[jbay808]

> But if you look deeper than that averaging it stops making sense to me. It's a completely different world.

I think you're less confused than you think you are!

As I posted elsewhere, it helps to think of entropy as a quantity that actually depends on how much you know about the system in question.

Typically when you calculate the entropy of a system at temperature X, that means all you know is that you stuck a thermometer in it and measured X. You don't know anything more than the average temperature. It could be in any state consistent with that temperature.

If you know more about the system, it has less entropy. If you know it down to the exact microstate, it has zero entropy.

[vishnugupta]

This is how I have come to understand entropy. The words disorder and order are a proxy for information content.

> If you know more about the system, it has less entropy.

One question though. When you say "it" does it include you as well as the system or just the system? To me "it" includes both because by it is "you" who's state has changed by acquiring more information. It could be in the form of neuronal rearrangement or bits being stored in some digital media etc., A new information content has thus been created.

There's an interesting side effect if one thinks deep enough here. The system will keep changing its state so the information one is out of date thus leading to more disorder (i.e., information loss) and increased entropy. One can keep the information updated but it takes energy. And I read somewhere that the energy thus used will lead to increase in overall entropy of the universe and thus the 2nd law.

[oh_my_goodness]

>This is how I have come to understand entropy. The words disorder and order are a proxy for information content

Does information content mean this? ... "How many bits of random-number generator would I need to make the number of micro-states in the macro-state?"

[vishnugupta]

That is my mental model, yes. More bits are needed to capture more detailed (or micro-states as you called it elsewhere in this thread, or finer-grained) information.

Let's say there's a stone, we want to know its details. If all we want to know is whether it weighs more than 100KG or not then one bit will do. 1 means > 100KG and 0 means < 100KG. If we want to know its colour (as one of 7 WIBGYOR) as well then we need 4 bits; 3 bits to encode 7 colours and 1 bit to encode yes/no for the weight. And so on..as we gain more and more information we need more bits to store that.

This is just for the storage though; in order to gain the information we need to expend energy. More information requires more energy leading to more disorder as expending energy releases heat and thus 2nd order of thermodynamics as well as arrow of time. IMO our perception of time is purely based on memory which is information content of event stored in Neurons.

Quite a bit of hand-wavy. But this is a mental model I've developed over the years of thinking and reading (and listening to lectures) about entropy, information, arrow of time, and energy and how they are interconnected.

[lubesGordi]

The article does say that some crystalline structures can have more entropy (information) than their fluid state. How could that be? Any ideas on what that fluid state might be? The information content in a crystal is really low.

[vishnugupta]

Unfortunately the author doesn't explain it beyond sharing a reference to this paper[1] which is way beyond my competence.

[1] https://www.nature.com/articles/nature08641

[lubesGordi]

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).

[andi999]

Entropy (differences) are an objective quantity which can be measured, there is no subjectivity about it. It is not which parameters you know it is about which parameters you hold fixed.

[jwarden]

Fascinating discussion. I see some parallel here to the Bayesian vs Frequentist view of probability.

They are perhaps both valid points of view depending on the situation.

If you take a frequentist view of an unbiased coin, then the probability that it will land heads on the next flip is objectively, by definition 50%. So the resulting calculation of entropy (log 50% = 1 bit) is also objectively defined. But if your 50% probability represents a subjective belief, the resulting entropy calculation should also be considered subjective, I would think.

[toxik]

Leonard Susskind disagrees with you. See his lectures on statistical mechanics, he is very clear that entropy is a matter of knowledge about the system. It has to be.

[oh_my_goodness]

Susskind says that entropy is determined by selecting a macro-state. He doesn't claim that the entropy of a macro-state depends on whether we know which macro-state the real system is really in.

If we happen to know, then, sure. For example we could pick a weird-ass observable state, and when we saw it we would know the entropy of the system was low. But the entropy of each macro-state just depends on how many micro-states we define it to contain. It doesn't depend on our knowledge of the system state.

[jbay808]

The concept of entropy wasn't invented so that we could calculate entropies of macrostates, it was so that we could calculate entropies of real systems and understand their behaviour. Macrostates are an accounting tool that helps us do this. You seem to be treating the calculation of macrostate entropy as an end-goal in itself, but also allowing yourself to somehow freely choose any macrostate you want. When it comes to applying thermodynamics in practice, you'll have to calculate the entropy of a real, or at least hypothetical, system.

The point of macrostates is that you ought to know which macrostate a given system is in. That's the thing that you know. You don't know which microstate it's in, but you do know which macrostate it's in.

For example, if I say "a cubic metre volume of air at room temperature and pressure", I've described a physical system. I've also described a macrostate.

If you're calculating the entropy of macrostates that are not consistent with a description of a system -- if you've defined your macrostates such that you don't know which macrostate a given system is in -- then in order to calculate that system's entropy you have to sum up over all such possible macrostates anyway, so you haven't saved yourself any work or earned any insights along the way.

So yes, you can calculate the entropy of a macrostate without knowing what macrostate a real system is in, but it kind of sounds like you're arguing that log(x) is not a function of variable x, because log(3) is a constant and log(4) is a constant, and you can divide up any x into constants of your choice.

[oh_my_goodness]

We seem to be stuck in a loop of explaining basic first-year statistical mechanics back and forth to each other repeatedly. I'm not sure why.

I'm making a pedagogical point. The OP addresses how difficult entropy is to understand. I'm responding to that. We don't need to talk about "knowledge" when you define entropy, or in an initial explanation of entropy. We could, but we could decide not to.

The log(x) example is a good one. First-time students who are learning about logarithms don't need to be told that a logarithm depends on 'knowledge' or on 'information.' It's ok to just tell them how logarithm is defined.

Sure, there is information. I'm saying it's confusing and unnecessary to introduce more big ideas like information, when the topic is "entropy is difficult to understand" or "logarithms are difficult to understand."

[jbay808]

Alright. I agree we seem to be stuck in a weird loop where we agree about all the observable facts but somehow are on different wavelengths in spite of that.

And I totally agree, entropy is a property of a macrostate. The information step comes inseparably in when you go from a system description to a macrostate. And you might just shuffle the confusion from not knowing what entropy is to not knowing what a macrostate is.

If you think it's clearer to teach students by explaining that the entropy of a macrostate is an objective property of that macrostate, that's fine. Just don't leave them believing that the entropy of a brick is an objective property of that brick.

[chermi]

I think you might enjoy cosma shalizi's paper "What is a macrostate?" https://arxiv.org/abs/cond-mat/0303625

[im3w1l]

I've been trying to reconcile these perspectives, and I think it really is both. And they are both physically relevant.

Consider the subjective entropy perspective. If you know the exact microstate of a system, then you can in theory play the part of Maxwell's demon. You could have a little gate that you open only for fast particles, and using your knowledge of the microstate, you can predict exactly when they will arrive.

But consider the objective perspective. If you take this very same system and put it in thermal contact with another system, then an objective entropy perspective is the relevant one. Those systems will equilibrize and your subjective knowledge is irrelevant to that process.

I haven't fully wrapped my head around it yet, but I do think that acknowledging both is a step in the right direction at least.

[jbay808]

> If you take this very same system and put it in thermal contact with another system, then an objective entropy perspective is the relevant one. Those systems will equilibrize and your subjective knowledge is irrelevant to that process.

The subjective view handles this scenario just fine, though, and makes more accurate predictions than the objective view.

For example, there are systems where some aspects of the original microstate survive thermal contact with another system. We use such systems to store data! I bet your hard drive is in thermal contact with its environment right now! It's very hard to reconcile this with an objective take on entropy.

And there are some systems that will rapidly be scrambled. The subjective perspective has no problem admitting that your knowledge of a system can become inaccurate and useless. Even without thermal contact, you'd need to perform a tremendous amount of (perhaps reversible) computation in order to make a functioning Maxwell's demon with your initial microstate conditions, because the microstate will evolve in time in a complicated way. The subjective view is still totally consistent with entropy of a system increasing over time!

[im3w1l]

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.

[jbay808]

No, it is subjective. We just only have such blunt instruments for practically measuring states, relative to the gargantuan amount of entropy in most real systems, that the subjective nature of entropy is easy to miss. But in a world where the frontiers of thermodynamics have moved from steam engines to lasers, computers, DNA, and black holes, the difference is increasingly obvious and important.

With steam engines, we got away with treating a volume of gas as having not only a few parameters that we knew and cared about, like mass, temperature and pressure, but we could further deceive ourselves into thinking that those were the only parameters that existed to describe the system. The only parameters that were knowable. But Boltzmann knew better.

Look at Boltzmann's formula, S = kB log W.

For any single particular system you describe to me, W will be 1, and so S will be 0. So it's only if you describe an ensemble of systems -- that is, if you describe a system vaguely, such that I am left to imagine the details -- that we have nonzero entropy. If you ask me to calculate the entropy of that "system", that macrostate, that ensemble, then sure, I'll end up with nonzero entropy. But if I ask you to keep transmitting more data about the scenario, then with each further description, you'll be narrowing the state space and thereby decreasing the entropy.

Look, since the entropy of a macrostate is nonzero, but the entropy of any single microstate which is consistent with that macrostate is zero, it's clear that entropy is not an intrinsic property of any real system. It's a property of how many other possible non-existent systems could be swapped out for the one in front of you, without you noticing the change.

If I swap out the air in your room for an equal volume of air at equal temperature and pressure, you probably won't notice.

If I swap out the hard drive in your laptop for an equal volume of hard drive at equal temperature and pressure, you probably will!

[raattgift]

Maybe better to say that the universe does not appear to pick out a single coarse-graining or fine-graining procedure for practically any system.

For instance, following your Boltzmannian example, I think one would notice swapping 1 µm³ of the r/w head and 1 µm³ of the recording surface of a new, freshly powered on HDD more than one would notice substituting the entire HDD for a new one of the same model and turning that on. And here I am already using units of length (cf. "equal volume"), and we know neither units nor lengths are generally picked out by the universe.

[deltaonefour]

Very few people know this but.

Information entropy and statistical mechanical entropy are two different things.

They share the same equation and the same name but they are two unrelated concepts. You have conflated the two. The person you are responding to is referring to statistical entropy.

Basically in this entire thread nobody, including you, is fully grasping the situation.

[chermi]

They are not at all unrelated. It is not easy to grasp, so I understand the confusion. https://en.m.wikipedia.org/wiki/Landauer%27s_principle

Fun rabbit hole would start with classic paper by jaynes

Many more recent examples relating bit erasure costs of computation. Some names to look up if interested include charlie Bennet,Dave wolpert, James crutchfield, Susanne still, for starters.

Edit -- a collection of ideas related to this problem and mixing in "complexity" can be found in SFI proceedings called "Complexity, entropy, and the physics of information"

[jbay808]

I respectfully disagree. Perhaps you'd like to present more than a mere assertion to make your case. I did.

If it helps, here's a paper that explains my stance in more detail. https://bayes.wustl.edu/etj/articles/theory.1.pdf

[kgwgk]

If you think there is no relation between the different things called entropy apart from the name maybe you're not fully grasping the situation either.

[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.

[oldsecondhand]

I think he is confusing the usage of entropy in physics and computer science. In computer science entropy is conditional probability and depends on what we know about a system.

[kgwgk]

As it does in physics!

"which parameters [thermodynamic variables] you know" ~ "which parameters [thermodynamic variables] you hold fixed"

(or know in average, like the energy for a system in a heat bath where the temperature is fixed)

https://bayes.wustl.edu/etj/articles/theory.1.pdf

http://nicf.net/articles/thermodynamics-statistical-mechanic...

[oldsecondhand]

You can observe the movement of molecules beyond macro properties like temperature.

[kgwgk]

Sure, at least in principle. And if you knew what every molecule was doing the entropy would vanish.

[IX-103]

I've thought that for a while, but I'm not a physicist. Do you know any prominent physicists that hold that view? It seems to contradict at least the popular narrative about entropy as a property of a system.

[deltaonefour]

No this is completely and utterly wrong. Entropy is not a function of knowledge.

Two people with varying and different levels of knowledge of a system does not mean the system has two different entropy values. Even if I knew the exact position of all atoms in a cup of water, the temperature of that water does not change due to that knowledge.

Entropy does rely on what your picked configuration of macro states and microstates. Temperature is an arbitrary choice of macrostate.

[jbay808]

> Even if I knew the exact position of all atoms in a cup of water, the temperature of that water does not change due to that knowledge.

It actually does! You would disagree with the other person about the temperature of that water. But I agree that this is admittedly not obvious at first.

[deltaonefour]

No it does not. The thermometer does not change based off of my knowledge or opinion.

[jbay808]

A thermometer doesn't measure temperature any better than a meterstick measures length. And we all know what Einstein had to say about the relativity of metersticks.

To paraphrase from the paper I linked in another reply to you, a thermometer is just a heat bath equipped with a pointer which reads its average energy, whose scale is calibrated to give the temperature T, defined by 1/T = dS/d<E>.

You can read the thermometer if you like, but if you know the exact microstate of the water to begin with, the thermometer reading will tell you much less than you already knew about the water. And precise knowledge of the water's microstate will (theoretically) allow you to extract much more work from that water than you would be able to with only the thermometer reading.

[deltaonefour]

But entropy does not change with this knowledge.

[jbay808]

You seem pretty convinced. Let me see if you're talking about the same pedantic distinction that oh_my_goodness was.

A: "an urn containing either a white ball or a black ball".

B: "I notice that the ball in the urn A is white".

I would say that initially the entropy of our ball-urn system is 1 bit, and that with observation B, we have reduced the entropy of our ball-urn system to 0 bits.

But if you are going to take the view that even knowing the ball in this particular urn is actually white doesn't change the fact that the entropy of <<"an urn containing either a white ball or a black ball">> is 1 bit and not 0, and that that's the entropy that we're discussing, then I won't argue about it any further.

Except for the obligatory xkcd: https://xkcd.com/221/

[kgwgk]

> Even if I knew the exact position of all atoms in a cup of water, the temperature of that water does not change due to that knowledge.

If you knew the exact position of all atoms in a cup of water you wouldn't assign any temperature to it. Not a thermodynamic temperature at least.

[deltaonefour]

The number of microstates does not change, even if you KNOW the the cup of water is in a specific microstate.

The boltzman equation is based on total accessible microstates.

[kgwgk]

"accessible" means something only given a set of constraints.

Like the temperature, if you keep the temperature of the water fixed. And the number of molecules if instead of a cup you have a close container to prevent it from evaporating. Then what you have is water at some temperature that you control. And you could have the water at a different temperature with exactly the same microstate.

Or imagine gas at some fixed temperature within a cylinder with one movable wall. If you knew the location of every molecule of the gas it wouldn't make sense to talk about its pressure - you could compress it (reducing the number of accessible microstates) without doing any work.

Edit: In summary, thermodynamics loses its meaning if you know the microstate and can act on that knowledge.

[deltaonefour]

>it wouldn't make sense to talk about its pressure -

If I have a pressure gauge that reads the same thing regardless of my knowledge how is pressure meaningless? The tool that reads pressure gives me an accurate pressure number regardless of what I know or don't know. This number is correct.

Your argument is basically saying that the pressure gauge becomes wrong once you have more knowledge of the system. No it doesn't. The pressure gauge is still giving you a number defined as "pressure."

The gas in that cylinder is at a specific microstate within the macrostate defined as pressure.

[kgwgk]

> The pressure gauge is still giving you a number defined as "pressure."

As long as you define “pressure” as “the reading of the manometer” and not as “the variable that together with temperature specifies the state of the gas and measures the quantity of energy required to compress it further”.

Thermodynamics is based on state variables giving a complete description of the system. Statistical mechanics is based on looking at the ensemble of microscopic descriptions possible given what is known about the system and their probabilities.

If all you know is a handful of thermodynamic variables that ensemble is huge. If you know already the microscopic description of the physical system your ensemble has one single possible configuration in it.

As in jbay808’s xkcd example, if you have a random number generator and you know the sequence of numbers that will be generated, do you have a random number generator? The random number generator is still giving you a number defined as “random”, right?

I guess that it’s still random if you “forget” that you know it in advance and that the macrostate is still meaningful as a complete description of the physical system if you “forget” that you have a perfect knowledge of its state.

Edit: the GPS receiver in my phone is giving me some coordinates defined as “position” that happen to be in the middle of the road. However, I know precisely where I am. Don’t you think that the meaning of that “position” is somehow affected by this additional information?

[jbay808]

If the cup of water is in a specific microstate at time t=0, and evolves over time according to deterministic equations of motion, how will it "access" other microstates that aren't along that specific trajectory in phase-space?

[deltaonefour]

It can't. But you're not typically defining ONLY microstates along that trajectory as accessible. You are defining all accessible configurations according to your defined macrostate.

Knowledge of future microstates does not change what was already defined as a macrostate. The definition and the rules you used to construct a macrostate are independent to knowledge of the system.

If you gain knowledge of the system and you would like to change your macrostate, then be my guest. You can certainly do that, but "entropy" as we know it does not actually change with more knowledge unless you change the parameters according to your gained knowledge.

Think of it this way. The thermometer ALWAYS reads the same thing EVEN if you have 100% knowledge of the current microstate. You can build a new thermometer using some other mechanism to get a different reading and to take advantage of your new found knowledge... but you'd be changing the definition of your macrostate.

[oh_my_goodness]

I think that works ok. But I think it's an unnecessarily tricky explanation. Entropy per macro-state decreases as we look at finer-grained macro-states. It feels simpler to associate the entropy of each macro-state with that macro-state, rather than assuming we know which macro-state the system is in, and then attributing the lower entropy to our knowledge of the macro-state.

I think it can probably be expressed either way. I just think the "knowledge" part is tricky and can be left out.

[jbay808]

Can you elaborate on the difference between a "fine-grained" macrostate and a macrostate that is not fine-grained?

I think you will find it hard to separate the concept of a macrostate from the state of knowledge (or ignorance) of an individual subjective observer.

[oh_my_goodness]

Sure. A fine-grained macro-state contains fewer micro-states. A coarse-grained macro-state contains more micro-states.

Say I flip 8 coins and I don't look at the results. A fine-grained macro state is TTTT TTTT. A coarser-grained macro state is TTTT xxxx. The one has 4 bits more entropy than the other. It works the same way in statistical mechanics. Call them spins.

We're just talking about some ensemble of micro states, and then we divide the ensemble up into macro-states. To do statistical mechanics at all, I think I have to define some macro-states according to which micro-states they contain. That doesn't mean I necessarily have any information about which macro-state the system is actually in.

[jbay808]

> A fine-grained macro state is TTTT TTTT.

This macrostate seems fine enough to be a microstate, but sure. For the macrostates with at least one 'x' in it, that 'x' seems to be a placeholder for the concept of subjective ignorance.

> That doesn't mean I necessarily have any information about which macro-state the system is actually in.

But the entire purpose of the exercise of assigning microstates to macrostates is so that you can match up a description of some system to a microstate ensemble and calculate its entropy! Otherwise there's no point to arbitrarily labelling various groups of microstates.

To follow your example more practically, let's say you have an 8-spin system, whose net spin is zero. (You know because you've measured its overall magnetic moment or something). I've just described a system that is in one of the following possible microstates:

TTTT HHHH, TTTH HHHT, ..., HHHH TTTT

Now you can go ahead and define the macrostates as fine-grained as you want, where TTTT HHHH and HHHH TTTT are in different macrostates, but to calculate the entropy of this system, you're going to have to sum up all of those macrostates anyway to get the one that's consistent with the described system.

[oh_my_goodness]

Good review of common ground. At this point hopefully the active folks in the discussion can see that we're all describing statistical mechanics exactly the same way.

What I'm saying is pedagogical. We need to define our macro-states. We don't need to go on and talk about our definitions being information or 'knowledge.' We could just use the definitions and calculate. We can talk about 'knowledge' but we don't need to.

The exception is when we actually have some information about what macro-state some system is really in. Obviously we then have to build that information into our model, and the entropy changes. What I'm saying is this: it's not necessary to mix that into our definition of entropy. That definition is not going to help folks who don't understand entropy, and it's unnecessary.

[kgwgk]

How is a macrostate TTTTxxxx different from having the information about the TTTT part and not about the xxxx part?

Talking bout the entropy of the macrostate TTTTxxxx makes sense only conditional on the TTTT information.

[oh_my_goodness]

It's different because I can define a macro-state without any information about which macro-state any system is actually in. As I think you're also saying, the only information I need is information about how I've defined my own macro-states.

If we just define the macro-states, we're good to go. We don't need to talk about 'knowledge'. We can talk about 'knowledge', it's fine, but that lets in unnecessary woo.

[kgwgk]

I'm not sure I see what's the point of that distinction.

The entropy of a macrostate is a measure of the indetermination about the microstate conditional on the macrostate. If you don't want to call that 'knowledge' the substance of the matter doesn't change.

A macrostate is not an intrinsict property of a physical system. It's related to our description of the system. In general, the same microstate of the system of interest may be compatible with multiple macrostates.

Given the thermodynamical description of some system I can calculate the entropy if T=T_1 and the entropy if T=T_2 without knowing what's the actual temperature specifying the macrostate. But in the first case the calculation is conditional on the hypothetical information T=T_1 and in the second case conditional on T=T_2.

[gota]

Oh my god, this explanation is gold. Thank you. I'm going to save it and refer to it in the future.