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A
measure of uncertainty regarding the state
of a system:
for example, a gas molecule
at an unknown location in a large volume has a higher entropy than one known
to be confined to a smaller volume. Free energy
can be extracted in converting a low-entropy state
to a high-entropy state:
the (time-average) pressure exerted by a gas molecule
can do useful work as
a small volume is expanded to a larger volume. In the classical configuration space picture, any molecular system
can be viewed as a single-particle gas in a high-dimensional space. In the quantum mechanical
picture, entropy is described as a function of the probabilities of occupancy
of different members of a set of alternative quantum
states. Increased information regarding the state of a system
reduces its entropy and thereby increases its free energy,
as shown by the resulting ability to extract more work
from it. An illustrative contradiction in the simple textbook view of entropy
as a local property of a material (defining an entropy per mole, and so forth) can be shown as follows: The third
law of thermodynamics states
that a perfect crystal
at absolute zero has zero entropy*; this is
true regardless of its size. A piece of disordered material, such as a glass,
has some finite entropy G0 > 0 at absolute zero. In the local-property view, N
pieces of glass, even (or especially) if all are atomically identical, must
have an entropy of NG0. If these N pieces of glass are arranged in a regular
three-dimensional lattice,
however, the resulting structure constitutes a perfect crystal (with a large unit cell);
at absolute zero, the third law states that this crystal
has zero entropy, not NG0. To understand the informational perspective on
entropy, it is a useful exercise to consider (1) what the actual entropy of
such crystal
is as a function of N, with and without information describing the structure
of the unit cell,
(2) how the third law can be phrased more precisely, and (3) what this more
precise statement implies for the entropy of well-defined aperiodic
structures. Note that any one unit cell
in the crystal
can be regarded as a description of all the rest.
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