# Entropy of a system of black balls and white balls. The black balls are heavier

Let's say I have a Jar filled with equal amounts of black balls and white balls that are the same shape and size. The position of each ball in the jar is a random coordinate. Completely arbitrary.

The black balls are slightly heavier than the white balls.

When I shake the jar after a long period of time the black balls settle to the bottom of the jar and the white balls rest on top effectively dividing the jar into two sections one with black balls the other with white balls.

Let us also presume that the jar is a perfect cage. Energy cannot enter and energy cannot escape. The only thing that can enter the jar is gravity and your magical all seeing eye.

Now consider an equivalent system where the balls have equal weight. I shake the jar and the balls all become randomly placed.

How do these two systems differ? How would you describe the entropy of one vs the other? There is significant self organization in one but not in the other. I presume entropy has increased for both systems yet one is organized the other is not. If entropy describes the disorder of a system what is going on here? Why does the jar with the heavier black balls seemingly have less entropy or less disorder?

• If energy cannot escape the jar would be warmer where the stratification took place. I am confident someone will answer in a more rigorous way. Apr 5 '19 at 8:05
• I shake the both jars with the same amount of force. The exact same amount so that the energy I put into the system (I know I said contradictorily said that energy couldn't go in, but let's make this shaking an exception) is exactly the same. The heat energy in the system is the same yet the Spatial placement of the spheres is different. If entropy describes this spatial disordering then obviously one jar has less entropy than the other. Is there something more subtle going on here or is my understanding of entropy incorrect? Did entropy for both systems increase or decrease? @Alchimista Apr 5 '19 at 8:09
• If energy cannot enter, how are you shaking the jar? This gives the balls kinetic energy, after all. Apr 5 '19 at 8:13
• Read my comment right above your comment. @probably_someone Basically I said that about the jar to get rid of arguments of the overall entropy of the universe increasing while the entropy of the jar decreases. Energy is only entering the jars through shaking. That's it. Apr 5 '19 at 8:14
• @BrianYeh Then you should edit your question to say that, instead of "Energy cannot enter". Apr 5 '19 at 8:19

TL;DR: This analogy is unlikely to help your understanding of entropy. There are way too many caveats.

The position of each ball in the jar is a random coordinate. Completely arbitrary.

Given that the balls all have finite size, this isn't strictly true, as the allowed positions of one ball are constrained by the positions of the other balls (i.e. it can't be inside another ball). If the balls are actually point masses, on the other hand, this changes the problem completely - you are now dealing with a mixture of two gases*, the particles of one of which are slightly heavier than the particles of the other. The fact that the balls' finite size constrains the system's movement is very important to the dynamics of the system, and such systems are the subject of active research at the moment. Long story short, the situation you have presented may be more complicated than you think, but for the rest of this discussion, we will proceed with the assumption you originally intended, namely, that the balls have a finite size.

When I shake the jar after a long period of time the black balls settle to the bottom of the jar and the white balls rest on top effectively dividing the jar into two sections one with black balls the other with white balls.

This also isn't strictly true. In reality, the results after a long time heavily depend on how vigorously you're shaking the jar. In particular, it depends on both the amplitude of the vibrations and the frequency. If you don't shake the jar with enough amplitude, then nothing happens at all, as the balls are not displaced enough to move past each other. On the other hand, if you shake the jar with a high amplitude, then the spread in kinetic energy given to the balls is much greater than the difference in gravitational potential energy between the two types, and (assuming there is enough free volume in the jar for the balls to move quasi-freely with enough kinetic energy) the balls in the jar will simply adopt random configurations without ever settling into layers.

In order for what you say to have any chance of happening, given a particular frequency of vibrations, you're going to have to shake the jar hard enough to get the balls to overcome static friction and move past their neighbors, but not hard enough for them to move much further than that, or else the configuration will just completely randomize. The frequency of the vibrations affects the threshold at which this happy medium happens - generally, the higher the frequency of the vibration, the lower the amplitude that this happens at. You can see this happen in real life with various demonstrations of sand or other materials that "flow" under the influence of certain vibrations. You can see an example of too-vigorous shaking when you shake up a jar of layered rainbow sand for a long time and destroy the rainbow layering.

Even then, though, the layers will not be perfect, even after a long time. There will always be some intermingling of black and white balls at the interface between the layers, where the difference in total energy between the system's various almost-layered configurations and the totally-layered one is essentially zero.

Also, if we assume that there is friction between the balls, the jar gets steadily warmer as you shake it. We will assume that the balls do not change their properties as this happens. (Why friction is necessary for this scenario to make sense will be covered in the next paragraph.)

Let us also presume that the jar is a perfect cage. Energy cannot enter and energy cannot escape. The only thing that can enter the jar is gravity and your magical all seeing eye.

By shaking the jar, you are adding energy to the balls inside. In real life, even without friction, eventually the balls in the jar have sufficiently high kinetic energy to exert forces on (and hence leak energy into) the shaker. But you have forbidden this from happening when you say "energy cannot escape." So there is no way for an equilibrium to be reached; the energy contained in the jar will keep increasing forever. The only way for any kind of kinetic equilibrium to be reached is if there is a sufficiently strong dissipative force within the jar that turns kinetic energy into heat. Some combination of friction and deformation of the balls would eventually, at a high enough kinetic energy, dissipate just as much heat as the energy input of the shaking of the jar. Given that, again, no energy can escape your jar, the balls will continue growing hotter and hotter indefinitely, but if we're only concerned about the spatial arrangement of the balls, we can ignore this detail as long as we assume that the balls' properties do not change at all as they heat up infinitely.

Now consider an equivalent system where the balls have equal weight. I shake the jar and the balls all become randomly placed.

Once again, this only happens if you shake the jar vigorously enough. If not, the configuration of the system will change only slowly, if at all, from its initial state.

How do these two systems differ? How would you describe the entropy of one vs the other?

Again, this depends on how vigorously you're shaking them. Let's assume that both jars have the same total energy, the same volume, and the same number of black and white balls in them. If both jars are being shaken very vigorously, then the difference in gravitational potential energy between the two kinds of balls at the same height is not likely to be important relative to their kinetic energy. In this case, both jars are adopting essentially randomly-arranged microstates, which are equally accessible in each case, so they should both have basically the same entropy. (The entropy is not exactly equal, but the difference is indistinguishably small at a high enough kinetic energy.)

On the other hand, if both are being shaken just enough for them to "flow," then the equal-weight jar will have a wider range of accessible microstates. After all, the energy of the equal-weight jar does not change if you arbitrarily swap black and white balls for a given configuration, while it definitely does for the unequal-weight jar (again, the reason this is significant now is because the total kinetic energy of the balls is low enough that the contribution from gravitational potential is non-negligible).

If, on the other hand, you intended to ask this question after you had stopped shaking both jars, then the situation is different again. In this case, none of the balls are moving in either jar. There is only 1 accessible microstate for each jar's macrostate (namely, the particular configuration of balls that it's already in), and so both jars have the same entropy.

There is significant self organization in one but not in the other.

Only in particular cases. See above.

I presume entropy has increased for both systems yet one is organized the other is not.

This is because entropy doesn't really have a whole lot to do with "organization," which is good, because organization is a subjective concept. Instead, entropy is a measure of how many microstates are accessible in a given macrostate. It's a measure of how many possible configurations the system can be in while also having a particular set of macroscopic quantities.

Also note that we have explicitly ignored the thermal-energy contribution to the total energy of the box, which would ordinarily contribute to the entropy, because if we didn't ignore it, your system would never be in equilibrium, statistical mechanics wouldn't apply, and entropy wouldn't even have a sensible definition.

If entropy describes the disorder of a system what is going on here?

It doesn't describe the disorder of a system. See above.

Why does the jar with the heavier black balls seemingly have less entropy or less disorder?

Because some of the configurations that are accessible to the equal-weight jar at a given energy are not accessible to the unequal-weight jar, because the unequal-weight jar's total energy changes if you swap black and white balls. It's literally a matter of counting up the possibilities that preserve the macroscopic properties of each system (here, this means total kinetic+gravitational potential energy, volume, and number of each kind of ball).

*A gas is characterized by two properties: 1) the separation between particles is much larger than their size, and 2) interaction between particles is very weak or nonexistent (see e.g. https://en.wikipedia.org/wiki/Gas). Property 1 is true by default, as point particles have zero size, and property 2 is true based on the fact that they were balls to begin with, which do not attract or repel each other beyond contact interactions.