The main point of this exercise is to make clear the
important distinction between mixing and probability.
Entropy is defined in terms of probability. This definition
is not going to change anytime soon. Mixing is not the
same as probability. If you want a high-entropy state,
it is usually necessary but never sufficient for the
occupied sites and the vacant sites to be well mixed
together. The crucial contribution to the entropy is
_not knowing_ where the particles are. If you know where
they are, it doesn't matter whether they are mixed or
not ... as you can demonstrate with the "Peek" button:
* Peeking sets the entropy to zero, but does not
unmix the particles.
* Purifying the system umixes the particles, which
is another way of setting the entropy to zero.
* For this system, the only way of adding entropy is
by stirring. (However, if the veil is all the way down,
you can stir until the cows come home and the entropy
will remain zero.)
* Conversely, lack of mixing does not cause the entropy
to decrease. If you turn off the mixing, the entropy does
not change. It will not decrease unless you do something
to find out where the particles are (e.g. purifying them
or peeking at them).
To say the same thing another way: Some people labor
under the misconception that entropy must somehow be a
highly dynamic, active process, requiring constant mixing
and re-mixing. There is no experimental or theoretical
basis for believing this, and counterexamples abound.
For example, suppose I hand you a chunk of glass (or if
you want to be fancy, a chunk of spin glass). It has a
very considerable amount of entropy that is frozen in,
completely non-dynamic. The point is, you don't know what
microstate it's in, and you can't easily figure it out,
so it really doesn't matter whether the microstate is
changing or not.
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