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not disconcert us. Most laws dealing with the macroscopic behavior of matter are Section 3.8
really statistical laws whose validity follows from the random behavior of huge num- Entropy, Time, and Cosmology
bers of molecules. For example, in thermodynamics, we refer to the pressure P of a
system. The pressure a gas exerts on the container walls results from the collisions of
molecules with the walls. There is a possibility that, at some instant, the gas molecules
might all be moving inward toward the interior of the container, so that the gas would
exert zero pressure on the container. Likewise, the molecular motion at a given instant
might make the pressure on some walls differ significantly from that on other walls.
However, such situations are so overwhelmingly improbable that we can with com-
plete confidence ascribe a single uniform pressure to the gas.

3.8 ENTROPY, TIME, AND COSMOLOGY

In the spontaneous mixing of two different gases, the molecules move according to
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Newton’s second law, F m d r/dt mdv/dt. This law is symmetric with respect
to time, meaning that, if t is replaced by t and v by v, the law is unchanged. Thus,
a reversal of all particle motions gives a set of motions that is also a valid solution of
Newton’s equation. It is therefore possible for the molecules to become spontaneously
unmixed, and this unmixing does not violate the law of motion F ma. However, as
noted in the previous section, motions that correspond to a detectable degree of un-
mixing are extremely improbable (even though not absolutely impossible). Although
Newton’s laws of motion (which govern the motion of individual molecules) do not
single out a direction of time, when the behavior of a very large number of molecules
is considered, the second law of thermodynamics (which is a statistical law) tells us
that states of an isolated system with lower entropy must precede in time states with
higher entropy. The second law is not time-symmetric but singles out the direction of
increasing time; we have dS/dt 0 for an isolated system, so the signs of dS and dt
are the same.
If someone showed us a film of two gases mixing spontaneously and then ran the
film backward, we would not see any violations of F ma in the unmixing process,
but the second law would tell us which showing of the film corresponded to how
things actually happened. Likewise, if we saw a film of someone being spontaneously
propelled out of a swimming pool, with the concurrent subsidence of waves in the
pool, we would know that we were watching a film run backward. Although tiny pres-
sure fluctuations in a fluid can propel colloidal particles about, Brownian motion of an
object the size of a person is too improbable to occur.
The second law of thermodynamics singles out the direction of increasing time.
The astrophysicist Eddington stated that “entropy is time’s arrow.” The fact that
dS/dt 0 for an isolated system gives us the thermodynamic arrow of time. Besides
the thermodynamic arrow, there is a cosmological arrow of time. Spectral lines in light
reaching us from other galaxies show wavelengths that are longer than the corre-
sponding wavelengths of light from objects at rest (the famous red shift). This red shift
indicates that all galaxies are moving away from us. (The frequency shift results from
the Doppler effect.) Thus the universe is expanding with increasing time, and this
expansion gives the cosmological arrow. Many physicists believe that the thermody-
namic and the cosmological arrows are directly related, but this question is still unde-
cided. [See T. Gold, Am. J. Phys., 30, 403 (1962); S. F. Savitt (ed.), Time’s Arrows
Today, Cambridge University Press, 1997.]
The currently accepted cosmological model is the Big Bang model: Much evi-
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dence indicates that about 13.7 billion (13.7 10 ) years ago, the universe began in
an extraordinarily dense and hot state and has been expanding ever since. It was for-
merly believed that the rate of expansion of the universe was slowing down due to