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isolated system is conserved, and it is the distribution of energy (which is related to Section 3.7
the entropy) that determines the direction of spontaneity. The equilibrium position cor- What Is Entropy?
responds to the most probable distribution of energy.
We shall see in Sec. 21.6 that the greater the number of energy levels that have
significant occupation, the larger the entropy is. Increasing a system’s energy (for
example by heating it) will increase its entropy because this allows higher energy lev-
els to be significantly occupied, thereby increasing the number of occupied levels. It
turns out that increasing the volume of a system at constant energy also allows more
energy levels to be occupied, since it lowers the energies of many of the energy lev-
els. (In the preceding discussion, the term “energy levels” should be replaced by
“quantum states” but we won’t worry about this point now.)
The website www.entropysite.com contains several articles criticizing the
increasing-disorder interpretation of entropy increase and promoting the increasing-
dispersal-of-energy interpretation.

Fluctuations
What light does this discussion throw on the second law of thermodynamics, which
can be formulated as S 0 for an isolated system (where dS dq /T)? The reason
rev
S increases is because an isolated system tends to go to a state of higher probability.
However, it is not absolutely impossible for a macroscopic isolated system to go spon-
taneously to a state of lower probability, but such an occurrence is highly unlikely. Hence
the second law is only a law of probability. There is an extremely small, but nonzero,
chance that it might be violated. For example, there is a possibility of observing the
spontaneous unmixing of two mixed gases, but because of the huge numbers of mol-
ecules present, the probability that this will happen is fantastically small. There is an
extremely tiny probability that the random motions of oxygen molecules in the air
around you might carry them all to one corner of the room, causing you to die for lack
of oxygen, but this possibility is nothing to lose any sleep over. The mixing of gases
is irreversible because the mixed state is far, far more probable than any state with sig-
nificant unmixing.
To show the extremely small probability of significant macroscopic deviations
from the second law, consider the mixed state of Fig. 3.12. Let there be N 0.6
d
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10 molecules of the perfect gas d distributed between the two equal volumes. The
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most likely distribution is one with 0.3 10 molecules of d in each half of the con-
tainer, and similarly for the e molecules. (For simplicity we shall consider only the dis-
tribution of the d molecules, but the same considerations apply to the e molecules.)
1
The probability that each d molecule will be in the left half of the container is .
2
Probability theory (Sokolnikoff and Redheffer, p. 645) shows that the standard de-
1 1/2 12
viation of the number of d molecules in the left volume equals N d 0.4 10 .
2
The standard deviation is a measure of the typical deviation that is observed from the
most probable value, 0.3 10 24 in this case. Probability theory shows that, when
many observations are made, 68% of them will lie within 1 standard deviation from
the most probable value. (This statement applies whenever the distribution of proba-
bilities is a normal, or gaussian, distribution. The gaussian distribution is the familiar
bell-shaped curve at the upper left in Fig. 17.18.)
In our example, we can expect that 68% of the time the number of d molecules in
12
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the left volume will lie in the range 0.3 10 0.4 10 . Although the standard
12
deviation 0.4 10 molecules is a very large number of molecules, it is negligible com-
24
pared with the total number of d molecules in the left volume, 0.3 10 . A deviation
12
12
24
of 0.4 10 out of 0.3 10 would mean a fluctuation in gas density of 1 part in 10 ,
which is much too small to be directly detectable experimentally. A directly detectable
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6
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density fluctuation might be 1 part in 10 , or 0.3 10 molecules out of 0.3 10 .