Where do people keep getting the idea that entropy is a measure of 'disorder' in a system, and that the second law says that things tend to become 'disordered'? I've taken a thermodynamics class that I think covered entropy and the second law of thermodynamics pretty thoroughly, and I've never seen anything to suggest it. I can see how something with more entropy might generally be considered more 'disordered' because it has a larger W (or omega, or however you want to do it) which would mean that there were more possible positions and energies for the particles in the system, but that doesn't have anything to do with 'order' in the way that we normally think of order, like a completed building being more ordered than a random pile of glass, steel and concrete. It's mainly just about the phase, temperature, and heat of the matter in the system.

I've even seen people describe entropy as the way in which a room tends to get messy, or the way in which complex machines tend to break down. This sort of analogy is confusing at best, misleading at worst. People don't seem to realize that entropy has to do with heat distribution, not vague notions of order or disorder. Where the heck do people come up with this stuff? Is it just the sort of thing that people who've never had a thermodynamics class repeat to each other?

Entropy is also spoken of in information theory, Shannon's entropy, which uses much the same descriptive statistics as thermodynamic probability. Perhaps the existence of two really quite opposed concepts of entropy may have led to the fuzzy application of the term. Consider Maxwell's demon, that is a cybernetic control system that lessens classical entropy through sustaining non-homogeneity. I think it would be characterized by a high degree of Shannon's entropy.

Seems to me imagining the complete engulfment of universe in a uniform and minimum heat distrubution "death" would be one in which minimal order existed. Order sustains diversity. Entropy tends towards uniformity. I suspect a general understanding of order verses disorder couched in terms of information coordination verses entropy has use in general discussion.

Hi Nasor,

I've taken a thermodynamics class that I think covered entropy and the second law of thermodynamics pretty thoroughly, and I've never seen anything to suggest it.

In order to fully understand the reason for that, you should be a little more patient and take a class in statistical mechanics, which is the "foundation" of thermodynamics through the laws of Newton and/or quantum mechanics.

I can see how something with more entropy might generally be considered more 'disordered' because it has a larger W (or omega, or however you want to do it) which would mean that there were more possible positions and energies for the particles in the system...

I would leave out the world "possible" in the previous sentence; I'll explain in a moment. For now, we'll talk about the Boltzmann definition of entropy, which is (roughly) given by:

S(W) = k_B ln |W|

where k_B is Boltzmann's constant, which I'll leave out (put equal to one) in what follows. W refers to a region in phase space, and I'll come back to that in a second too.

... but that doesn't have anything to do with 'order' in the way that we normally think of order, like a completed building being more ordered than a random pile of glass, steel and concrete. It's mainly just about the phase, temperature, and heat of the matter in the system.

Now you are mixing two things. You should clearly distinguish between two different notions of entropy:

you have entropy as defined in thermodynamics by the second law. There are various ways of putting the second law into words, but I prefer how Callen does it in his book Thermodynamics and an introduction to thermostatistics: Thermodynamic entropy is defined to be some quantity with the interesting property that it increases in time (for a closed system), is positive, additive and a convex function of its coordinates. Using the Maxwell transformation formulas, you can express entropy as a function of temperature, heat, volume, number of particles, ...

It is important to note that all these observables are macroscopic observables. From a Newtonian point of view, there is no such thing as "heat" (what is the temperature of one particle ? what is the volume of a point particle?). It is only when you put loads of particles together that you can talk about temperature, heat, volume, pressure, ... or as a whole: thermodynamics.

In this sense, entropy is, as you said, only a function of the phase, heat, temperature, ...

You can also talk about entropy as a "bridge" between the microscopic and the macroscopic world.

If you believe that Newton's laws (or if you want more refinement: quantum mechanics) lies at the basis of how particles interact, then there must be some way to explain pressure, heat, temperature by just looking at Newton's equations. Since I stressed the word macroscopic in the previous paragraph, it will probably not surprise you that this is only possible if you take many particles (Avogadro's number, N = 10²³). As you know, to describe a particle with Newton's laws, all you need to know are its positions and momenta, and then you can calculate everything. (Forces deliver the acceleration to move from a given position x₀ with initial momentum p₀ to somewhere else for each particle).

So the problem is obviously: if we only have positions and momenta, how can we relate them to macroscopic properties such as temperature and heat ? The answer is exactly entropy: Boltzmann realised that if you put

S(x) = ln |W(x)|

that the result of this formula is exactly the entropy of thermodyamics. And once you know the entropy (and the internal energy U which is the sum of all kinetic and potential energies of all particles, and ofcourse N, the number of particles) you know that it is a matter of applying Maxwell's transformations to get free energies, enthalpies, the whole thermodynamics thing. The beauty of the formula is that it relates the microscopic world, given by coordinates x to the thermodynamica (macroscopic) world, S.


So what does the Boltzmann entropy formula mean ? First of all, you should know that x = (q,p) is a short notation for the two vectors q and p which hold all the positions and momenta of the N particles:

q = (q₁ , q₂, ... q_N)
p = (p₁ , p₂, ... p_N)

Such an x describes the entire system (remember that knowing all positions and momenta is sufficient, and this is exactly what x gives us). If you look at the mathematical space where this variable lives in, then you'll see that it is a subset of R^6N, the 6N dimensional real space. This mathematical space is refered to as phasespace.

Now there is a slight problem. If you specify one "macroscopic" configuration, e.g. you say that you have N particles with a total energy U = <font face="symbol">S</font> p_i² / 2 m_i + V(q_i) , then you immediatelly see that you can usually find loads of configurations x which are compatible with this constraint. For instance, if U = 10 J and you have N particles (assume no potential energy), you can put all but one particles to rest, and put one of the N particles with momentum so that it gives 10 J in total. This already gives you N possibilities, and if you remember that N = 10²³, you'll probably agree that the number of configurations that is compatible with a given energy is huge ;).

So if we specify the macroscopic/b] parameters U, N and V (the volume where the particles are restrained to) then we probably have trillions and trillions of microscopic parameters x which are compatible with the macroscopic parameters.

This divides the entire phasespace R^6N^(*) into regions which group microscopic states that are compatible with a given macroscopic state. Such a group is refered to as W(x):

W(x) is the macroscopic state that corresponds to the microscopic state x.

^(*) Or rather V^N x R^3N where V is the 3 dimensional volume where the particles are restricted to, the momenta of the particles are not restricted so they are inR^3N.

I hope you are still with me at this point ;) ... Now, what is the experimental problem ? We usually have access only to macroscopic states, and never know the exact microscopic state that a system is in. If we were able to distinguish individual microscopic states, then we would never have to talk about temperature or entropy; if we always knew the exact microstate, then you could entirely describe the state of a glass of water by pinpointing its microstate in the phasespace. In practice this is ofcourse not possible, you specify the state of a glass of water by saying the volume it is contained in, how much water it is in weight and what temperature it has (related to energy).

So in practice, we are forced to work with macrostates and never with microstates. This leads to a small technical problem in the description of nature: if we only specify the macrostate W' = (U,N,V), then we have this trillions and trillions of microstates xto choose from which all have W(x) = W', so which one do we pick to let Newton's laws act on ?

This is where statistical mechanics comes in. It turns out that if you simply take all the microstates into account, that you get a very good description of what happens! (This is called the Gibbs ensemble theory). There is absolutely no fundamental justification in this but to say that "it just works". If you statistically assign equal weights to each microstate x, i.e. you say that given your macroscopic values, each of the microstates is equally possible the real microstate of the system, then this ensemble theory will predict up to a good accuracy the temperature and pressure and everything you want to know about your system.

What weight do you have to assign to each microstate in order to do this ? Very simple, you count the number of microstates that are compatible with your given macrostate W'; since we are in a continous space (real numbers) this is simple the so-called Liouville weight of the macrostate:

| W' | = &int; _R^6N dx I[ W(x) = W' ]

where I is the indicatorfunction (takes the value 1 if, in this case, W(x) = W' and zero otherwise), and the integral goes over the entire phasespace.

Now we're nearly there...

The Boltzmann entropy, by definition, is the natural logarithm of this Liouville weight. If you want to know the entropy associated to a given macrostate W', you just count the number of microstates that are compatible and take the logarithm of this number.

Now let's talk low and high entropy:

So what does it mean if the entropy is low ? For example, what happens if S = 0 ? If the logarithm is zero, this means that |W| must be equal to one, i.e. there is only one microstate x which is compatible with the given restrictions.

I hope you feel that this is a very very very very very unlikely thing to happen --- this means you have chosen an energy U which is such that there is only one way for the system to realise; in the case of a harmonic potential energy (V(q) = <font face="symbol">S</font> q_i²), U = 0 is such an example. You can only get this by putting ALL particles in the origin, q_i = 0 and they must have momentum p_i = 0. This is the only microstate compatible with that macroscopic energy value.

And what does high entropy mean ? This means that the logarithm is large ... and since the logarithm is a strictly monotone increasing function this means that |W'| must be large, or ... that there are a load of microstates that are compatible with a given macrostate. Basically this means that "there are loads of ways you can arrange your system and still get the same entropy, energy, volume, pressure, temperature, ...".

This is what is meant by disorder: usually it is the case that when there are loads of possible ways to arrange things, that all of those possibilities are "messy". Think of a free gas of particles: the equilibrium situation is where the gas is homogenously spread out in the entire volume. This is exactly the situation with the highest entropy -- if you would confine all the particles in a smaller region somewhere in the volume (i.e. [b]ordering the particles) then the entropy would decrease.

It is not limited to this situation alone: if you have a high entropy, then usually this means that you are looking at a system which is statistically "homogenously distributed in phasespace" (remember the equal weights, this is a homogeneous distribution), which also means that usually you have loads of possible momenta at your disposal ... many possible momenta = many ways of moving particles around and that leads again to the "idea of disorder".


In that particular sense entropy is related to order and disorder. You see that it is nothing to do with the thermodynamic entropy that you refered to, and only makes sense when you go a step further and interpret entropy through Boltzmann's formula.

I would also like to remark that the "usually" that I used here and there is very important. High entropy does not always mean disorder. One can prove for example that for systems where gravity is important, the highest entropy state is actually where matter clusters together. Example: the universe clusters into solar systems and galaxies because at universal scales, gravity is important, and clustered matter has a higher entropy than evenly spread out gas (situation after the big bang). This is the second law (entropy increases) at action.

And finally this: Mr. Chips remarked how in information theory and statistics sometimes one also talks about Shannon entropy. This is a totally different concept of entropy, but I can give you a slight hint on where it enters the story: I told you that you give equal weights to all microstates in a given macrostate (uniform distribution). This is a statistical operation... and that is also how you can introduce the statistical concept of entropy, the Shannon entropy. Note that Boltzmann entropy has nothing to do per se with statistics! ... I'll add that if you use this recipe, i.e. assigning equal weights to all microstates within a macrostate, that then, in equilibrium, "all entropies coincide" ... The Boltzmann entropy has the same value as the thermodynamic (Clausius) entropy, and also the Shannon entropy is the same.

The whole story on the later is a bit ... long to tell here.

Anyway, I hope this attempt to clarify helps. If you have any more questions, feel free to ask... you have just entered my domain of expertise ;).

Bye!

Crisp

Nasor, things tend to become more disordered through time (one way of showing that time is unidirectional). Now, this phenomenon must be explained in some way. Many textbooks have showed that the phenomenon is related to entropy and ITS change through time so, although its not a complete definition of entropy, it is still useful in helping 1st year physicists get some kind of comprehension before they understand exactly what it is.

Crisp:

Thanks for the detailed reply.

The thing is, this notion of order and disorder doesn't seem to relate to anything beyond the molecular level. Yet whenever I hear people talking about it, they relate it to macroscopic things that don't necessarily have anything to do with entropy. For example I could transform a building into a pile of rubble and although the resulting rubble pile would be more disordered than the original building, it's entropy could easily be the same.

Here's another way of looking at it: Imagine I have a system that consists of a box of 100 perfect crystals that are very near absolute zero. This system would have very low entropy. Now if I were to arrange the crystals so that they spelled my name or made some sort of complicated geometric design on the bottom of the box, they would undoubtedly be more ordered - but would the entropy be any different than if the crystals were lying on the bottom of the box randomly?

I can see how a system in which a large number of microstates could give the observed macrostate could be considered to be more 'disordered,' and how forcing a system into a state where only a few microstates could give the corresponding macrostate could be considered 'giving order' to the system. But you very rarely hear people talking about it like that.

For example, check out this web page that uses stacked vs. randomly piled brick as an example of entropy. http://hyperphysics.phy-astr.gsu.edu/hbase/therm/entrop.html

Would the entropy of the random brick pile actually be any different from the entropy of the carefully stacked bricks? I don't see any obvious reason why there would be more possible microstates in the macrosrate of the random pile vs. the stack.

You seem to understand this much better than I do, so please correct me if I'm missing something here.

Nasor:

Think of your building and all the bricks in it. The question is: how many ways can you put the individual bricks together to make a building? Obviously, there are many different ways. Now think: how many ways can you put the individual bricks together to make a pile of rubble (with no particular restrictions on its shape)? Certainly, there are more ways of doing this than there are ways of putting the bricks together to make a building.

In thermodynamic terms, there are more microstates corresponding to the "rubble" macrostate than there are miscrostates corresponding to the "building" macrostate. Leave things to themselves, and buildings will tend to become piles of rubble, but the reverse will almost never happen.

Now, most people will claim that a pile of rubble is more "disordered" than a building, especially a pile of rubble which can have a number of different shapes yet still contain the same bricks.

This is a rough example of why entropy is orften associated with disorder.

Nasor:

Think of your building and all the bricks in it. The question is: how many ways can you put the individual bricks together to make a building? Obviously, there are many different ways. Now think: how many ways can you put the individual bricks together to make a pile of rubble (with no particular restrictions on its shape)? Certainly, there are more ways of doing this than there are ways of putting the bricks together to make a building.

In thermodynamic terms, there are more microstates corresponding to the "rubble" macrostate than there are miscrostates corresponding to the "building" macrostate. Leave things to themselves, and buildings will tend to become piles of rubble, but the reverse will almost never happen.

Now, most people will claim that a pile of rubble is more "disordered" than a building, especially a pile of rubble which can have a number of different shapes yet still contain the same bricks.

This is a rough example of why entropy is orften associated with disorder.I can see how there would be many more possible microstates if you consider any randomly collapsed pile of rubble to be equivalent, but in reality each individual piece of destroyed building material in a real pile of rubble will still have just as discrete a location, position, phase, temperatures, etc. as when the building was still standing.

If you actually calculated the entropy of any given pile of rubble vs. the entropy of the building before it collapsed, would the entropy of the pile necessarily be any higher from an S=K ln(w) perspective?

If you actually calculated the entropy of any given pile of rubble vs. the entropy of the building before it collapsed, would the entropy of the pile necessarily be any higher from an S=K ln(w) perspective?

Yes. The problem is that there are just way too much things to count to get an intuitive idea (if you want to talk about how you can arrange 1000 bricks, then there are trillions of possible ways). It is easy to accidently leave out a few trillion when just thinking about it :).

I think the problem is the following: if you say "how is the entropy for a GIVEN pile of bricks higher than the entropy for a GIVEN building", then the answer is clearly: it isn't, both entropies are zero! If you know exactly how all the bricks are arranged in the pile, then you know the exact microstate. In that case, the macrostates and microstates coincide, so |W(x)| = 1 always, so entropy S(x) = 0 always.
[This is a very very very subtle, dangerous point, and an enormous mind breaker if you draw this reasoning a bit further. I will not do it (yet) here to prevent confusion ;)].

It is really because you do not precisely now how all the bricks are arranged, that there are an overwhelming number of possibilities which correspond to "bricks in a pile" and a lot less with "bricks as a building". A ridiculous example is perhaps two or three bricks labelled 1, 2 and 3. Brick 1 is a large brick (foundation), brick 2 is a bit smaller, and brick 3 is tiny, so they should be stacked in order to give a nice building.


There is only one way to make a "building" (all bricks stacked upwards nicely) with three bricks.
There are 3! = 6 ways to stack them in a pile (all three lying next to eachother on the ground), since now, which brick lies next to which brick is not relevant anymore.


Just for three bricks, this number is already a factor 6 different. If the only way to make a building would be to stack bricks vertically, and the only way to make a pile would be to lay them on the ground next to eachother (which is even quite "ordered" compared to a pile of bricks) then for N bricks the numbers would differ a factor N! ... If you consider that N! grows exponentially with N, then perhaps you already get a hint at why there are overwhelmingly more ways to arrange N bricks in a pile, compared to arranging them in a building.

Nasor:

If you want to consider all piles of bricks to be unique, then the entropy of each pile will be equal to the entropy of the building, which is itself just another kind of pile of bricks. However, in most cases we don't wish to distinguish between all the different piles, but we do want to distinguish between a "random" pile and a building. That's why we define two different macrostates of the bricks - the "building" state and all the states corresponding to piles.

Yes. The problem is that there are just way too much things to count to get an intuitive idea (if you want to talk about how you can arrange 1000 bricks, then there are trillions of possible ways). It is easy to accidently leave out a few trillion when just thinking about it :).

I think the problem is the following: if you say "how is the entropy for a GIVEN pile of bricks higher than the entropy for a GIVEN building", then the answer is clearly: it isn't, both entropies are zero! If you know exactly how all the bricks are arranged in the pile, then you know the exact microstate. In that case, the macrostates and microstates coincide, so |W(x)| = 1 always, so entropy S(x) = 0 always.
[This is a very very very subtle, dangerous point, and an enormous mind breaker if you draw this reasoning a bit further. I will not do it (yet) here to prevent confusion ;)].

It is really because you do not precisely now how all the bricks are arranged, that there are an overwhelming number of possibilities which correspond to "bricks in a pile" and a lot less with "bricks as a building". A ridiculous example is perhaps two or three bricks labelled 1, 2 and 3. Brick 1 is a large brick (foundation), brick 2 is a bit smaller, and brick 3 is tiny, so they should be stacked in order to give a nice building.


There is only one way to make a "building" (all bricks stacked upwards nicely) with three bricks.
There are 3! = 6 ways to stack them in a pile (all three lying next to eachother on the ground), since now, which brick lies next to which brick is not relevant anymore.


Just for three bricks, this number is already a factor 6 different. If the only way to make a building would be to stack bricks vertically, and the only way to make a pile would be to lay them on the ground next to eachother (which is even quite "ordered" compared to a pile of bricks) then for N bricks the numbers would differ a factor N! ... If you consider that N! grows exponentially with N, then perhaps you already get a hint at why there are overwhelmingly more ways to arrange N bricks in a pile, compared to arranging them in a building.This seems to imply that entropy is determined by how well we can observe the system, not an inherent quality of the system. I thought that the possible microstates that could result in a given macrostate all had to be indistinguishable when observing the macrostate – as in fundamentally indistinguishable, not just indistinguishable to whatever method of observation we happen to be using.

Am I wrong about this?

This seems to imply that entropy is determined by how well we can observe the system, not an inherent quality of the system. I thought that the possible microstates that could result in a given macrostate all had to be indistinguishable when observing the macrostate – as in fundamentally indistinguishable, not just indistinguishable to whatever method of observation we happen to be using.

Am I wrong about this?

Exactly the subtle point that I was trying to avoid :)

You see, the problem is that your ability to observe the system defines the macrostates, and hence also how many microstates are grouped into one macrostate, and thus entropy. The most clarifying example I heard in that context was the following:

Suppose that you have a transparant box of 100 red and green balls, and you mix them all. You can clearly talk about order and disorder here, and define macrostates by that; e.g. the number of macrostates where are the red balls are on top and all the green balls are at the bottom of the box is clearly a lot smaller than where they are all disordered. This will lead you to conclude that the macrostate with all balls as randomly mixed as possible is the one with highest entropy.

Now suppose that somebody who is colorblind looks at the box...

In that sense, entropy is very seriously limited to the choice of macroscopic observables that you take to describe your system. Luckily, for experimental purposes, the set of variables that you can choose is usually always the same: the typical experiment is performed at a given pressure, temperature, volume, number of particles (or mass). These are the "canonical" macroscopic observables that are used to divide phasespace into macrostate-slices. So in that sense Boltzmann's formula for the entropy must be supplemented with a "good set of macroscopic variables" to look at.

In a sense it is also not really surprising that entropy depends on the way you look at the system; after all, if you know that your experiment is performed at normal atmospheric pressure, then your system is not free to evolve over entire phasespace: there is a constant of motion, namely the pressure! This constrains your microscopic evolution to a hypersurface in phasespace, and it is not surpriseable that this changes the maximum entropy state! I'll let you think on that one for a while ;).

Bye!

Crisp

But after all this, do we all agree that the spontaneous disordering of a system though time is coupled with its increasing entropy?

But after all this, do we all agree that the spontaneous disordering of a system though time is coupled with its increasing entropy?

I do :D

[edited to lengthen message to at least 10 characters]

Heat-death is highly ordered - same amount of heat everywhere. Not so?

In any case, I think the original argument was more about an Eternal Champion (or if you like, information theory) sense of order/disorder. Since encoding human-context information into a volume of matter necessarily disorders it from its previous condition, (since entropy always increases), we can associate an increase in information (order) with an decrease in total energy gradient (order). We can then erase the information by randomizing the volume of matter, and thereby associate a decrease in information (order) with an decrease in total energy gradient (order).

So, it would follow that the two are not linked.

Heat-death is highly ordered - same amount of heat everywhere. Not so?

Not quite; "same amount of heat everywhere" is the most disordered possible state. Only if all the heat was concentrated in one single point, then it would be highly ordered.

Since encoding human-context information into a volume of matter necessarily disorders it from its previous condition, (since entropy always increases), we can associate an increase in information (order) with an decrease in total energy gradient (order). We can then erase the information by randomizing the volume of matter, and thereby associate a decrease in information (order) with an decrease in total energy gradient (order). So, it would follow that the two are not linked.

If you were to replace the words "energy gradient" by "entropy" in the above text, then I would totally agree. I do not understand where "energy gradient" comes in when you talk about the amount of information stored somewhere. Perhaps you in fact mean entropy but never labelled it as such ?

You see, the problem is that your ability to observe the system defines the macrostates, and hence also how many microstates are grouped into one macrostate, and thus entropy. So what about other things that are related to entropy, like Gibbs free energy? Can it also be changed simply by changing the way you observe your system?

For example: Say I have a system that consists of a sealed box of gas particles. I also have an amazingly high-tech scanning device that allows me to track the exact position and velocity of each gas particle in the box to maximum extent allowed by the uncertainty principle.

If entropy is really only affected by how precisely we can observe the system, then the entropy of the gas particles in the box should be relatively high before I turn my scanner on, since I'll know very little about their disposition. This would indicate that the Gibbs energy would be relatively low via G=H-TS

Since the the entropy would have to drastically decrease when I turn the scanner on and measure the position and velocities of the particles, this would also increase the Gibbs free energy...but how could the Gibbs energy change when nothing has really changed inside the box? Especially since the Gibbs energy can be used to calculate things like gas pressure and temperature – things that shouldn't depend on how much information I have about the contents of the box. I mean, the gas in the box should always have some objective temperature and pressure that's independent of whether or not I've 'taken a look,' right?

Even ignoring all that, I still don't see how entropy could necessarily be tied to notions of order and disorder even if it's dependent on what we can observe about the system. Let's say I have two systems that consist of two identical boxes. One box contains a working pocket watch that weighs 50 grams. The other box contains 50 grams of gas that's uniformly distributed throughout the box. If I can't see inside the box and the entropy really only depends on what I can observe about the macrostate, then I would have to conclude that the entropy of both systems was the same; each system consists of an identical box that contains 50 grams of indistinguishable matter. But I think everyone would agree that a pocket watch is more ordered than uniformly distributed gas. Am I missing something here?

Thanks for the detailed replies. It's obvious that my class didn't cover entropy as throughly as I thought it did.

Edit: Do you see why I'm having a hard time accepting that entropy is both a measure of disorder and affected by how well we're able to observe a system? I can certainly see how there is likely to be a correlation between how disordered a system is and how well we're able to extract information by observing it, but it seems to me that how 'ordered' a system is should be an independent, objective reality that exists regardless of our perception. That's what my example about the box with pocket watch vs. box with uniform gas was supposed to show.

So what about other things that are related to entropy, like Gibbs free energy? Can it also be changed simply by changing the way you observe your system?

Yes, but don't let this be a surprise. The catch is that implicitly in all these thermodynamic statements there are some assumptions regarding the way you measure the system. One of those is that you measure always such that you are in rest with respect to the volume the gas is contained in, that you use the same macroscopic observables to compute the entropy with (U, V, N), ...

Say I have a system that consists of a sealed box of gas particles. I also have an amazingly high-tech scanning device that allows me to track the exact position and velocity of each gas particle in the box to maximum extent allowed by the uncertainty principle.

In that case the entropy is trivially zero -- all the macroscopically accessible states, which define the macrostates in your system, are exactly the "microscopic" states, i.e. |W(x)| = 1.

... this would also increase the Gibbs free energy...but how could the Gibbs energy change when nothing has really changed inside the box? Especially since the Gibbs energy can be used to calculate things like gas pressure and temperature – things that shouldn't depend on how much information I have about the contents of the box. I mean, the gas in the box should always have some objective temperature and pressure that's independent of whether or not I've 'taken a look,' right?

Yes, let's not introduce some sort of measurement problem for "the gas in the box" :).

There is one caveat which is why the analysis is not applicable: first of all, you have not used that you are using a macroscopically large system (that N = 10²³ or of that order). From an experimental point of view, this prevents you from trivially having an entropy zero -- it is practically impossible to measure the exact microstate. Also, if you want to apply thermodynamics like saying G = H - TS, then you need to have many particles (thermodynamics is only valid for macroscopic systems -- when you study statistical mechanics you will see that thermodynamic relations are retrieved only when you take the limit N -> oo).

Secondly, if you want to use thermodynamics, you should also use the variables which it applies to; you should divide phasespace into slices according to your knowledge of U, V and N. This once again limits you from knowing the exact microstate, because thermodynamics simply does not allow you to make that kind of measurements within its applicable domain.

I agree that this seems extremely sloppy. The truth is also that these matters are far from resolved yet. Statistical mechanics and its relation to thermodynamics (entropy, how non-equilibrium situations can evolve in time towards thermodynamic equilibrium) is still an active research topic at any self-respecting university. The questions on "how can we relate thermodynamics to quantum or Newtonian mechanics" are pretty well understood. From the moment you drag in entropy and time evolutions however, you enter a domain with many different opinions ;).

Even ignoring all that, I still don't see how entropy could necessarily be tied to notions of order and disorder even if it's dependent on what we can observe about the system.

You should seperate this dependency on what we can observe from entropy as a notion of order or disorder. Even if you are limited in the observation, what seems to you the most disordered state is still the one with highest entropy. Perhaps (probably even) with a different value than with someone who has better instruments, but still the highest.

Let's say I have two systems that consist of two identical boxes. One box contains a working pocket watch that weighs 50 grams. The other box contains 50 grams of gas that's uniformly distributed throughout the box. If I can't see inside the box and the entropy really only depends on what I can observe about the macrostate, then I would have to conclude that the entropy of both systems was the same; each system consists of an identical box that contains 50 grams of indistinguishable matter. But I think everyone would agree that a pocket watch is more ordered than uniformly distributed gas. Am I missing something here?

If you cannot see inside the box, if god himself prevents you from looking inside the box, then you would not be able to make any conclusions about what is inside and then you are limited to what you see. A statement about entropy would be quite trivial then, since entropy in the case of two objects is quite limited in application :).

This again seems strange, but you have already positioned yourself as a superior being than the observer in the scenario, because you know what is inside the box. This already implies that you have better measurement devices than the actual observer you are talking about.

[I can certainly see how there is likely to be a correlation between how disordered a system is and how well we're able to extract information by observing it, but it seems to me that how 'ordered' a system is should be an independent, objective reality that exists regardless of our perception. That's what my example about the box with pocket watch vs. box with uniform gas was supposed to show.

Two remarks:
- The relation between order and disorder is only exactly valid in some scenarios; when gravity is involved, "more ordered" systems correspond to higher entropy. It is not a scientific fact that "entropy expresses visible order" (what you see) -- in some situations the order or disorder in phasespace, what entropy is really about, is just reflected to the visual representation of what we see.
- How ordered a system is, is really dependent on your observational qualities. You can look at a building and say that it is a nicely ordered "pile of bricks". On the other hand, if you turn on your color vision and then, to your astonishment, detect that all bricks have different colors, then your conclusion would probably be "my my, someone made a mess of properly arranging the bricks in that building such that all red bricks are clustered, all black bricks are clustered, ... "

Entropy, just like energy and countless other physical quantities, is observer dependent.

This is hard to understand because within conventional thermodynamics, there is no word about "the observer" (as I said before, everything is assumed to be at rest with respect to the observer, implicitly a "typical" set of variables is assumed, ...). When you drag in mechanics, where observers suddenly come in, there seems to be a paradox with thermodynamics, which can only be resolved by making these assumptions about the observation. They may seem absurd assumptions, but they are necessary to distille thermodynamics, a rather well working theory, from a mechanical perspective.