🚀 go-pugleaf

In article <1992Oct1.223858.10776@hellgate.utah.edu>,
tolman%asylum.cs.utah.edu@cs.utah.edu (Kenneth Tolman) writes:

>There are two spheres in the following area divided into "macrostates" and
>"microstates":
>
>-------------
>|     |     |
>|     |     |
>|     |     |
>-------------
>|     |o    |
>|     |     |
>|     |   o |
>-------------
>
>The bottom right corner is a special state, the others are not interesting.
>The probability of it being in this state is 1/16... what is its entropy?

In statistical mechanics the entropy of a macroscopic state is given
by k log W, where k is Boltzman's constant, and W is the number of
microscopic states that give rise to the same macroscopic state.
Before you can determine the entropy of a system, you need to know
what a (sufficiently) complete microscopic description of the state
is, and what a complete macroscopic description of the state is.
Usually, in classical statistical mechanics the microscopic state is
determined by giving the positions and momenta (to a certain
accuracy). The macroscopic state is given by the total energy, volume,
and number of particles.

In your case, you haven't really said what a complete microscopic and
macroscopic description is. Just for definiteness, I will assume that
the particles are distinguishable, and that a complete microscopic
description consists of saying for each sphere in which of the four
boxes the sphere can be found. Since you declared the lower right box
"special", I will assume that the macroscopic description consists
simply of giving the total number of spheres in the lower right box,
and the total number of spheres in all other boxes (we don't care
which box a sphere is in, as long as we know that it is not in the
lower right box).

Given all this, there is only one possible state consistent with the
macroscopic state "Two spheres in lower right, none elsewhere". So the
entropy is zero.

>Now consider the system at some time later:
>
>-------------
>|     |  o  |
>|     |     |
>|     |     |
>-------------
>|     |o    |
>|     |     |
>|     |     |
>-------------
>
>What is the entropy of this, and why?

The macroscopic state is given by "One sphere in lower right, one
elsewhere". Assuming distinguishable spheres, there are 6 possible
microscopic states. The first sphere can be in the lower right, while
the second sphere can be in one of three boxes, or the second sphere
can be in the lower right, while the first is in one of three boxes.
Thus the entropy is k log 6.

If instead you assume that the spheres are indistinguishable, then the
number of possible microscopic states is 3---one sphere is in the lower
right, while one sphere is in one of the 3 other boxes.

Daryl McCullough
ORA Corp.
Ithaca, NY