because we are confining the ice cube molecules substantially. Their lo-
cation and velocity is highly restricted, whereas an individual water
molecule is free to zip around and go anywhere. The second law of thermo-
dynamics, in this viewpoint, is a statement about probability; it is more
likely that if a system evolves from one state to another, it will move to-
ward a state with higher probability. When we throw a die, it is less likely
to land less than 3 than greater than 3 because there are only two states
less than 3 (1 and 2) but three states greater than 3 (4, 5, and 6).
This gives a probabilistic explanation of why the ice cubes melt: there
are fewer states with ice cubes and hot water than with lukewarm water at
a uniform temperature. It also points out why systems tend toward equi-
librium: these are the states of highest probability, and any deviation
therefrom will naturally tend to evolve back toward a state of higher prob-
ability.
However, the statistical view of the second law opens a door that is hid-
den in the classical formulation. A system is not compelled to be in its
maximum-probability state, it is just more likely to be there than any-
where else. Unlikely though it may be, a glass of water at uniform tem-
perature may undergo a highly unlikely series of transitions, resulting
in a glass with rectangular cubes of ice immersed in hot water—or, even
more unlikely, with ice cubes shaped like miniature replicas of the Par-
thenon. When I was younger, I read One, Two, Three... Infinity, a won-
derful book by the physicist George Gamow.^8 In it, he describes a similar
situation in which all the air molecules in a room migrate to an upper
corner, leaving the unfortunate inhabitants gasping for breath. He then
does the calculation, and shows that we would have to wait roughly for-
ever for this to happen. I confess that I was definitely relieved to hear that
this was one more thing I wouldn’t have to worry about—but I probably
wouldn’t have worried about it had I not read the book.
Order and Disorder
Everyday life offers us a way to visualize the number of microstates asso-
ciated with a given macrostate. Linda, my wife, and I have a very different
view of the purpose of my closet. Linda thinks that the closet exists for
clothes to be hung in their proper places. Left to her own devices, she
sorts the hangers from left to right; shirts on the left, pants on the right.
The shirts and pants are further subdivided by whether they are work
clothes (characterized by whether they have stains from the colored over-
head markers I use when lecturing) or dress clothes (those that are still
pristine; all shirts and pants were initially purchased to be dress clothes,
192 How Math Explains the World