...but mix is an active verb.

Entropy is not disorder because mixing is not the same as mixed up. When you stop the mixing, clearly the block on the left is more ordered than the one on the right. The cells on the right are "all mixed up." Yet both blocks have the same entropy, 0 J/K. For entropy to be greater than 0, you have to have energy. You have to have exchange. The system has to be dynamic. All entropy is the entropy of mixing. But the opposite of mixing is not "order" -- the opposite of mixing is not mixing.

The Entropy of Mixing So, what is this "entropy of mixing?" All this is saying is that when you have particles and energy together, you have a dynamic system that is constantly mixing. The number of ways the system can be found (the thermodynamic probability, W) depends upon how much energy you have, how many particles you have, and what types they are. What's interesting here is that "what types of particles you have" includes how much energy the particle has as well as how exactly that particle has been excited -- translationally, rotationally, vibrationally, or electronically.

Calculation of W

Two dynamic, mixing systems will have different entropies if they have different numbers of ways of being found. In this case, there are only two types of atoms on the left, but four types on the right. The entropy of the system on the right, with twice as many types of atoms, has almost exactly twice the entropy of the system on the left. This is because of how we calculate W:    W = N!/Na!Nb!Nc!... (The smaller numbers in the denominator for the calculation of W for the system on the right clues us that W on the right will be larger.) W = 100!/50!50! W = 100!/25!25!25!25!

But what is mixing?

As you look at the mixing system, what do you see? Are the particles themselves exchanging position, or are they just trading colors? The fact is, we can't know. (OK, actually we can know. If you look at the code, they are trading color.) And, even better, it doesn't matter. Take your pick. The calculation is the same; the result is the same. If you want to think of all of entropy in relation to the exchange of energy, feel free. If you can "see" it better as particles exchanging position, then that is just as good. You will get exactly the same result.

System and Surroundings

Reset In this case what is happening is that we are allowing the system on the left to exchange energy with the system on the right. What do you think is going to happen when we consider the TOTAL entropy of these two systems? Click "Reset" a few times, and see what happens. Notice that the overall entropy (the sum of system and surroundings) goes up. Also, there is a lot of fluctuating going on. Because energy is being exchanged, our numbers that go into calculating W are changing as well.

Caveate -- Energy Distributions

Technically, if we were considering energy in the other cases, we would see fluctuation there as well, but we were ignoring that. What we are seeing in those early examples is only one possible distribution of energy in a system. Here we are seeing a more complex picture, because each time energy is exchanged, we get a new distribution of it in both the system and the surroundings. What you see here is more like what you would see in a real system, even all by itself, if we were considering the distribution of energy. With larger systems the fluctuations tend to get smaller and less noticeable, but they are (theoretically) still there.