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Maxwell's Demon revisited 
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#37
Feb2612, 11:41 PM

P: 789

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 Turingmachine 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. 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 informationtheoretic explanation of how this happens, consistent with the second law. Earman and Norton have three objections to Landauer's principle  1) (pages 1416) 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?" 


#38
Feb2612, 11:59 PM

P: 789

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. 


#39
Feb2712, 02:10 AM

P: 98

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? 


#40
Feb2712, 07:25 AM

P: 789

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 measurementtrapdoor 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
Feb2712, 07:53 AM

P: 98

I think I can describe the microstates of my dogleg 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
Feb2712, 07:58 AM

P: 273

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
Feb2712, 08:31 AM

P: 98




#44
Feb2712, 07:26 PM

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P: 3,080




#45
Feb2812, 12:59 AM

P: 789




#46
Feb2812, 02:00 AM

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#47
Feb2812, 02:29 AM

P: 1,583

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
Feb2812, 09:08 AM

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#49
Feb2812, 10:46 AM

P: 1,583




#50
Feb2812, 12:24 PM

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#51
Feb2812, 02:37 PM

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#52
Feb2812, 05:45 PM

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#53
Feb2812, 06:20 PM

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#54
Feb2812, 08:16 PM

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P: 3,080

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 "nonteleological" 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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