What is the connection between Maxwell's Demon and Liouville's Theorem?

In summary, the argument makes no reference to the size of the air molecules, and assumes that the demon himself can be neglected.
  • #36
I also agree that we don't care about equilibrium - that's for PM machines to worry about, not us.

I'd still like to quibble this enormous temperature business. If we don't care about equilibrium, I can use balls from the fridge. They don't have to move very fast so there's no relativistic doppler effect to worry about, so I think they will look to the demon like normal cool balls. He won't see them glowing like a quasar.

You guys keep saying "the entropy of the demon will increase." I'd like to see one piece of evidence for that which does not use the 2nd law as an assumption. This is E+N's point, and apparently Bennet accepts it. Without using the second law, there's no argument saying that there will be any increase of entropy anywhere. All that stuff (whether you think observing or forgetting is associated with +ve or -ve entropy or whatever) sits on top of the assumption that the demon does not break the 2nd law, so you can hardly turn it around and use it to defend the 2nd law.

I think Ken is saying there will be something else defending the 2nd law from the demon if we think hard enough about it, and then all that information theory will be valid. But my mind is blank. Yes we must examine the whole system including the demon's brain, and we'd be home and dry if we could prove (without invoking the 2nd law) that this brain must increase entropy somewhere else. But how? I'm at a loss as to how this should work. Presumably there's some light in the system so he can see what's coming, but I think I can let that light be totally disordered from start to finish, so no problem there. I think I can make his brain a steady state so no problem there either.

I think it might be helpful to get the whole temperature thing out of the way because it's just causing distraction. How about a gas of aromatic molecules in a mixture of left handed and right handed forms. We don't give a monkey's about the temperatures - we just want to sort the two forms. The chemical reflects circularly polarised light in the matching handedness. (A bit of detail is called for here but I think it can work.) So if the demon sees a lot of right-circular light in his vicinity he opens he door. We can use oodles of light because we no longer care about the temperature.

If there are no losses to the environment, the light energy won't get lost but it will tend to turn from a few high energy photons to a large number of low energy photons. But I think we can probably arrange for the interesting mode (the one where the chirality plays a role) to be the lowest frequency mode available. Failing that, we only need to show that we can sort a molecule without degrading a useful photon into two lower energy ones.

If this light stuff fails we can also make a chemical detector, like a nose.

I'm back to square one now - there may be losses, but I don't see how I can equate them with the entropy reduction without invoking the 2nd law.
 
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  • #37
This all seems to be making a mountain out of a molehill. The demon and the two gases are to be considered a single system. If an outside observer looks at the system, measuring only the thermodynamic states of the two systems and the state of the demon - a combination of logical state, mechanical state, and thermodynamic parameters, the outside observer has to ask, how many ways could such a set of measurements occur? Equivalently, how many microstates could yield this observation? The entropy will be k log(W) where W is that number of ways. The second law says that will increase (or stay the same, but probably not).

The only way the demon is going to violate the second law is if the number of ways the demon could give rise to our measured state of the demon is less than the number of ways the entropy of the two gases could have been reduced. I think the second law will hold, so the number of ways will be larger, but part of these ways is a description of the logical state of the Turing-machine demon. That means that information theory must be in part invoked in the calculation of the state of the demon. Information theory applied to the "logical state" of the demon is just a way of figuring the contribution to the calculation of the total number of ways the demon could exist in his final state. If the logical state of the demon is the same as before it made the decision, and the mechanical parts are all in the same place, then heat must have been generated.

This all turns on what we define as the "macrostate of the demon". The crux of the problem is here, I think.

AdrianMay said:
OK let me quote from that paper.

First the top line: Bennet, 2011.

...

So then we excitedly skip to the third section:

Earman and Norton have pointed out with some jus-
tice that Landauer’s principle appears both unnecessary
and insufficient as an exorcism Maxwell’s demon, be-
cause if the Demon is a thermodynamic system already
governed by the Second Law, no further supposition
about information and entropy is needed to save the
Second Law. On the other hand, if the Demon is not
assumed to obey the Second Law, no supposition about
the entropy cost of information processing can save the
Second Law from the Demon.
I would nevertheless argue that Landauer’s princi-
ple serves an important pedagogic purpose ...

That means he agrees. Pedagogic purposes are not logic.

You guys know me! Would I ever presume to question what a giant like Bennet says ?

Ok, I'm still trying to understand these papers, and if I can summarize the above, I read that E&R point out that Landauer's principle is based on the second law and therefore applied to the demon, it will of course save the second law. Bennett agrees. But what does it mean to save the second law? What does it mean to "exorcise the demon"? Does it mean that the demon cannot reduce the entropy of the two gases (i.e. will fail in its purpose) or does it mean that it will (or can sometimes) succeed because the entropy of the demon increases as the entropy of the two gases decreases, yielding a net increase in entropy?

I think they mean the latter.

If the demon can succeed in reducing the entropy of the two gases, then what are the details of how its entropy increases? It seems to me that Landauer is giving an information-theoretic explanation of how this happens, consistent with the second law. Earman and Norton have three objections to Landauer's principle -

1) (pages 14-16) seems to me to be "who cares about the details, Landauer's principle upholds the second law, so everything is fine". Since I care about the details, I reject that.

2 and or 3) (pages 16 - 20) I could not find a delineation of two separate problems here, its all mushed together. I definitely do not understand these pages in detail. To begin with, E&N say Bennett requires that at the end of each decision by the demon, the demon (a Turing machine) must be reset to its original state, which involves erasure and the creation of entropy by Landauer's principle. (I don't think that's necessary, but ok, let's go with that.) They then give a case where erasure supposedly does not happen. They say that Zurek and Caves (Z&C) dealt with this problem by a more complicated definition of entropy involving algorithmic complexity of the Turing machine demon. etc. etc. etc. That's as far as I got.

One question I have is "are they assuming the temperature of the gases and the demon are the same or not?"
 
  • #38
AdrianMay said:
How about a gas of aromatic molecules in a mixture of left handed and right handed forms. We don't give a monkey's about the temperatures - we just want to sort the two forms.

This would avoid lugita15's problem which needs to be considered, if not just to dispose of it.

Note that the two actually can in principle be separated. There can exist in principle, a membrane that is permeable to one, but not the other. If you have a volume of mixture, you put the membrane at one end and start slowly pushing on it. It gets to the middle, and the pressure on the pushed side is lower than on the other side. You replace the membrane with an impermeable one, let the pressures equilibrate, put in the permeable membrane, and push again. This process will eventually "distill" one type from the other. When you calculate up the total work done by pushing, divide by temperature, you will get the new (lowered) entropy of the system. The entropy (and energy) that the system has lost has gone into the pushing machine. If its your muscles, they will heat up, etc.

lugita15 said:
One of the formulations of the second law is the amount of usable energy in the universe can only decrease. Usable energy is essentially the kind of energy that a low-entropy system has, and entropy is essentially a measure of what percentage of the universe's energy isn't usable. So if Maxwell's demon isn't expending that much energy, is he increasing the amount of usable energy in the universe?

To put it another way, can Maxwell's demon extract more usable energy than a Carnot engine? If we take the fumes coming out of the (ideal) combustion engine of a car, can we feed it into Maxwell's demon and get more energy out? (Let the molecules of the car fumes be cannonballs... It's a giant car, OK?)

Hmmmm
 
  • #39
We should be able to express all this in the terms lugita is asking for, but I'm not sure how. If we dodge it with my aromatic gas for now, we have to come back to it. He's quite right that usable energy is an important consequence of this whole entropy thing, but I lost track of that when it turned out that I couldn't make a power station out of my cannonball gas. I thought I would be able to because of the usable energy thing.

If we can clarify this, we might get a line of sight on the business of how much work the demon will have to do, i.e., how much energy needs to be pumped into keep him running, and I think that'll turn out to be the crux of it.

> what does it mean to save the second law?

I think it means to come up with a catch in the demon challenge, without resorting to the second law. The catch ought to be mechanistic or statistical, and it should show that this demon system will observe the 2nd law of its own accord. Then, all that Landauer stuff would rest on the 2nd law and be valid. But that's not happening right now, rather, they are calculating the performance limits of the demon such that it can do all things permitted under the 2nd law but nothing that's forbidden.

> the outside observer has to ask, how many ways could such a set of measurements occur? Equivalently, how many microstates could yield this observation?

Now we have a precise definition, and I think it's the correct one. But I see no reason why the demon needs more states in its entire life cycle than you can count on the fingers of one hand, so how could it contribute anything significant?
 
  • #40
AdrianMay said:
Now we have a precise definition, and I think it's the correct one. But I see no reason why the demon needs more states in its entire life cycle than you can count on the fingers of one hand, so how could it contribute anything significant?

No not the states in its life cycle, the number of microstates that could give its present macrostate. For a particle gas, classically, a microstate is the specification of the position and momenta of each particle in the gas. The macrostate is three thermodynamic parameters, like temperature, pressure, and number of particles.

For the demon, the microstate is the position and momenta of every molecule that makes up the demon. If the demon is a computer that measures nearby gas particle positions and momenta, makes some calculations using some mechanism, and then opens or shuts a trap door, and generates some heat, then what is its microstate? I think you can describe its microstate by the state of the registers in it and the position and momenta of every particle that would be considered "heat" but for the fact that we now know them. The "state of its registers" is where information theory steps in and contributes knowledge of the number of microstates the demon could be in. If we follow Bennett and require that the demon be in the same logical state after a measurement-trapdoor opening as it was before, then, if the second law holds, the whole calculation process had to generate some heat, and that heat represents the entropy generated by the demon. Landauer's principle is more specific, it says that erasure causes heat. So we want to look at the whole process by which the demon arrives at its conclusion and does what it does, and count up the erasures and add up the heat and hopefully say that it all fits together. I think it is the details of this process that Bennett and E&N are discussing.

This has, of course, ignored the contributions to entropy of the measurement process and the trapdoor opening process, which we are more or less ignoring, and maybe we should not be. Szilard's principle says the measurement process produces entropy, I think Bennett says no, not necessarily, me, I don't know, but if the inquiry into the demon's entropy production fixes things, then it doesn't matter. The trapdoor opening and closing - well, I'm willing to ignore that for the same reason.

The crux of the problem, and the thing that puzzles me, is the macrostate and microstate of the demon. It seems we can make different assumptions here, and get different results, but I expect all those results are in accord with the second law. Entropy is "missing information" and depending on how you define the demon's macrostate, you get different entropies. Its should be all ok nevertheless, though. Even with gases, you can suppose different levels of information knowledge, thus get different macrostates for the same situation, thus get different entropies, but no matter which description you use, the second law still holds.
 
  • #41
I think I can describe the microstates of my dog-leg protein. If the molecule is in a given macrostate (e.g. open, closed, thinking about it, etc) then the microstates are just the waqys of arranging thermal phonons over the molecule in that configuration and at that temperature. This is just a function of temperature if we're considering a specific configuration. Different configurations have different specific heats.

I think it's that simple. You'd only get inflation of the microstates per macrostate if the temperature was increasing. But we haven't shown that it will, at least, not without invoking the second law as Landauer, Bennet and friends do.

So I don't see a route in that line of thought. It still boils down to finding some mechanical reason why the demon has to get hotter. Then we'd be home and dry. Even if the demon then dissipated his heat into the chambers, we'd still be home and dry because that would still be the sought entropy increase.

But why can't he just be frictionless and flout the second law, leaving Landauer etc without a leg to stand on?
 
  • #42
AdrianMay said:
I think I can make his brain a steady state so no problem there either.
I'd like to see that :)

I'm pretty sure it's impossible and this is exactly the reason why the whole setup is not going to work.

If your computations return to back to exactly the same state, they are reversible and there is no particular reason for them to go forward, they are just as likeky to run backwards and release the molecules instead of capturing them.
 
  • #43
Delta Kilo said:
I'd like to see that :)

If your computations return to back to exactly the same state, they are reversible and there is no particular reason for them to go forward, they are just as likeky to run backwards and release the molecules instead of capturing them.

Hmm. Maybe there's something in that.
 
  • #44
lugita15 said:
One of the formulations of the second law is the amount of usable energy in the universe can only decrease. Usable energy is essentially the kind of energy that a low-entropy system has, and entropy is essentially a measure of what percentage of the universe's energy isn't usable. So if Maxwell's demon isn't expending that much energy, is he increasing the amount of usable energy in the universe?
The energy is already usable, because you have a container filled with hot gas (the temperature is associated with the average kinetic energy of the cannonballs, AdrianMay-- that doesn't mean the balls themselves are hot, indeed we can imagine they are at absolute zero internal temperature for all the difference it would make here), sitting in an environment of lower T. If you don't have that, say the whole universe is at that T, then the Demon's environment is also at that T, and then the Demon cannot work (it needs to be at a much lower effective T so it can increase entropy without using up all the energy being made extractable from the gas).
To put it another way, can Maxwell's demon extract more usable energy than a Carnot engine?
A Carnot engine can extract all the energy from a gas, it's all a question of the effective temperature of the brain making the decisions (or in the standard setup, the temperature of the exhaust reservoir). Like all thermodynamic engines, Carnot cycles participate with our intentions only so long as the total entropy increases in the cycle (or stays nearly the same). Frankly, I don't see what Bennett and Landauer etc. are on about, the situation all seems pretty clear. Unless their intention is to delve into the constraints on modeling how the Demon thinks, i.e., on just how it manages to increase the entropy of its environment, that's the only issue that seems subtle. I would say it is perfectly obvious that the Demon obeys the second law, and the only reason we might imagine giving it a "pass" on that law is that we are treating it as magic-- rather than as an actual physical system.
 
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  • #45
Ken G said:
The energy is already usable, because you have a container filled with hot gas (the temperature is associated with the average kinetic energy of the cannonballs.

A Carnot engine can extract all the energy from a gas, it's all a question of the effective temperature of the brain making the decisions (or in the standard setup, the temperature of the exhaust reservoir).

I think what he means is that you cannot run an engine between two reservoirs at the same temperature. If the demon causes the two temperatures to differ, then and engine can extract "usable" energy. One of the gas systems is serving as the exhaust reservoir.

Ken G said:
Frankly, I don't see what Bennett and Landauer etc. are on about, the situation all seems pretty clear. Unless their intention is to delve into the constraints on modeling how the Demon thinks, i.e., on just how it manages to increase the entropy of its environment, that's the only issue that seems subtle.

Yes, I think that's just what they are doing. I think its very interesting to try to understand how the demon increases its "entropy". I put that in quotes, because unless you can define a microstate and a macrostate, you cannot define entropy, and I am not always sure what constitutes a macrostate of the demon. If you demand that the demon return to its original state, then the macrostate is just a matter of temperatures, that's why they demand it, it simplifies things. But on the other extreme, if you say the demon never erases, just keeps using new memory, then what? Temperature does not change? In order not to violate the second law, the entropy has to be k times the information entropy of the demon's memory? That means the macrostate of the demon is one in which we cannot read its internal logical state and its entropy is the number of ways its memory could be configured given what we know about the two gases at that point. Does the second law therefore imply that there is no way we can communicate with the demon and read its logical state? Forgetful or not, the demon is not thermally connected to the system, yet it is acquiring or creating entropy to match the entropy it is removing, not by thermodynamic means, but by "logical" means.
 
  • #46
Rap said:
I think what he means is that you cannot run an engine between two reservoirs at the same temperature. If the demon causes the two temperatures to differ, then and engine can extract "usable" energy. One of the gas systems is serving as the exhaust reservoir.
What I'm saying is that you can run a heat engine between two temperatures, and you can run a Demon to create that temperature difference, and both stages increase entropy (the Carnot cycle can increase it very little). This isn't a problem with free energy, free energy depends on the context. A hot gas inside more of the same hot gas has no free energy, but a hot gas inside a cold or low pressure environment, or connected to a Demon that effectively has access to a cold environment, has lots of free energy. Free energy just means that there is some other class of states that are more numerous (possible just barely more numerous) in which that energy is somewhere else. That is just as true when the Demon is creating a T difference as when we are running an engine off that T difference. They both happen because entropy increases.

Does the second law therefore imply that there is no way we can communicate with the demon and read its logical state? Forgetful or not, the demon is not thermally connected to the system, yet it is acquiring or creating entropy to match the entropy it is removing, not by thermodynamic means, but by "logical" means.
The second law is way easier than that, I really don't know why they want to make it so complicated. It doesn't have anything to do with memory, unless memory is the way the entropy is increasing. If the memory isn't there, then it would just be some other way. The Demon's brain simply doesn't function unless it increases entropy, that's how it works. The entropy increase can be used to make decisions that drop the entropy somewhere else, but it can't operate to drop total entropy, or the behavior simply doesn't happen. More likely configurations do not evolve into less likely ones, that's all the second law says.
 
  • #47
Ken G said:
The second law is way easier than that, I really don't know why they want to make it so complicated. It doesn't have anything to do with memory, unless memory is the way the entropy is increasing. If the memory isn't there, then it would just be some other way. The Demon's brain simply doesn't function unless it increases entropy, that's how it works. The entropy increase can be used to make decisions that drop the entropy somewhere else, but it can't operate to drop total entropy, or the behavior simply doesn't happen. More likely configurations do not evolve into less likely ones, that's all the second law says.
Ken G, I'm a bit uneasy about the philosophical attitude you seem to have concerning the 2nd law. It is not some iron-clad cosmic law of the universe, like the first law. It is, as you said, a statement that things tend to go from less likely configuration to more likely configurations. You make it sound as if machines cannot even in principle function if they decrease the total entropy of the system, as if the second law is what is driving them. But the second law is an effect, not a cause, of physical behavior, and it is a statistical rule, not a deterministic law; see the fluctuation theorem in statistical mechanics, which gives the probability that the entropy of a system increases or decreases.

For this reason, I think that instead of analyzing Maxwell's demon based on what the second law "permits" it to do, it's more useful to analyze it without assuming the second law, and thus finding out what makes the second law work in this case.
 
  • #48
lugita15 said:
You make it sound as if machines cannot even in principle function if they decrease the total entropy of the system, as if the second law is what is driving them.
Yes, that statement is correct-- no machine can function reliably unless its net effect is to increase entropy. It makes no difference if we imagine this constraint is "driving" them-- it's just a true constraint.
But the second law is an effect, not a cause, of physical behavior, and it is a statistical rule, not a deterministic law; see the fluctuation theorem in statistical mechanics, which gives the probability that the entropy of a system increases or decreases.
I did not say it is a "deterministic" law-- it is a constraint. You can't use it to determine what will happen because you can't always know what is most likely to happen without significant knowledge of the details. However, you can know what will not happen without any knowledge of the details-- the Demon will not work unless he increases entropy. Not having to know the details is the raison d'etre of thermodynamics.
For this reason, I think that instead of analyzing Maxwell's demon based on what the second law "permits" it to do, it's more useful to analyze it without assuming the second law, and thus finding out what makes the second law work in this case.
The reason that is not terribly useful is because you have a different analysis for every different Demon you can construct. The purpose of thermodynamics is to make general statements, not statements tied directly to some particular mechanism (which must be analyzed using mechanics, not thermodynamics). If you want to use mechanics to test thermodynamics, then you need a model of the Demon, but you cannot get "paradoxes" by imagining magical properties of the Demon-- you need an actual mechanism, and when you get that, you will see that the Demon either increases entropy, or doesn't work.
 
  • #49
Ken G said:
The reason that is not terribly useful is because you have a different analysis for every different Demon you can construct. The purpose of thermodynamics is to make general statements, not statements tied directly to some particular mechanism (which must be analyzed using mechanics, not thermodynamics). If you want to use mechanics to test thermodynamics, then you need a model of the Demon, but you cannot get "paradoxes" by imagining magical properties of the Demon-- you need an actual mechanism, and when you get that, you will see that the Demon either increases entropy, or doesn't work.
But the second law of thermodynamics is a mere statistical consequence of the laws of mechanics. Thus, if the second law tells you that Maxwell's demon cannot reliably reduce total entropy, without actually knowing the inner workings of the demon, it should also be possible to draw the same statistical inference directly from the laws of mechanics. In other words, you are in essence redoing the proof of the second law from Newton's laws in the particular case where a device has the properties Maxwell's demon claims to. It is this kind of analysis that has allowed people to figure out some of the underlying reasons why the second law is an emergent property of mechanical systems, for instance the connection between information and entropy.
 
  • #50
lugita15 said:
But the second law of thermodynamics is a mere statistical consequence of the laws of mechanics.
I would say it must be consistent with any laws of mechanics, but it is independent of those laws, and it doesn't even really require any laws of mechanics be in place. Mechanics are the details, thermodynamics is what you can do without even knowing the details.
Thus, if the second law tells you that Maxwell's demon cannot reliably reduce total entropy, without actually knowing the inner workings of the demon, it should also be possible to draw the same statistical inference directly from the laws of mechanics.
Yes, if you have a model for the Demon. Do we have mechanical models of brains?
In other words, you are in essence redoing the proof of the second law from Newton's laws in the particular case where a device has the properties Maxwell's demon claims to. It is this kind of analysis that has allowed people to figure out some of the underlying reasons why the second law is an emergent property of mechanical systems, for instance the connection between information and entropy.
The "why" of the second law is independent of mechanics, it is thermodynamics. It all boils down to, entropy is our way of counting which collections of configurations contain more equally likely states, and hence are what will happen.
 
  • #51
Ken G said:
I would say it must be consistent with any laws of mechanics, but it is independent of those laws, and it doesn't even really require any laws of mechanics be in place. Mechanics are the details, thermodynamics is what you can do without even knowing the details..
We can easily have a universe in which the laws of mechanics do not lead to the second law of thermodynamics. For instance, the laws could dictate that systems try to attain a specific ordered state.
Yes, if you have a model for the Demon. Do we have mechanical models of brains?
My point was that if we can use the 2nd law of thermodynamics to conclude that Maxwell's demon cannot reliably decrease the total entropy, and we can conclude this without examining in detail how it works, then we should be able to reach a similar conclusion using Newton's laws, again without looking at a specific model for the demon. If nothing else, we can adapt the proof of the second law of thermodyamics given in e.g. the later chapters of Feynman Lectures volume 1, for the case of a unspecified device which is assumed to have the properties we ascribe to Maxwell's demon. If we do something like this, we can find out some of what make the second law "tick", such as the role of information.
The "why" of the second law is independent of mechanics, it is thermodynamics. It all boils down to, entropy is our way of counting which collections of configurations contain more equally likely states, and hence are what will happen
I agree that the definition of entropy is independent of the laws of physics. But I disagree with your assertion that thermodynamics is independent of mechanics. The fact that entropy tends to increase with time more often than it decreases seems like a contingent fact of the universe. And in fact, even in our universe with our laws of physics, the fluctuation theorem suggests that if entropy gets really high, it is possible in principle for the second law of thermodynamics to go in reverse.
 
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  • #52
lugita15 said:
We can easily have a universe in which the laws of mechanics do not lead to the second law of thermodynamics. For instance, the laws could dictate that systems try to attain a specific ordered state.
"Disorder" simply means "more ways of being", which means "more likely", and that's the second law in a nutshell. The sole assumption is that you can just count the ways of being (the number of configurations)-- this is the crux of statistical mechanics, that every individual state is equally likely. That's the only assumption behind the second law, and if it weren't true, it would only mean that we would need to have a more sophisticated concept of what entropy is, beyond just ln(N), if some states were "preferred" by the mechanics. But any mechanics without that property yields the second law, quite generally.
My point was that if we can use the 2nd law of thermodynamics to conclude that Maxwell's demon cannot reliably decrease the total entropy, and we can conclude this without examining in detail how it works, then we should be able to reach a similar conclusion using Newton's laws, again without looking at a specific model for the demon.
And that is what is not true. Newton's laws are about the details, thermodynamics is what you can do without anything like Newton's laws. That's why the main principles of thermodynamics were discovered independently of Newton's laws (like the work of Carnot and Clausius), and sometimes even prior to them (like Boyle's law).
If nothing else, we can adapt the proof of the second law of thermodyamics given in e.g. the later chapters of Feynman Lectures volume 1, for the case of a unspecified device which is assumed to have the properties we ascribe to Maxwell's demon. If we do something like this, we can find out some of what make the second law "tick", such as the role of information.
Right, with no reference to any mechanism or mechanics of the Demon. This is crucial-- the mechanics only serve as informative examples of the second law, they are not part of the derivation of it. The derivation proceeds along the lines I gave above, and with no mention of any laws of mechanics, other than that they do not pick out preferred states. One might thus say that the second law arises from the universe being "non-teleological", which still remains the hardest thing for many to accept about it. It might not even be true-- it's only a law after all! But we won't know until we can really model thought, to see if it brings in some kind of teleology that could motivate essentially "magical" treatments of the Demon.
 
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  • #53
Ken G said:
No, that is exactly what cannot be true, in any theory of mechanics exhibited by large systems. That's pretty much the whole point of thermodynamics! Again, "disorder" simply means "more ways of being", which means "more likely", and that's the second law in a nutshell. The sole assumption is that you can just count the ways of being (the number of configurations)-- this is the crux of statistical mechanics, that every individual state is equally likely. That's the only assumption behind the second law, and if it weren't true, it would only mean that we would need to have a more sophisticated concept of what entropy is, beyond just ln(N).
Are you saying that it is literally impossible to have laws of physics in which all the particles work together to produce a particular ordered state?
And that is what is not true. Newton's laws are about the details, thermodynamics is what you can do without anything like Newton's laws. That's why the main principles of thermodynamics were discovered independently of Newton's laws (like the work of Carnot and Clausius), and sometimes even prior to them (like Boyle's law).
Sure, just like Kepler's laws were discovered before Newton's law of gravitation and the Balmer series was discovered before the Schrodinger's equation. Phenomena of nature can be discovered independently even if they derive theoretically from a common source.
Right, with no reference to any mechanism or mechanics of the Demon. This is crucial-- the mechanics only serve as informative examples of the second law, they are not part of the derivation of it. The derivation proceeds along the lines I gave above, and with no mention of any laws of mechanics.
I was envisioning a different sort of procedure. I'm suggesting doing the statistical mechanics derivation of the second law of thermodynamics from Newton's laws of motion, as outlined in the Feynman lectures and fleshed out by Boltzmann, but restricting the proof to the case where you have a Maxwell's demon with unspecified mechanism. So the rest of the scenario will be analyzed according to mechanics, it is only the demon that is a black box.
If you set F=mv instead of ma, as the ancients imagined, you still get the second law of thermodynamics, without any difference. Indeed, this is the second law in highly dissipative situations, and it's still just thermodynamics.
I don't think this is too surprising; (this part of) Aristotelian physics is just Newtonian physics in the limit of strongly dissipative forces.
 
  • #54
I confess that I did some editing of my last post, just after I posted it, so some of the points were clarified and you might want to look at the improved version. I'm not saying it's impossible to have laws that create ordered states, I'm saying that only a very general assumption about the laws is required to rule that out (the assumption needed is that all possible states are equally likely, so none are picked out by the laws as special in some way). I would call that "non-teleological" laws, similar to what we get in relativity where there are no preferred reference frames. The key point about the Demon is that it is not given a pass to violate this rule-- it cannot target specific states, it must "throw darts" like everything else, and what it hits, ultimately, is simply the largest target, i.e., the highest entropy. Anything else is "magic". Now, of course we must include the entire "target", not just the gas and its entropy, so that's why violations of the second law invariably result from not recognizing the full space of possible outcomes that are being affected by the Demon. That's how the Demon works-- by gaining access to some other set of possible outcomes that can mitigate the "smaller target" of the reduced entropy in the gas. So my point is, this is not some technicality about how the Demon functions, it is how the Demon functions, thermodynamically speaking. No magic, no teleology, and you have the second law, regardless of the mechanics.

Now of course, this is the thermodynamics view-- I don't say reality actually contains no magic, and no teleology. It is physics that doesn't have those things, and does well without them, and that seems to be the reason that thermodynamics works so well. We don't have any reason to think a mind, demonic or human, can violate the second law, but we can notice that the second law stems from how our mind analyzes nature, so the law is as much a product of our minds as it is something that is a rule of nature itself. Hence we don't have to say that our minds result from the action of the second law, we can always assert the converse if we prefer. But either way, the two come together, and I would say the responsibility is on any who would claim a Demon can do something that the patent office rejects as plausible.
 
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  • #55
Its the temperature and entropy thing again - When the second law talks about entropy, it is always in the context of temperature. Thermodynamic entropy is information entropy, but not all information entropy is thermodynamic entropy. Thermodynamic entropy always involves temperature.

Lets assume we can ignore the measurement process as far as its contribution to the second law balance sheet is concerned. Measuring the cannonballs in the cannonball gas suggests to me this is true. Let's ignore the trapdoor process as well. Then we have two containers of gas and a demon. In the beginning, the two gas systems are at the same temperature. Some time later, they are not. There has been a thermodynamic entropy reduction, if we ignore the demon. Let's say the second law holds, and the demon is some kind of computer, a Turing machine for simplicity. This means that the demon's thermodynamic (!) entropy must be increased, or, if it must remain at some low temperature, that its increasing entropy is continually dumped outside. The information entropy of the demon's logical state is not the subject of the second law, unless you can define some "logic temperature" and multiply it by k times the info entropy to get an energy. Let's say you cannot. Then it follows that there can exist no Turing machine that accomplishes the demon's purpose without the dissipation of the amount of lost gas entropy as HEAT. If you want to throw a monkey wrench into the workings of the second law, then you have to design a Turing machine that accomplishes the demon's purpose while dissipating less than that lost entropy. If you can design a Turing machine that is as general as possible, which dissipates no heat, you will have surely thrown a monkey wrench into the second law (given the arguable assumptions already made).

This is partly what Bennett and E&N are discussing, trying to figure out the thermodynamics of a Turing machine, or some other universal computer. If you break the computer down into individual logical operations, then one or more of those operations must dissipate heat. The heat dissipation, or absence of it, for various logical operations, is what they are discussing, and I see it as an interesting discussion, given the implications.

They also discuss Szilard's principle which says entropy is lost in the measurement process. As far as I can see, Bennett disagrees and E&N, I don't know.
 
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  • #56
Rap said:
Its the temperature and entropy thing again - When the second law talks about entropy, it is always in the context of temperature. Thermodynamic entropy is information entropy, but not all information entropy is thermodynamic entropy. Thermodynamic entropy always involves temperature.
Actually I think it's the other way around-- the meaning of temperature stems from thermodynamical entropy. The best way to think about what temperature is is to multiply it by k and say that kT is the energy a system must take in in order to increase its access to undifferentiated states by a factor of e. But one can talk about the latter without ever referring to the former, if you simply don't care about energy (given that it's conserved, we usually do choose to care about it, but we don't have to and we still have a second law). So T comes in not so much with entropy, but when you want to connect entropy to energy. That means it relates to the importance of entropy, because energy measures the consequences of what the entropy is doing. But we still have a second law even if we have never heard of either energy or temperature.
Lets say the second law holds, and the demon is some kind of computer, a Turing machine for simplicity. This means that the demon's thermodynamic (!) entropy must be increased, or, if it must remain at some low temperature, that its increasing entropy is continually dumped outside.
Be careful not to confuse entropy with energy. If the Demon is at a very low T, then a very small energy change can correspond to a huge entropy change, and the Demon can remain at low T yet still increase its entropy without dumping anything to the outside. Indeed, if the Demon is nearly at absolute zero T, then it can increase its own entropy arbitrarily, and still be at nearly absolute zero T because it is taking on very little energy. But I agree that in practice, most Demons are not going to have an effective T that is incredibly small, so they are going to need an environment they can dump non-negligible heat into and increase the entropy of.

If you want to throw a monkey wrench into the workings of the second law, then you have to design a Turing machine that accomplishes the demon's purpose while dissipating less than that lost entropy. If you can design a Turing machine that is as general as possible, which dissipates no heat, you will have surely thrown a monkey wrench into the second law (given the arguable assumptions already made).
But you will also run afoul of the basic axioms of thermodynamics, which will not be possible to avoid. Let's say you do make such a Turing machine, whose functioning is to reduce the entropy in the gas by more than it increases the entropy in the environment in which it is functioning. Then the net result of the action of your machine is to go from more likely configurations to less likely ones. What keeps the machine from running backward? Remember, all the laws of physics involved are time reversible, so if I send t to -t, I get a version of your Turing machine that makes the opposite decisions and causes the gas to come to the same T, and does so in a way that increases entropy. So what is going to make your Turing machine not turn into mine? I claim that is just exactly what it will do. Remember, the Turing machine is not a magical genie, even if you give it a program it can follow that program in either order of time, and it will follow it in whatever order increases entropy, because the arrow of time comes from less likely configurations being replaced by more likely ones. The arrow of time isn't magic either.
 
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  • #57
Ken G, if you believe that the 2nd law of thermodynamics is so natural, and that it is an obvious consequence of not only Newton's laws but any laws remotely like them, then why is Boltzmann's H-theorem so hard to prove, involving advanced mathematics and clever pieces of reasoning? Your view of the 2nd law as a near-tautology "systems are more likely to evolve into more likely states" should allow for a much easier proof of the theorem.
 
  • #58
Ken G said:
Actually I think it's the other way around-- the meaning of temperature stems from thermodynamical entropy. The best way to think about what temperature is is to multiply it by k and say that kT is the energy a system must take in in order to increase its access to undifferentiated states by a factor of e. But one can talk about the latter without ever referring to the former, if you simply don't care about energy (given that it's conserved, we usually do choose to care about it, but we don't have to and we still have a second law). So T comes in not so much with entropy, but when you want to connect entropy to energy. That means it relates to the importance of entropy, because energy measures the consequences of what the entropy is doing. But we still have a second law even if we have never heard of either energy or temperature.

Hmmm - ok, right.

Ken G said:
Be careful not to confuse entropy with energy. If the Demon is at a very low T, then a very small energy change can correspond to a huge entropy change, and the Demon can remain at low T yet still increase its entropy without dumping anything to the outside. Indeed, if the Demon is nearly at absolute zero T, then it can increase its own entropy arbitrarily, and still be at nearly absolute zero T because it is taking on very little energy. But I agree that in practice, most Demons are not going to have an effective T that is incredibly small, so they are going to need an environment they can dump non-negligible heat into and increase the entropy of.

Yes. I am keeping close track of the difference between entropy and energy, and I agree with the above.

Ken G said:
But you will also run afoul of the basic axioms of thermodynamics, which will not be possible to avoid. Let's say you do make such a Turing machine, whose functioning is to reduce the entropy in the gas by more than it increases the entropy in the environment in which it is functioning. Then the net result of the action of your machine is to go from more likely configurations to less likely ones. What keeps the machine from running backward? Remember, all the laws of physics involved are time reversible, so if I send t to -t, I get a version of your Turing machine that makes the opposite decisions and causes the gas to come to the same T, and does so in a way that increases entropy. So what is going to make your Turing machine not turn into mine? I claim that is just exactly what it will do. Remember, the Turing machine is not a magical genie, even if you give it a program it can follow that program in either order of time, and it will follow it in whatever order increases entropy, because the arrow of time comes from less likely configurations being replaced by more likely ones. The arrow of time isn't magic either.

Good point. It shows that in order for a Turing machine to have a forward direction, it must generate entropy. Let's assume the Turing machine is made from macroscopic mechanical parts. The arrangement of its mechanical parts is its "logical state". This logical state has nothing to do with thermodynamics, it constitutes a macrostate, it does not factor into the many microstates which give rise to this configuration. The entropy the demon generates is thermodynamic entropy, which is heat (assuming its temperature is not absolute zero). Now, this entropy, by the second law, must be greater than the entropy it removed from the two gases. The problem now is to prove this, hopefully by conceptually breaking down the Turing machine into separate logical operations, and showing which logical operations are generating the entropy, and what is the least amount that they generate, in principle. Then show that when you add them all up for a program that contains the minimum number of entropy-generating logical steps, but which still implements the demon's purpose, that value of entropy is greater than the entropy removed from the two gases.
 
  • #59
lugita15 said:
Ken G, if you believe that the 2nd law of thermodynamics is so natural, and that it is an obvious consequence of not only Newton's laws but any laws remotely like them, then why is Boltzmann's H-theorem so hard to prove, involving advanced mathematics and clever pieces of reasoning? Your view of the 2nd law as a near-tautology "systems are more likely to evolve into more likely states" should allow for a much easier proof of the theorem.
This is certainly a valid challenge, and I'll give it some thought, but on the surface I would say the difference is in understanding what the second law is basically saying, in contrast with how to place that fairly intuitive statement into a more rigorous mathematical structure. In the mean time consider these words by de Broglie:
"When Boltzmann and his continuators developed their statistical interpretation of Thermodynamics, one could have considered Thermodynamics to be a complicated branch of Dynamics. But, with my actual ideas, it's Dynamics that appear to be a simplified branch of Thermodynamics. I think that, of all the ideas that I've introduced in quantum theory in these past years, it's that idea that is, by far, the most important and the most profound."
 
  • #60
Rap said:
The problem now is to prove this, hopefully by conceptually breaking down the Turing machine into separate logical operations, and showing which logical operations are generating the entropy, and what is the least amount that they generate, in principle. Then show that when you add them all up for a program that contains the minimum number of entropy-generating logical steps, but which still implements the demon's purpose, that value of entropy is greater than the entropy removed from the two gases.
But what I would say is, once you have accomplished that breakdown, you still don't know which way your machine will function until you either invoke the second law, or claim to have experience in similar machines. Both are determined by the answer to the entropy issue, not the other way around. So our job is not to prove that the machine increases entropy, it is to figure out what the machine will do, given that it increases entropy. The problem with the "Demon" is that we borrow from the function of our brains to give the Demon magical properties, but we only do that because we think we are familiar with thought and decisions-- as soon as you have to provide a machine instead, broken down as you say, then you immediately do not know what that machine will do (in particular, what sense will time have for your machine). That comes from entropy, there is no other way to calculate what the machine will actually do once you build it, unless you invoke experience with similar machines (but then it's not a theoretical analysis any more).

If you do go that latter way, and rely on your experience with machines (say computers) instead of with brains, then the way the second law works in our experience is already built into the analysis. So the choice is, invoke experience and use it to show that the second law is behind that experience (which is what you want to do), or invoke the second law and figure out what will happen without the benefit of experience (which is what the Patent Office does when it refuses to consider perpetual motion machines). But there's no reason to think that using a machine to separate gas into different Ts is going to reduce the entropy, I think it's pretty clear that's a dead duck and the details of the machine (like if it has memory) are not terribly important or even advisable to analyze (since you cannot individually analyze every possible machine, just like the Patent Office cannot). But granted, you want to be convinced it's a dead duck, so for that you will need to invoke a lot of experience in how machines work, and if you see it for a few examples, you can develop the faith you seek in the second law. You can't add to my faith by analyzing a few more examples, you could only have an effect by finding a counterexample (certainly a noble effort, most likely doomed to fail but instructive in how it fails each time).
 
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  • #61
I'm tending to think that the 2nd law honestly doesn't apply to every conceivable system. Why should it? It says that we expect systems to go from less likely states to more likely ones, but it was all worked out in the context of gases, which exhibit vast numbers of microstates and blindly wandering molecules. That means we can get away with using very naive probability theory. Why should that be generalisable to computers with very deliberate wiring and small numbers of states? Not micro-, not macro-, just states.

A computer like that doesn't have anything we can *relevantly* identify as temperature, and why should it? We can have TdS+pdV systems that don't need FdL, or TdS+FdL without pdV, so why not pdV+FdL without any mention of temperature or entropy. A computer at 0 Kelvin would probably work rather well. If the program is basically cyclic then there is no accumulation of entropy in its logical state. If you want to claim that it's oozing entropy, then you'd have to mean that it dissipates heat somewhere in the universe, but is there really any prospect of finding a *mechanical* reason for putting a numerical limit on how much heat it has to dissipate? Maybe it runs on neutrinos or something.

What does likely mean anyway? In thermodynamics it means microstates per macrostate, but we seem to have agreed that macrostates are in the eye of the beholder. There's been no discussion of conditional probability, or dependent probabilities. The whole treatment of probability has been restricted to what dumb gases need.

There's been some suggestions that Turing machines can just as easily run backwards as forwards, but I don't see this either. I just need a 2 bit counter and I've got a one way system. I'm probably going to need to feed it some energy, but the cannonball gas argument already showed that this energy can be made negligible compared with the energy being switched around by the demon, so we concluded that it's not about the energy anyway. If it's not about the energy though, I can have my 2 bit counter, it can be arbitrarily efficient, and the entropy I need to generate elsewhere to keep it running can be made arbitrarily small.

With that cannonball gas we can actually shift a hell of a lot of entropy per decision. To see that, let the balls be ever so slightly inelastic and let them eventually dissipate their KE by warming the cannonballs over a long, long time. The demon was supposedly struggling to run on a limited entropy budget, but with big cannonballs he can afford to drink a bit of electricity. This has been my bottom line worry all along. We can hand-wave about this stuff, but can we write equations for it in units that match up?

I really think it's on the cards that computers could walk all over the 2nd law, but what would be the damage? Would we have to bin all the books? Nope. Gases would still behave just the same, and computers would still be ten orders away from finding out. I can't think of an area of physics that would totally implode if computers could break the 2nd law.
 
  • #62
lugita15 said:
Ken G, if you believe that the 2nd law of thermodynamics is so natural, and that it is an obvious consequence of not only Newton's laws but any laws remotely like them, then why is Boltzmann's H-theorem so hard to prove, involving advanced mathematics and clever pieces of reasoning? Your view of the 2nd law as a near-tautology "systems are more likely to evolve into more likely states" should allow for a much easier proof of the theorem.

Boltzmann's H-theorem holds for equilibrium only (or if you prefer, the tendency for a system to approach equilibrium). The theorem itself, dH/dt ≤ 0, is not trivial but not 'hard to prove', either- classical and quantum mechanical proofs are available from many sources.
 
  • #63
AdrianMay said:
I'm tending to think that the 2nd law honestly doesn't apply to every conceivable system.

The second law can be broken for short times, in terms of the fluctuation-dissipation theorem (S is allowed to fluctuate, just like any other physical quantity), and there are challenges using systems far from equilibrium, but to date no meaningful violation of the second law of thermodynamics has ever been observed.

http://prl.aps.org/abstract/PRL/v89/i5/e050601
http://www.mdpi.org/entropy/papers/e6010001.pdf
 
  • #64
AdrianMay said:
I'm tending to think that the 2nd law honestly doesn't apply to every conceivable system. Why should it? It says that we expect systems to go from less likely states to more likely ones, but it was all worked out in the context of gases, which exhibit vast numbers of microstates and blindly wandering molecules. That means we can get away with using very naive probability theory. Why should that be generalisable to computers with very deliberate wiring and small numbers of states? Not micro-, not macro-, just states.
We should expect it to apply to computers because computers also represent a vast number of states, not a small number. Now, if you build a quantum computer, you have microstates interacting with macrostates, and you might be able to isolate the microstates and get into the area of quantum thermodynamics (which still has some questions associated with the different interpretations and so forth). Some hold that thermodynamics is just a kind of special case and might not hold for quantum systems, others (like that de Broglie quote) take the opposite view that thermodynamical thinking is quite fundamental, and even things like wave functions and spacetime are merely instances of deeper thermodynamic (entropy controlled) engines.
 
  • #65
lugita15 said:
Ken G, if you believe that the 2nd law of thermodynamics is so natural, and that it is an obvious consequence of not only Newton's laws but any laws remotely like them, then why is Boltzmann's H-theorem so hard to prove, involving advanced mathematics and clever pieces of reasoning? Your view of the 2nd law as a near-tautology "systems are more likely to evolve into more likely states" should allow for a much easier proof of the theorem.

I would say the second law is not a near tautology. It's a simple concept, "the more likely a situation, the more likely it is to occur" gets right to the core of the second law, but to state it precisely can get rather complicated.

You have to have the concept of a microstate and a macrostate to begin with, and what macrostate is associated with each microstate. This can be less than obvious, and can be different for different observers. You have to have a mechanism by which each microstate changes, in time, into another microstate. Well, no, not exactly, you have to have a mechanism by which ALMOST EVERY microstate changes into another microstate. That requires a very large number of microstates. Then you have to know that this mechanism allows, by a series of steps, almost every microstate to evolve into almost every other microstate. You have to know or assume that, as a result of this process, almost every microstate is just as likely to occur as any other microstate. You have to show that almost every microstate yields the same macrostate, (the equilibrium macrostate). Only then can you say that almost every microstate which does not yield the equilibrium macrostate, will evolve in time in a way that it approaches that equilibrium macrostate. This is a statement of the second law. The entropy is defined as proportional (lets say equal to) the logarithm of the number of microstates that yield a given macrostate. That means that the entropy of a non-equilibrium macrostate will be lower than that of the equilibrium macrostate, and its entropy will tend to increase to that of the equilibrium macrostate. This is another way of stating the second law. It also means that the entropy of the equilibrium macrostate is almost equal to the logarithm of the total number of microstates. Even this description is not complete. Its these details that cause the H-theorem to be so complicated.


Ken G said:
But there's no reason to think that using a machine to separate gas into different Ts is going to reduce the entropy, I think it's pretty clear that's a dead duck and the details of the machine (like if it has memory) are not terribly important or even advisable to analyze (since you cannot individually analyze every possible machine, just like the Patent Office cannot). But granted, you want to be convinced it's a dead duck, so for that you will need to invoke a lot of experience in how machines work, and if you see it for a few examples, you can develop the faith you seek in the second law. You can't add to my faith by analyzing a few more examples, you could only have an effect by finding a counterexample (certainly a noble effort, most likely doomed to fail but instructive in how it fails each time).

I think we agree on how things work, we disagree on what is interesting or important. I am interested in the idea that some fundamental statements can be made about the thermodynamics of computing. You may have to treat every real case as a separate example, but I think it is very interesting if some statements can be made about the thermodynamics of individual logical operations, like Landauer's statement that only irreversible logical operations will unavoidably generate thermodynamic entropy which must be greater than some minimum value. The patent office rejects perpetual motion machines because they are not in the business of finding where somebody screwed up when they are sure that they have screwed up. I think there may be situations where finding out where they screwed up can be interesting and informative, and yield a new or more complete understanding of the second law.

AdrianMay said:
I'm tending to think that the 2nd law honestly doesn't apply to every conceivable system. Why should it? It says that we expect systems to go from less likely states to more likely ones, but it was all worked out in the context of gases, which exhibit vast numbers of microstates and blindly wandering molecules. That means we can get away with using very naive probability theory. Why should that be generalisable to computers with very deliberate wiring and small numbers of states? Not micro-, not macro-, just states.

The fundamental (classical) statistical mechanics question is "how do you describe the evolution of a physical system when you don't have complete knowledge of its state?". You don't have complete knowledge of the initial state or any intermediate state. The second law does not apply to situations in which you have complete knowledge of the initial state.

If you don't have complete knowledge of initial conditions, one approach is to assign probabilities to every conceivable initial condition, and then, using physical principles, calculate the probabilities of a what you will finally measure. The second law says that if you have a situation where a particular final measurement is almost certain, then that's almost certainly what you will measure. Or maybe, more weakly the second law says that if you have a situation where a particular final measurement is most likely, then that's most likely what you will measure. Otherwise, the second law is not applicable.
 
  • #66
I agree with you that analyzing why the second law continues to apply no matter how hard you try to "trick it" into not applying is informative. It's kind of like the Lorentz symmetry in relativity, or the uncertainty principle in quantum mechanics-- laws that people tried very hard to "get around" with all kinds of examples, but eventually gave up the effort and instead just accepted the law. Indeed that is how physics works, to a large extent-- we never know our laws are correct, we just eventually gain faith in them after trying hard enough to refute them (and usually we eventually do refute them in ways that are very informative indeed). I'm just saying that the easiest way to analyze situations involving very high velocities, or very small systems, or Maxwell's Demons, is to accept that Lorentz symmetry, and the uncertainty principle, and the second law of thermodynamics, are going to tell you what will happen there. So the question then becomes, why are they going to work, not why are they not going to work. I feel we benefit more from understanding and accepting the law, that has so much empirical support in all these contexts, then we do by constantly doubting it, even though I admit maintaining constant doubt is a key part of scientific progress. So what I really mean is, when we analyze these Demons, we should be looking for the places that the entropy goes, given that we know the entropy has to go somewhere or the Demon just won't work (and indeed the time-reversed version of the Demon will work instead). In this way, we have a guide to keep us from overlooking some entropy destination-- rather than looking for why the second law isn't going to work, which some parts of this thread started to get the flavor of (I'm not saying that was your approach).
 
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  • #67
Ken G said:
In this way, we have a guide to keep us from overlooking some entropy destination-- rather than looking for why the second law isn't going to work, which some parts of this thread started to get the flavor of (I'm not saying that was your approach).

Well, I had not looked at Maxwell's demon too much before this thread, and I thought that the answer was that the demon would not work due to the second law. This has changed.

I have been trying to understand the concept of indistinguishable particles in a classical gas, using what I called a "billiard ball" gas and supposing for clarity that each was imprinted with a unique serial number that had no effect on collisions, and why the thermodynamics of this gas is not changed by erasing the serial numbers, you still need to make the indistingushable particle assumption. As long as no thermodynamic process is a function of those serial numbers, so its not a quantum effect, QM just says its a matter of principle, rather than a happenstance. So when I saw a "gas of cannonballs" I was definitely interested.

Regarding the second law, its still not fixed in my mind. Not so much its validity, but its range of application, the trouble in defining a "macrostate" and the extent to which the macrostate is somewhat arbitrary, depending on the capabilities of the person doing the measurement, rather than on the system itself, pointing out again the arbitrariness of the entropy. Entropy is missing information, and if you manage to gain more information without disturbing the system (classical, again), then you reduce the entropy of the system, without having altered it in any way.
 
  • #68
Rap said:
Well, I had not looked at Maxwell's demon too much before this thread, and I thought that the answer was that the demon would not work due to the second law. This has changed.
Yes, the Demon does work. Whether or not it is a practical way to get free energy is not so clear, maybe it's just technologically unfeasible.
I have been trying to understand the concept of indistinguishable particles in a classical gas, using what I called a "billiard ball" gas and supposing for clarity that each was imprinted with a unique serial number that had no effect on collisions, and why the thermodynamics of this gas is not changed by erasing the serial numbers, you still need to make the indistingushable particle assumption.
I'm not clear on what you mean by still needing to make the indistinguishable assumption. It seems to me this will simply depend on your goals, and you can make it in some situations and not make it in others. You can treat distinguishable (classical) particles as indistinguishable if it doesn't matter to the outcomes you have in mind, and you can treat distinguishable particles as indistinguishable too, it depends on what you care about, or more correctly, what you can get away with not caring about. That's generally true of the entropy concept-- the order is, we choose what we care about and what we know, that controls the entropy, and the entropy gives us a second law.
As long as no thermodynamic process is a function of those serial numbers, so its not a quantum effect, QM just says its a matter of principle, rather than a happenstance.
Quantum mechanics just brings in another type of situation we might need to care about, because it brings in entanglement. In some situations, particles become entangled in ways where indistinguishability is of fundamental importance, and we have to care about it or we miss the necessary entanglements. In other words, sometimes nature tells us what we need to care about, rather than us telling her what we want to care about.
Regarding the second law, its still not fixed in my mind. Not so much its validity, but its range of application, the trouble in defining a "macrostate" and the extent to which the macrostate is somewhat arbitrary, depending on the capabilities of the person doing the measurement, rather than on the system itself, pointing out again the arbitrariness of the entropy. Entropy is missing information, and if you manage to gain more information without disturbing the system (classical, again), then you reduce the entropy of the system, without having altered it in any way.
I would say that concept is not supposed to be "fixed" in your mind, it is supposed to be highly fluid in your mind! Entropy is malleable, it is whatever we need it to be to describe the relative probability of various categories of outcomes. In my view, entropy is nothing but a classification scheme for states, and some classification schemes are more useful than others. So the trick to using entropy, and the second law, is simply finding a good classification scheme, and that's not always easy.
 
  • #69
I'm not sure I understand these brain arguments against the problem. It seems that the argument is that the mental processes required by the demon increase the entropy to preserve the second law of thermodynamics (unless I am misunderstanding).

I don't know if this has been brought up or not, but what if the demon were an extraordinarily stupid and simple being who had no idea what he was doing, but he still manipulated the system with such luck as to match exactly what the thinking demon would have done? I suppose this is similar to a sequence of atoms spontaneously forming into crystals, but that doesn't seem impossible either (though certainly unlikely).

I don't see how this could be considered anything other than statistical, though I am certainly no expert.
 
  • #70
Acala said:
I'm not sure I understand these brain arguments against the problem. It seems that the argument is that the mental processes required by the demon increase the entropy to preserve the second law of thermodynamics (unless I am misunderstanding).

I don't know if this has been brought up or not, but what if the demon were an extraordinarily stupid and simple being who had no idea what he was doing, but he still manipulated the system with such luck as to match exactly what the thinking demon would have done? I suppose this is similar to a sequence of atoms spontaneously forming into crystals, but that doesn't seem impossible either (though certainly unlikely).

I don't see how this could be considered anything other than statistical, though I am certainly no expert.

You get around this problem by assuming the demon is a computer. That way you have a definite physical system to deal with.
 
<h2>1. What is Maxwell's Demon and Liouville's Theorem?</h2><p>Maxwell's Demon is a thought experiment proposed by physicist James Clerk Maxwell in 1867. It involves a hypothetical being that can selectively control the movement of particles in a closed system, seemingly violating the second law of thermodynamics. Liouville's Theorem, on the other hand, is a mathematical principle that states the volume of a given region in phase space is conserved over time in a closed system.</p><h2>2. What is the connection between Maxwell's Demon and Liouville's Theorem?</h2><p>The connection between Maxwell's Demon and Liouville's Theorem lies in the fact that both concepts deal with the behavior of particles in a closed system. Maxwell's Demon challenges the second law of thermodynamics, which states that entropy (disorder) in a closed system will always increase over time. Liouville's Theorem, on the other hand, provides a mathematical basis for this law by showing that the total volume in phase space, which represents all possible states of a system, remains constant over time.</p><h2>3. How does Maxwell's Demon relate to the second law of thermodynamics?</h2><p>Maxwell's Demon challenges the second law of thermodynamics by proposing a scenario in which a being can selectively control the movement of particles in a closed system, leading to a decrease in entropy (increase in order) over time. This goes against the idea that entropy always increases in a closed system, as stated by the second law.</p><h2>4. Can Maxwell's Demon be reconciled with Liouville's Theorem?</h2><p>No, Maxwell's Demon cannot be reconciled with Liouville's Theorem. While Maxwell's Demon challenges the second law of thermodynamics, Liouville's Theorem provides a mathematical basis for this law. The two concepts are incompatible and cannot coexist in a closed system.</p><h2>5. What are the implications of Maxwell's Demon and Liouville's Theorem in physics?</h2><p>Maxwell's Demon and Liouville's Theorem have significant implications in the field of physics. Maxwell's Demon challenges our understanding of the second law of thermodynamics and raises questions about the nature of entropy and order in closed systems. Liouville's Theorem, on the other hand, has practical applications in statistical mechanics and quantum mechanics, providing a mathematical foundation for the laws of thermodynamics.</p>

1. What is Maxwell's Demon and Liouville's Theorem?

Maxwell's Demon is a thought experiment proposed by physicist James Clerk Maxwell in 1867. It involves a hypothetical being that can selectively control the movement of particles in a closed system, seemingly violating the second law of thermodynamics. Liouville's Theorem, on the other hand, is a mathematical principle that states the volume of a given region in phase space is conserved over time in a closed system.

2. What is the connection between Maxwell's Demon and Liouville's Theorem?

The connection between Maxwell's Demon and Liouville's Theorem lies in the fact that both concepts deal with the behavior of particles in a closed system. Maxwell's Demon challenges the second law of thermodynamics, which states that entropy (disorder) in a closed system will always increase over time. Liouville's Theorem, on the other hand, provides a mathematical basis for this law by showing that the total volume in phase space, which represents all possible states of a system, remains constant over time.

3. How does Maxwell's Demon relate to the second law of thermodynamics?

Maxwell's Demon challenges the second law of thermodynamics by proposing a scenario in which a being can selectively control the movement of particles in a closed system, leading to a decrease in entropy (increase in order) over time. This goes against the idea that entropy always increases in a closed system, as stated by the second law.

4. Can Maxwell's Demon be reconciled with Liouville's Theorem?

No, Maxwell's Demon cannot be reconciled with Liouville's Theorem. While Maxwell's Demon challenges the second law of thermodynamics, Liouville's Theorem provides a mathematical basis for this law. The two concepts are incompatible and cannot coexist in a closed system.

5. What are the implications of Maxwell's Demon and Liouville's Theorem in physics?

Maxwell's Demon and Liouville's Theorem have significant implications in the field of physics. Maxwell's Demon challenges our understanding of the second law of thermodynamics and raises questions about the nature of entropy and order in closed systems. Liouville's Theorem, on the other hand, has practical applications in statistical mechanics and quantum mechanics, providing a mathematical foundation for the laws of thermodynamics.

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