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All mathematical formulas representing physical phenomena are called models. Some models are more accurate then other models. Entropy is one such model. The ma

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 4y ago
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 https://news.ycombinator.com/item?id=31201129 (somehow I've missed that comment until now)
- deltaonefour 4y ago
  >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. Yes. That is what entropy is as defined. >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. If the input is macrostate. And you don't know the macrostate. Then you can't calculate the value. That's pretty basic and this applies for ANY model. If you don't know the input variables, you can't calculate anything. Nobody talks about mathematical models this way. This applies to everything. I don't think you picked up on my model argument either. You seem to think you made progress on us agreeing that entropy is a "model." I'm saying every single math formula that representing physical phenomena on the face of the earth is a "model." Thus it's a pointless thing to bring up. It's like saying all mathematical formulas involve math. If entropy uniquely has a parameter called knowledge that affects it's outcome, citing properties universal to everything doesn't lend evidence to your case. Let's "reconcile" everything: You're implying that there is some input parameter modeled after knowledge. And that input parameter affects the outcome of the entropy calculation. I am saying no such parameter exists. Now your saying that knowledge of the input parameter itself is what your talking about. If you don't know the input parameter you can't perform the calculation. The above is an argument for everything. ANY model on the face of the earth if you don't know the input parameters you can't derive the output. Entropy is not unique for this property and obviously by implication we're talking about how you believe entropy is uniquely relative to knowledge. >Alice can calculate S1 and S2 and the final entropy is SA=S1+S2. Who says you can add these two entropies together? S1 and S2. The macrostates are different and Mixing the two gases likely produces a third unique set of macrostates indpendent of the initial two.
  - kgwgk 4y ago
    > You seem to think you made progress on us agreeing that entropy is a "model. I thought we had agreed that entropy is something you calculate with a model, in fact. > You're implying that there is some input parameter modeled after knowledge. I was trying to say that the inputs to S(...) are the things that we know because we did measure them or set their values. It seems that we agree on that because it's extremely obvious. Hopefully we also agree that if there are other other relevant things that we know in addition to the inputs to that model we could refine our model. I fully acknowledge that we may choose to ignore the additional knowledge and keep using the old model - and it may be good enough for some uses. (We may also choose to incorporate the additional knowledge. Maybe it rules out some microstates and we could be using a smaller macroset to represent what we know about the system.) When all we know is the macrostate, the macrostate is the most detailed description - and gives the most precise predicitions - available to us regarding the system. However, if we know more the original macrostate is no longer "complete". Because we do know - and we can predict - more precise things. There is a fundamental change from "the macrostate represents all we know and is the basis of everything we can predict" to "not the case anymore". Which also seems obvious. Probably we agree on that as well! (Sure, it applies to everything. Anytime one ignores information one has a suboptimal model compared to the model one could have. The improved model may or may not be better for a particular purpose.) > Who says you can add these two entropies together? S1 and S2. Alice, who considers two equal volumes of an ideal gas at the same temperature and pressure. > The macrostates are different They were the same in my example. Same volume. Same temperature. Same pressure. > and Mixing the two gases likely produces a third unique set of macrostates indpendent of the initial two For an ideal gas doubling the volume and the number of particules (so the pressure remains the same for a fixed temperature) doubles the entropy. If you have two identical systems the total entropy doesn't change when you put together the two containers resulting in a single container twice as large with twice as many particles. If you thought that the number of microstates - and the entropy - increases when you bring toghether two identical systems because they will mix with each other that's not correct. (Even though there are still debates about this issue 120 years later.) https://en.wikipedia.org/wiki/Gibbs_paradox https://en.wikipedia.org/wiki/Gibbs_paradox The entropy would increase however if they are different ideal gases (it doesn't matter how different). Bob - who knows that they are different - would calculate the correct entropy. It could be the other way. Maybe they're actually the same gas but Bob treats them as different because he isn't aware and keeps the general case. He calculates an increase in entropy due to the mixing. While for Alice, who knows that they are the same gas, the total entropy hasn't changed. Ax Maxwell wrote: "Now, when we say that two gases are the same, we mean that we cannot separate the one from the other by any known reaction. It is not probable, but it is possible, that two gases derived from different sources but hitherto regarded to be the same, may hereafter be found to be different, and that a method be discovered for separating them by a reversible process." If we think that the two gases are the same the entropy is 2S but if we discover later a way to tell apart one from the other the entropy is higher (there are more microstates for the same macrostate that we thought initially).

kgwgk 4y ago

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 https://news.ycombinator.com/item?id=31201129 (somehow I've missed that comment until now)

deltaonefour 4y ago
>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. Yes. That is what entropy is as defined. >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. If the input is macrostate. And you don't know the macrostate. Then you can't calculate the value. That's pretty basic and this applies for ANY model. If you don't know the input variables, you can't calculate anything. Nobody talks about mathematical models this way. This applies to everything. I don't think you picked up on my model argument either. You seem to think you made progress on us agreeing that entropy is a "model." I'm saying every single math formula that representing physical phenomena on the face of the earth is a "model." Thus it's a pointless thing to bring up. It's like saying all mathematical formulas involve math. If entropy uniquely has a parameter called knowledge that affects it's outcome, citing properties universal to everything doesn't lend evidence to your case. Let's "reconcile" everything: You're implying that there is some input parameter modeled after knowledge. And that input parameter affects the outcome of the entropy calculation. I am saying no such parameter exists. Now your saying that knowledge of the input parameter itself is what your talking about. If you don't know the input parameter you can't perform the calculation. The above is an argument for everything. ANY model on the face of the earth if you don't know the input parameters you can't derive the output. Entropy is not unique for this property and obviously by implication we're talking about how you believe entropy is uniquely relative to knowledge. >Alice can calculate S1 and S2 and the final entropy is SA=S1+S2. Who says you can add these two entropies together? S1 and S2. The macrostates are different and Mixing the two gases likely produces a third unique set of macrostates indpendent of the initial two.
- kgwgk 4y ago
  > You seem to think you made progress on us agreeing that entropy is a "model. I thought we had agreed that entropy is something you calculate with a model, in fact. > You're implying that there is some input parameter modeled after knowledge. I was trying to say that the inputs to S(...) are the things that we know because we did measure them or set their values. It seems that we agree on that because it's extremely obvious. Hopefully we also agree that if there are other other relevant things that we know in addition to the inputs to that model we could refine our model. I fully acknowledge that we may choose to ignore the additional knowledge and keep using the old model - and it may be good enough for some uses. (We may also choose to incorporate the additional knowledge. Maybe it rules out some microstates and we could be using a smaller macroset to represent what we know about the system.) When all we know is the macrostate, the macrostate is the most detailed description - and gives the most precise predicitions - available to us regarding the system. However, if we know more the original macrostate is no longer "complete". Because we do know - and we can predict - more precise things. There is a fundamental change from "the macrostate represents all we know and is the basis of everything we can predict" to "not the case anymore". Which also seems obvious. Probably we agree on that as well! (Sure, it applies to everything. Anytime one ignores information one has a suboptimal model compared to the model one could have. The improved model may or may not be better for a particular purpose.) > Who says you can add these two entropies together? S1 and S2. Alice, who considers two equal volumes of an ideal gas at the same temperature and pressure. > The macrostates are different They were the same in my example. Same volume. Same temperature. Same pressure. > and Mixing the two gases likely produces a third unique set of macrostates indpendent of the initial two For an ideal gas doubling the volume and the number of particules (so the pressure remains the same for a fixed temperature) doubles the entropy. If you have two identical systems the total entropy doesn't change when you put together the two containers resulting in a single container twice as large with twice as many particles. If you thought that the number of microstates - and the entropy - increases when you bring toghether two identical systems because they will mix with each other that's not correct. (Even though there are still debates about this issue 120 years later.) https://en.wikipedia.org/wiki/Gibbs_paradox https://en.wikipedia.org/wiki/Gibbs_paradox The entropy would increase however if they are different ideal gases (it doesn't matter how different). Bob - who knows that they are different - would calculate the correct entropy. It could be the other way. Maybe they're actually the same gas but Bob treats them as different because he isn't aware and keeps the general case. He calculates an increase in entropy due to the mixing. While for Alice, who knows that they are the same gas, the total entropy hasn't changed. Ax Maxwell wrote: "Now, when we say that two gases are the same, we mean that we cannot separate the one from the other by any known reaction. It is not probable, but it is possible, that two gases derived from different sources but hitherto regarded to be the same, may hereafter be found to be different, and that a method be discovered for separating them by a reversible process." If we think that the two gases are the same the entropy is 2S but if we discover later a way to tell apart one from the other the entropy is higher (there are more microstates for the same macrostate that we thought initially).