Maxwell's Demon revisited

lugita15

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?)

AdrianMay

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.

Rap

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.

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 thats necessary, but ok, lets 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?"

Rap

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.

Hmmmm

AdrianMay

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 in to 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?

Rap

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.

AdrianMay

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?

Delta Kilo

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.

AdrianMay

Hmm. Maybe there's something in that.

Ken G

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).
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.

Rap

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.

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.

Ken G

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.

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.

lugita15

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.

Ken G

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.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.
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.

lugita15

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.

Ken G

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.
Yes, if you have a model for the Demon. Do we have mechanical models of brains?
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.

lugita15

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.
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.
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.