[alecst]

U, the internal energy is objective.

The free energy F = U - TS is the maximum amount of work you can extract from the system. This depends on how much you know about the system. S does indeed depend on what you know about the system.

See the Gibbs Paradox for more information.

[mannykannot]

If two people disagree on the maximum amount of work that could be extracted from a given system (with both of them basing their figure on their own evaluation of S), are there any cases where it would it be impossible to empirically demonstrate that at least the proponent of the lower figure was wrong?

If only changes in S (and F) have measurable consequences, would that not merely mean that assigning an absolute value is an arbitrary choice, which would not mean the same as it being subjective (there could still be an objective conversion between one basis and another, as there is for kinetic energy in different inertial reference frames.)

In the Gibbs Paradox, there is no subjectivity in whether the gases being mixed are the same or different, and no subjectivity in what the change of entropy is in either case. The paradox is that it does not feel right that identity makes an objective difference between the two cases, but the empirically-demonstrable distinction between fermions and bosons shows that this intuition does not hold in general. I believe Von Neumann came up with a QM resolution of the paradox.

[jbay808]

> are there any cases where it would it be impossible to empirically demonstrate that at least the proponent of the lower figure was wrong?

Isn't it more interesting to examine a situation where it would be possible to empirically demonstrate that the proponent of the lower figure was wrong?

[mannykannot]

In that case we could objectively say that one value of S does not yield F for that system (given that F is defined as a maximum), but this would not resolve the general question of subjectivity.

[jbay808]

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.

[mannykannot]

> 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.