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is uniform. Once again, the sorted condition (fast molecules on one side,
slow on the other) is a state of low probability, and the- uniform
distribution is a state of high probability. Upon contact, the system
evolves spontaneously from a state of low probability to one of high

probability.
3. A stirred cup of coffee. Suppose that you stir a cup of coffee and then
remove the spoon. Over time, the circulation of the liquid decreases and

viscosity causes the energy of the rotating fluid to be dissipated as internal
energy of the molecules. In the initial state there is an ordered motion of
the swirling coffee. In the final equilibrium state there is a random
molecular motion. Once again, the ordered circulation of the molecules is

a state of low probability, while the random disordered motion is a state of
high probability. In this natural process, the system has gone from a state
of low probability to a state of high probability.

In all three cases above, the system has gone spontaneously from a state
of low probability to one of high probability. All three of these situations
are irreversible natural processes that are characterized by an increase in

the entropy of the system. It is, therefore, reasonable to conclude that
there is a quantitative relationship between probability and entropy. This
relationship, which was proposed by Boltzmann, is

S  k ln P
Here k is the Boltzmann constant, and S is the entropy of the system. P, to
define it loosely, is the number of different molecular arrangements that
correspond to the same macroscopic state. If, for example, the

circumstances are so special that only one arrangement is possible, then P
= 1 and S (= k In P) = 0, and we have a completely ordered state. Larger
values of P, such as that corresponding to the more uniform distribution

of molecules in the container, give larger values of the entropy. When we
say that state A of a system is more probable than state B, we simply mean
that state A has a larger value of P.
An increase in entropy has also been said to be a measure of the increase

in the disorder of a system, and, thus, entropy is a measure of disorder.
The term "disorder" does not have a precise mathematical definition, but
it is qualitatively related to the probability. A state of low disorder is a

state in which the components of a system have been carefully sorted,
such as by placing all the molecules with low speeds in one part of a
system. A state of high disorder is a random state in which no sorting has

occurred. The increase in entropy of a system in natural processes can,
thus, also be regarded to as an increase in the level of disorder of the

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