It took me 10 years to understand entropy

If that doesn't, I'm not sure what would. Maybe it would help if I taboo the word "subjective". Are you familiar with Maxwell's demon? Let's set up a variation of that experiment.

I have a partitioned box full of air at room temperature and pressure in both partitions. There's a frictionless door that can be open and closed by an ultrafast servomechanism. The servo is connected to a computer which will read a very long bitstring from a magnetic hard-drive platter at a high frequency and open the door when the bit is '1' and close it when the bit is '0'.

Admittedly, this mechanism would be hard to construct in practice, but I hope it's clear enough as a thought experiment.

Now if you're familiar with Maxwell's demon, you'll agree that there are particular, albeit rare, joint configurations of gas microstate and hard drive bitstring, such that after the servo has finished its last motion, the gas will have been separated into hot and cold on either side of the partition. This temperature difference can be used to extract work.

For each possible bitstring on the drive, there are certain corresponding microstates of gas that will maximize the free energy extracted by this process.

And for each possible microstate of the gas, there are certain corresponding bitstrings that will maximize the free energy extracted by this process.

(For the vast majority of other combinations of hard drive bitstring and gas microstate, the operation will have no effect).

The claim "entropy is subjective" is basically just an acknowledgement that the energy extractable from the gas is dependent on both the state of the gas itself and also the data written to the hard drive. It means that two experimenters, tasked with writing the initial data on the hard drive to extract as much work as possible from the gas, will have different levels of success depending on whether they know the particular microstate of the gas (and can thus select the corresponding optimal bitstring) or if they don't know the microstate of the gas beyond "a box at room temperature and pressure", and have to guess a bitstring based on only that. And when the operation is successful, we can describe the data on the hard drive as "information about the gas microstate that was used to extract work".

This experiment, of course, is so impractical that it sounds ridiculous. But we can make a more controlled version of it on the small scale, with excited trapped atoms, and actually make it work.

Beyond me too, but I'm going to assume that this is sort of an edge case where fluid crystalline structure can have less entropy than the static version (sort of sounds like the laminar flow state has less complex structure than its packed solid state). I doubt it contradicts your description above (which is similar to my understanding as well).

+1. The commenter above also wanted cared about bijective mappings, and squaring a random variable in [-1, 1] is not bijective. Squaring a random variable defined over positive real numbers would lead to a bijective mapping and the distribution would still remain uniform.

Actually, I find it hard to come up with a bijective mapping that leads to a non uniform distribution that's useful for anything practical.

> If that doesn't, I'm not sure what would.

It is not clear to me in which direction you think the question would be resolved, given a situation in which it would be possible to empirically demonstrate that the proponent of the lower figure (for S) was wrong. Maybe that is because I do not see the connection between your thought experiment and this issue: neither of the candidate values for S entail a particular distribution of states, let alone a particular sequence of future times when a molecule will approach the gate in a particular direction.

As I see it, your experiment is a difficult-to-perform way to demonstrate that, due to the inherent randomness of thermal processes, the entropy of a closed system may decrease when the conditions are right. This is explicitly covered in the article (see also "Monkeys typing Hamlet.")

Furthermore, in the case where the microstates of the system are measured in detail and the arrival times and velocities of the molecules at the gate are computed, one must add in the change in entropy resulting from those measurements and calculations. I am pretty sure this has been done, and is in accordance with the 2nd. law.

Your definition of 'subjective' in your penultimate paragraph is contrary to both common usage and what is being discussed in this thicket of threads, and appears to be closer to 'stochastic'. The outcome of the spin of a roulette wheel does not become subjective when different gamblers place different bets on it, or even when someone who has recorded statistics for its outcomes is able to place better-than-random bets.

I wrote two replies to this that I both deleted. Then I had a good long thunk, and here's what I came up with.

The temperature of an object can be determined through 1/T = dS / dE. What is this S? How can it exist if you know the system perfectly? And here is where the great insight comes. The thermometer! You apply a thermometer to a system you perturb it! The system may have started in one particular microstate, but the very nature of thermal contact involves random influence. Those random tiny influences from the thermometer allow the object (harddrive in our case) to enter a bunch of microstates with certain probabilities. And that's what S measures.

So our subjective knowledge does actually not matter. (Classically speaking) the system is in a particular microstate we may know it or not, and it still manages to have a temperature. That is due to the states it could hypothetically enter (but haven't yet)!

If we think back to the harddrive and it's contents: Very gently touching a harddrive with a thermometer while not scramble its contents. So we may say that microstates corresponding to different files than the ones you put there are actually not accessible. And they don't contribute to the entropy we used for the temperature.