A friend of mine doesn't know what macrostates and microstates are, so I'm spending this journal entry to give a quick little description of what they are. What we conventionally think of as discrete events, are actually categories of events. When someone falls off a 12 story building, there are many variants to that fall, not just in who's falling and which building, but in the exact position and state of every molecule and unit of space in that 12 story fall. It doesn't matter if his hair's mussed one way instead of another, he's still falling off a freaking building. And that is why physicists came up with the concept of macrostate and microstate.
The classic example is to imagine a cardboard box lid full of blue and red marbles. If you draw a line on the box lid and place all the red marbles so they're to the right of that line, you could say "All of the red marbles are to the right of the line." That's a macrostate. Now swap the position of two red marbles--has anything changed? The macrostate is the same, because the macrostate is not the exact situation, but the category of situation in which it falls. You can mix around the marbles all you want, and as long as they're all red on that half of the box, you can say, "Half of this box lid has only red marbles."
A microstate is the exact situation. Two microstates are two arrangements of positions of marbles, even if the arrangements are indistinguishable. It's the exact position of all the molecules and particles and all the energy levels, every insignificant detail makes up a microstate. In the marbles example, each microstate would be all the positions of each marble.
It's important to note now that macrostates are categories of exact situations. Another way to say that is macrostates are categories of microstates. You can say 12 microstates are all the same, and this macrostate is how they are all the same. Then you can say these 23 microstates are all the same, but they're different from the other 12, and that's a second macrostate. Stuff like that. Macrostates are what we observe in the real world, when we drink tea that's a macrostate. When we get on a bus that's a macrostate. The event could have been any of a zillion microstates, but since they're all "the same" we don't care which microstate it is. The macrostate determines what we perceive as the real event.
So physicists and statisticians together will tell you their greatest assumption: that all microstates are equally probable. It's like rolling the dice. You have just as much change of getting exactly a 3, 6, and 1, as you do of getting exactly a 2, 3, and 4. This works with marbles too: if you dump them randomly into the box lid you have the same chance of one arrangement of marbles happening as any other specific arrangement of marbles. That might not make sense, but that's because we can't perceive microstates. Since they're all the same, they affect us in categories of similarity, not in exact, discrete states.
And that's the odd thing right there. All microstates are equally probable, but some macrostates have more microstates than others. The odd thing is that why should the most entropic macrostate have any more microstates than the others?
In the marbles example, if you have 20 marbles randomly distributed in the box lid, and you examine the macrostate "all red marbles are on the right." Then you have exactly 10! ways to arrange the red marbles on the right side, and 10! ways to arrange the blue marbles on the left side, for a total of 2*10! ways to arrange the marbles, or 2 * 10 * 9 * 8 etc...
If you examine the macrostate, "Three red marbles are on the left, and three blue marbles are on the right," then you end up with 3 positions already determined. If the 3 odd marbles are moved, then that changes the macrostate because we can see the difference between red and blue. So for one side you're left with permutations of the 3 odd marbles, and permutations of the 7 normal marbles, or 3! * 7!, which is less than 10! for one side as described above. The most entropic state is of course with the marbles evenly distributed, but in that case it would be 5! * 5!, which is even less than 3! * 7!
| Type | Equation | How many microstates |
|---|---|---|
| Least entropy (all 1 color on 1 side) | 10!*2 | 7257600 |
| More entropy (3 marbles mixed) | 3!*7!*2 | 60480 |
| Most entropy (all intermixed evenly) | 5!*5!*2 | 28800 |
So I wonder then, why is it that when I throw the marbles randomly in the box, they don't land with all of one color on one side more often than not? There sure seem to be more microstates with that as the outcome! I know you know that's totally untrue and you can easily prove my pitiful grasp of statistics wrong, but these are the kinds of things I cling to in hopes that someday I'll figure out how to make all the marbles fall into order, and that I will be able to change into something small, cute and furry, in a universe immortal.
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