Left Right Left Right # Pairs in
Loafer Loafer Sneaker Sneaker Correct Box
LB SB SB SB 1
SB LB LB LB 0
SB LB LB SB 0
SB LB SB LB 0
SB LB SB SB 1
SB SB LB LB 0
SB SB LB SB 0
SB SB SB LB 0
SB SB SB SB 1There are three macrostates: 2 pairs in the correct box (the most ordered
macrostate), 1 pair in the correct box, and 0 pairs in the correct box
(the most disordered macrostate). There is 1 microstate corresponding
to the most ordered macrostate, 6 microstates corresponding to the next-
most-ordered macrostate, and 9 microstates corresponding to the most
disordered macrostate. It doesn’t always work out as neatly as this, but the
more the possible number of microstates, the more likely it is that the or-
dered macrostates are far less probable than the disordered ones. What
makes it so unlikely that the air molecules in the room will all migrate to
the upper three inches is that there are on the order of 10^25 air molecules
in the room, and the number of microstates in which all these molecules
are up near the ceiling is almost infinitesimal when compared with the
number of microstates in which the molecules are spread out all over the
room (to the great relief of the inhabitants of that room).
Entropy and Information
We are living in what has been described as the Information Age. Amer-
ica, the country whose vast industrial complex once churned out the au-
tomobiles and refrigerators that contributed so greatly to its wealth, has
nearly abandoned the production of these material commodities to places
where it can be done more efficiently (thanks to more modern production
equipment) or more cheaply (thanks to an abundant supply of labor). Yet
America still retains much of its leadership of the industrial world be-
cause the new coin of the realm is neither automobiles nor refrigerators,
but information; and America is at the forefront of the manufacture and
distribution of information.
But how does this relate to the concepts we have been examining? It was
194 How Math Explains the World