# Does entropy depend on the observer?

Entropy as it is explained on this site is a Lorentz invariant. But, we can define it as a measure of information hidden from an observer in a physical system.

In that sense, is entropy a relative quantity depending on the computation, measurement and storage capacity of the observer?

• Entropy measures the number of microstates that correspond to some set of (not necessarily macroscopic) order parameters. If two observers agree on what the order parameters are then they must also agree on the entropy. So the question is whether the relevant order parameters (e.g. $N$, $V$, $T$ for canonical thermodynamic entropy) are invariant. Feb 27, 2017 at 9:31
• @lemon Why not necessarily macroscopic? I thought that macroscopic order was necessary Feb 27, 2017 at 10:15
• @lemon If length is not invariant, does this mean volume isn't either? Is there such a thing as relativistic entropy?
– user140374
Feb 27, 2017 at 10:24
• Related: physics.stackexchange.com/q/193677/50583 and its linked questions. Feb 27, 2017 at 10:37
• @veronika One can select any set of parameters that represent a simplified description of the system (e.g. average bond length). An entropy may then be assigned. Feb 27, 2017 at 11:15

## 2 Answers

E.T. Jaynes agrees with you, and luckily he is a good guy to have on your side:

From this we see that entropy is an anthropomorphic concept, not only in the well-known statistical sense that it measures the extent of human ignorance as to the microstate. Even at the purely phenomenological level, entropy is an anthropomorphic concept. For it is a property, not of the physical system, but of the particular experiments you or I choose to perform on it.

This is a quote from his short article Gibbs vs Boltzmann Entropies'' (1965), which is a great article on the concept of entropy in general, but for this discussion in specific you can turn to section VI. The "Anthropomorphic" Nature of Entropy. I will not try to paraphrase him here, because I believe he already described himself there as succinctly and clearly as possible. (Note it's only one page).

I was trying to find another article of him, but I couldn't trace it at the moment. [EDIT: thanks to Nathaniel for finding it]. There he gave a nice example which I can try to paraphrase here:

Imagine having a box which is partitioned in two equally large sections. Suppose each half has the same number of balls, and they all look a dull grey to you, all bouncing around at the same velocity. If you now remove the partition, you don't see much happen actually. Indeed: if you re-insert the partition, it pretty much looks like the same system you started with. You would say: there has been no entropy increase.

However, imagine it turns out you were color blind, and a friend of yours could actually see that in the original situation, the left half of the box had only blue balls, and the right half only red balls. Upon removing the partition, he would see the colors mix irreversibly. Upon re-inserting the partition, the system certainly is not back to its original configuration. He would say the entropy has increased. (Indeed, he would count a $\log 2$ for every ball.)

Who is right? Did entropy increase or not? Both are right. As Jaynes nicely argues in the above reference, entropy is not a mechanical property, it is only a thermodynamic property. And a given mechanical system can have many different thermodynamic descriptions. These depend on what one can --or chooses to-- measure. Indeed: if you live in a universe where there are no people and/or machines that can distinguish red from blue, there would really be no sense in saying the entropy has increased in the above process. Moreover, suppose you were color blind, arrive at the conclusion that the entropy did not increase, and then someone came along with a machine that was able to tell apart red and blue, then this person could extract work from the initial configuration, which you thought had maximal entropy, and hence you would conclude that this machine can extract work from a maximal entropy system, violating the second law. The conclusion would just be that your assumption was wrong: in your calculation of the entropy, you presumed that whatever you did you could not tell apart red and blue on a macroscopic level. This machine then violated your assumption. Hence using the 'correct' entropy is a matter of context, and it depends on what kind of operations you can perform. There is nothing problematic with this. In fact, it is the only consistent approach.

• "turns out you were color blind, and a friend of yours could actually see that in the original situation, the left half of the box had only blue balls" - I'd recommend changing that to red and green. Most color-blind people have trouble distinguishing red from green. None of the three types of dichromats have trouble distinguishing red from blue. A condition with no hue discrimination whatsoever is extremely rare (1 in 40000 vs 1 in 40). Feb 27, 2017 at 12:56
• How exactly could work be extracted from the pools of pure red and blue balls? Feb 27, 2017 at 16:06
• Yes, I get that part, what I don't get is that fact that if there is a distinguishing trait, there is automatically a way to exploit that to generate work. Why does one follow from the other? Feb 27, 2017 at 23:17
• @MikeWise For a way to generate work think osmosis (and osmotic pressure). Box has red balls in left compartment, blue in right,equal volume, number average kinetic energy and therefore pressure. Now let the membrane be permeable only to red balls. Red balls will go to the right until equilibrium is reached. Eventually left compartment will have only red balls, while the right will have red and blue balls, and it will have higher pressure. Now allow your membrane to slide, turning it into a piston and you have preformed work! Feb 28, 2017 at 9:07
• To be clear, I agree with all the actual physical content of what you/ Jaynes are saying. But the obvious conclusion to me is not that entropy is subjective, but that one can sometimes get away with ignoring irrelevant contributions to it (as one can do with energy). Mar 12, 2017 at 19:21

I think that Shannon-von Neumann definition of Entropy pass this anthropocentric paradox by establishing the minimum amount of information that cannot be reversibly be exchanged between two states of the same system, no matter if there is an agreement or even the presence of observers. In such way Entropy is indeed a physical characteristic and not an observer artifact, plus establishes a unique direction for the flow of information, hence causality, flow of time etc..

I know I am simply placing postulates against each other and I am in no position to establish or hint at the correctness of one or another, but I prefer keeping my physics understanding within the boundaries of experimental verification.