They emerge as we zoom out from the microscopic level to the level of the
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everyday. To appreciate why we seem to live in a world of causes and pur-
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poses, while nature deep down is a story of impersonal Laplacian patterns,
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we need to understand the arrow of time.
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To understand time, it helps to start with space. Here on the surface of the
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Earth, you would be forgiven for thinking that there is an intrinsic differ-
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ence between the directions “up” and “down,” something deeply embedded
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into the fabric of nature. In reality, as far as the laws of physics are con-
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cerned, all directions in space are created equal. If you were an astronaut,
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floating in your spacesuit while you performed an extravehicular activity,
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you wouldn’t notice any difference between one direction in space and any
09
other. The reason why there’s a noticeable distinction between up and down
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for us isn’t because of the nature of space; it’s because we live in the vicinity
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of an extremely influential object: the Earth.
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Time works the same way. In our everyday world, time’s arrow is unmis-
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takable, and you would be forgiven for thinking that there is an intrinsic
14
difference between past and future. In reality, both directions of time are
15
created equal. The reason why there’s a noticeable distinction between past
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and future isn’t because of the nature of time; it’s because we live in the af-
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termath of an extremely influential event: the Big Bang.
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Remember Galileo and conservation of momentum: physics becomes
19
simple when we ignore friction and other bothersome influences, and con-
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sider isolated systems. So let’s think of a pendulum rocking back and forth,
21
and for convenience let’s imagine that our pendulum is in a sealed vacuum
22
chamber, free of air resistance. Now someone records a movie of the pendu-
23
lum rocking, and shows it to you. You are not very impressed; you’ve seen
24
pendulums before. Then they reveal the surprise: they were actually playing
25
the movie backward. You hadn’t noticed because a pendulum rocking back-
26
ward in time looks exactly like one rocking forward in time.
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That’s a simple example of a very general principle. For every way that a
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system can evolve forward in time in accordance with the laws of physics,
29
there is another allowed evolution that is just “running the system backward
30
in time.” There is nothing in the underlying laws that says things can evolve
31
in one direction in time but not the other. Physical motions, to the best of our
32
understanding, are reversible. Both directions of time are on an equal footing.
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That seems reasonable enough for simple systems: pendulums, planets
34
moving around the sun, hockey pucks gliding on frictionless surfaces. But
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when we think about complicated macroscopic systems, everything in our
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experience tells us that certain things happen as time moves from past to
02
future, but not in the other direction. Eggs break and get scrambled but
03
don’t unscramble and unbreak; perfume disperses into a room but never
04
retreats back into its bottle; cream mixes into coffee but never spontane-
05
ously unmixes. If there is a purported symmetry between past and future,
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why do so many everyday processes occur only forward and never backward?
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Even for these complicated processes, it turns out, there is a time- reversed
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process that is perfectly compatible with the laws of physics. Eggs could un-
09
break, perfume could go back into its bottle, cream and coffee could unmix.
10
All we have to do is to imagine reversing the trajectory of every single par-
11
ticle of which our system (and anything it was interacting with) is made.
12
None of these processes violates the laws of physics— it’s just that they are
13
extraordinarily unlikely. The real question is not why we never see eggs un-
14
breaking toward the future; it’s why we see them unbroken in the past.
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•
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Our basic understanding of these issues was first put together in the nine-
18
teenth century by a group of scientists who invented a new field called sta-
19
tistical mechanics. One of their leaders was the Austrian physicist Ludwig
20
Boltzmann. It was he who took the concept of entropy, which was recog-
21
nized as a central idea in the study of thermodynamics and irreversibility,
22
and reconciled it with the microscopic world of atoms.
23
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25
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Ludwig Boltzmann, master of entropy and
27
probability, 1844– 1906. (Courtesy of
28
Goethe University of Frankfurt)
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Before Boltzmann came along, entropy was understood in terms of the
01
inefficiency of things like steam engines, which were all the rage at the time.
02
Anytime you try to burn fuel to do useful work such as pulling a locomo-
03
tive, there is always some waste generated in the form of heat. Entropy can
04
be thought of as a way of measuring that inefficiency; the more waste heat
05
emitted, the more entropy you’ve created. And no matter what you do, the
06
total entropy generated is always a positive number: you can make a refrig-
07
erator and cool things down, but only at the cost of expelling even more
08
heat out the back. This understanding was codified in the second law of
09
thermodynamics: the total entropy of a closed system never decreases, stay-
10
ing constant or increasing as time passes.
11
Boltzmann and
his colleagues argued that we could understand entropy
12
as a feature of how atoms are arranged inside different systems. Rather than
13
thinking of heat and entropy as distinct kinds of things, obeying their own
14
laws of nature, we can think of them as properties of systems made of at-
15
oms, and derive those rules from the Newtonian mechanics that applies to
16
everything in the universe. Heat and entropy, in other words, are conve-
17
nient ways of talking about atoms.
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Boltzmann’s key insight was that, when we look at an egg or a cup of
19
coffee with cream, we don’t actually see the individual atoms and molecules
20
of which it is made. What we see are some observable macroscopic features.
21
There are many possible arrangements of the atoms that give us exactly the
22
same macroscopic appearance. The observable features provide a coarse-
23
graining of the precise state of the system.
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Given that, Boltzmann suggested that we could identify the entropy of
25
a system with the number of different states that would be macroscopically
26
indistinguishable from the state it is actually in. (Technically, it’s the loga-
27
rithm of the number of indistinguishable states, but that mathematical
28
detail won’t concern us.) A low- entropy configuration is one where rela-
29
tively few states would look that way, while a high- entropy one corresponds
30
to many possible states. There are many ways to arrange molecules of cream
31
and coffee so that they look all mixed together; there are far fewer arrange-
32
ments where all of the cream is on the top and all of the coffee on the
33
bottom.
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With Boltzmann’s definition in hand, it makes perfect sense that
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01
entropy tends to increase over time. The reason is simple: there are far more
02
states with high entropy than states with low entropy. If you start in a low-
03
entropy configuration and simply evolve in almost any direction, your en-
04
tropy is extraordinarily likely to increase. When the entropy of a system is
05
as high as it can get, we say that the system is in equilibrium. In equilib-
06
rium, time has no arrow.
07
•
08
09
What Boltzmann successfully explained is why, given the entropy of the
10
universe today, it’s very likely to be higher- entropy tomorrow. The prob-
11
lem is that, because the underlying rules of Newtonian mechanics don’t
12
distinguish between past and future, precisely the same analysis should
13
predict that the entropy was higher yesterday, as well. Nobody thinks the
14
entropy actually was higher in the past, so we have to add something to our
15
picture.
16
The thing we need to add is an assumption about the initial condi-
17
tion of the observable universe, namely, that it was in a very low- entropy
18
state. Philosopher David Albert has dubbed this assumption the Past Hy-
19
pothesis. With that assumption, and an additional (much weaker) assump-
20
tion that the initial conditions weren’t finely tuned to make the entropy
21
decrease even further with time, everything falls into place. The reason why
22
the entropy was lower yesterday than it is today is simple: because it was
23
even lower the day before yesterday. And that’s true because it was even
24
lower the day before that. This reasoning proceeds stepwise all the way back
25
14 billion years into the past, right to the Big Bang. That may or may not
26
have been the absolute beginning of space and time, but it’s certainly the
27
beginning of the part of the universe we can observe. The origin of time’s
28
arrow, therefore, is ekinological: it arises from a special condition in the
29
far past.
30
Nobody knows exactly why the early universe had such a low en-
31
tropy. It’s one of those features of our world that may have a deeper explana-
32
tion we haven’t yet found, or may just be a true fact we need to learn to
33
accept.
34
What we know is that this initially low entropy is responsible for the
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“thermodynamic” arrow of time, the one that says entropy was lower
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toward the past and higher toward the future. Amazingly, it seems that this
01
property of entropy is responsible for all of the differences between past and 02
future that we know about. Memory, aging, cause and effect— all can be
03
traced to the second law of thermodynamics and in particular to the fact
04
that entropy used to be low in the past.
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Memories and Causes
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Every person’s life is caught in the relentless grip of time. We are born
young, grow older, and die. We experience moments of surprise and
delight, as well as periods of profound sadness. Our memories are
cherished records of the past, and our aspirations help us map our future
18
plans. If we want to situate our everyday lives as human beings in a natural
19
world governed by physical laws, one of our first goals must be to under-
20
stand h
ow the flow of time relates to our individual lives.
21
You may be willing to believe that something straightforward and me-
22
chanical, such as increasing entropy, can be responsible for something
23
equally straightforward and mechanical, such as how cream mixes into cof-
24
fee. It seems harder to establish that entropy is responsible for all of our
25
experience of the flow of time. For one thing, the past and future seem not
26
only like different directions but also like completely different kinds of
27
things. The past is fixed, our intuition assures us; it has already happened,
28
while the future is still unformed and up for grabs. The present moment,
29
the now, is what actually exists.
30
And then along came Laplace to tell us differently. Information about
31
the precise state of the universe is conserved over time; there is no funda-
32
mental difference between the past and the future. Nowhere in the laws
33
of physics are there labels on different moments of time to indicate “has
34
happened yet” and “has not happened yet.” Those laws refer equally well
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to any moment in time, and they tie all of the moments together in a
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unique order.
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We can highlight three ways that the past and future seem radically dif-
01
ferent to us:
02
03
• We remember the past, but not the future.
04
• Causes precede their effects.
05
• We can make choices that affect the future, but not the past.
06
07
All of these features of how time works can ultimately be reconciled
08
with the fact that the universe runs according to time- symmetric laws by
09
the additional fact that the past had a lower entropy than the future. Let’s
10
look at the first two now, postponing for the moment the contentious issues
11
of choice and free will. We will get there (I predict).
12
13
•
14
There are few more important manifestations of time’s arrow than the phe-
15
nomenon of memory. We have impressions in our minds— not always per-
16
fectly accurate, but often quite good— of events that have happened in the
17
past. We do not, most of us agree, possess analogous impressions of the
18
future. The future may be predicted, but it cannot be remembered. This
19
imbalance accords quite well with our intuitive feeling that the past and the
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