution, or the appearance of complicated structures— must ultimately be
15
traced to the arrow of time and therefore to the second law. The increase of
16
entropy over time literally brings the universe to life.
17
Apparent complexity doesn’t capture all of what people have in mind
18
when they admire the workings of a clock or a human eye. What makes
19
those remarkable is how the different pieces work together in harmony to
20
help achieve what appears to be some sort of purpose. We’ll have to work a
21
bit harder to see how such behavior can arise through the action of mind-
22
less matter obeying simple laws. The answer, unsurprisingly, can be traced
23
once again to the growth of entropy and the arrow of time.
24
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•
26
As we work our way up from quantum fields and particles to human beings,
27
the subjects we will tackle are going to become more and more difficult, and
28
our statements correspondingly less definitive. Physics is the simplest of all
29
the sciences, and fundamental physics— the study of the basic pieces of real-
30
ity at the deepest level— is the simplest of all. Not “simple” in the sense that
31
the homework problems are easy, but simple in the sense that Galileo’s trick
32
of ignoring friction and air resistance makes our lives easier. We can study
33
the behavior of an electron without worrying about, or even knowing much
34
about, neutrinos or Higgs bosons, at least to a pretty good approximation.
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The rich and multifaceted aspects of the emergent layers of our world are
02
not nearly so accommodating to the curious scientist. Once we start dealing
03
with chemistry, biology, or human thought and behavior, all of the pieces
04
matter, and they matter all at once. We have made correspondingly less
05
progress in obtaining a complete understanding of them than we have, for
06
example, on the Core Theory. The reason why physics classes seem so hard
The Big Picture Page 40