[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?

[deltaonefour]

>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?

No it is not affected by it. The meaning of position is never changed. Your knowledge of your position can change, but your actual position exists regardless of your knowledge or inaccuracies of your tools.

>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?

Random number generators are a rabbit hole. There's not even a proper mathematical definition for it. We're not sure what a random number is... we just have an intuition for it. Case in point, the xkcd article could not define it mathematically. This is the reason why the joke exists, because we're not even truly sure what it is or if random numbers are a thing. We have intuition for what a random number is but this is likely some kind of illusion similar to the many optical illusions produced by our visual cortex. If formalization of our intuitions are not possible then there is likelihood that the intuition is not even real.

>Statistical mechanics is based on looking at the ensemble of microscopic descriptions possible given what is known about the system and their probabilities.

ok take a look at this: https://math.stackexchange.com/questions/2916887/shannon-ent...

They're talking about deriving the entropy formula for fair dice. But they talk about it as if we don't have knowledge about physics, momentum and projectile motion. We have the power to simulate the dice in a computer simulation and know the EXACT outcome of the dice. The dice is a cube and easily modeled with mathematics. So then why does the above discussion even exist? What is the point of fantasizing about dice as if we have no knowledge of how to mechanically calculate the outcome? The point is they chose a specific set of macrostates that have uniform distribution across all the outcomes. It is a choice that is independent of knowledge.

[kgwgk]

Thanks for your reply!

You didn't address the first line in my comment about the definition and meaning of "pressure" so maybe we actually agree.

To ellaborate a bit, one may define "pressure" as the reading of a device that measures its exchange of momentum with the particles of gas averaged over time. The last bit is important because those microscopic impacts are discrete events. If we know [in a classical mechanics framework] the state of every particle in the gas we can predict when they will happen - and succesfully calculate the (averaged) "pressure" measurement.

However, one may also define and interprete "pressure" as a variable that - together with volume and temperature - characterizes completely the behaviour of an ideal gas in equilibrium. But if we have a precise knowledge of the physical state we could in principle do impossible things - like compressing the gas without effort or creating a temperature gradient.

If we have a fish contaminated with mercury and the concentration of 0.01% characterizes completely its toxicity we won't eat it. If we also know that the mercury is only on the surface we won't eat it either but in principle we could if we are careful. The content of arsenic in the fish remains the same although the meaning of that number changes - but of course if we're a bear unable to clean our fish the additional information doesn't change anything at all.

> They're talking about deriving the entropy formula for fair dice. But they talk about it as if we don't have knowledge about physics, momentum and projectile motion. We have the power to simulate the dice in a computer simulation and know the EXACT outcome of the dice. The dice is a cube and easily modeled with mathematics. So then why does the above discussion even exist? What is the point of fantasizing about dice as if we have no knowledge of how to mechanically calculate the outcome? The point is they chose a specific set of macrostates that have uniform distribution across all the outcomes. It is a choice that is independent of knowledge.

I can make a model where the moon is made of cheese. That model is independent of any knowledge about the true nature of the moon. But if I visit the moon and find that - surprisingly! - it's made of lunar rock I may re-evaluate the pertinence of that model.

The model where all the outcomes of the die are equally likely it's particularly useful when all the outcomes of the die are equally likely. If you have no additional knowledge - apart from the number of outcomes - you have no reason to prefer one outcome to another. All of them are equally likely - to you. You can calculate the entropy of one event assuming that there are six equally-probable possible outcomes.

If I know exactly the future outcomes of the die - 4, 2, 5, 1, ... - I can also calculate the entropy of each event assuming that there is one single possible outcome that will happen with certainty. You have one model. I have one model. Are all models created equal? If we play some game you'll painfully realize that my model was better than yours - or at least you'll believe than I'm incredibly lucky.

[deltaonefour]

All mathematical formulas representing physical phenomena are called models. Some models are more accurate then other models.

Entropy is one such model. The mathematical input parameter that goes into this model is a macrostate. We are also fully aware that the model is an approximation Just like how we're aware newtonian mechanics and probability itself is an approximation.

If you feel entropy is too vague of a description then you can choose to use another model for the system. One with billions of parameters and can record the exact state of the system. Or you can use Entropy, which has it's uses just like how classical mechanics still has uses.

[kgwgk]

Ok, we agree then. Models may or may not represent a physical reality. They may be in conflict with reality - as in "the moon made of cheese". They may be incomplete - as in "the fish is 0.01% mercury". Those inaccuracies may or may not have practical relevance. Fundamentally it makes a difference though. In principle, someone with a better model of the die can consistently win bets contradicting the predictions of the "fair die" model and someone with a better model of the gas can do things forbidden by the "entropy is a measure of the energy unavailable for doing useful work" interpretation.

To reconcile those views in the context of your first comment: "Entropy is not a function of knowledge."

Entropy is a function of the macrostate. The macrostate is defined by state variables (the constraints on the system). Those state variables represent what is known about the system. Given P1, T1 we calculate S(P1, T1). Given P2, T2 we calculate S(P2, T2). The entropy obviously change with our knowledge in the sense that if we know that the pressure is P1 and the temperature is T1 we calculate one value and if we know that the pressure is P2 and the temperature is T2 we calculate a different value. If we don't know P and T we cannot calculate _one_ "entropy value" for the system at all because the corresponding macrostate is not defined.

"Two people with varying and different levels of knowledge of a system does not mean the system has two different entropy values."

What is the “entropy value of the system”?

Imagine that the system is composed of two containers with equal volumes of an ideal gas at the same temperature and pressure that are then put together - the volume is now the sum of the volumes, the pressure and temperature don’t change.

Alice can calculate S1 and S2 and the final entropy is SA=S1+S2.

Bob knows something that Alice ignores: that it was hydrogen in one container and helium in the other. They will mix and he can calculate that in the end SB>S1+S2.

What is the “entropy value of the system”? It seems to be more a property of the description of the system than of the system itself.

I'll say more about that in a reply to https://news.ycombinator.com/item?id=31201129 (somehow I've missed that comment until now)