Monday, December 17, 2012

Exorcising Maxwell's daemons

And the lowest allowed power consumption of PCs

In our discussions about information and heat, James Gallagher said some of the usual wrong things about irreversibility – for example, he believes that the proof of the H-theorem is invalid because of the molecular chaos assumption (this assumption is a pure technicality allowing explicit calculations but the overall conclusion, the increasing entropy, is independent of any such Ansatz!).

However, he has also made a statement about an algorithm to reduce the entropy with the help of his PC:
I mean I can simulate deterministic dynamical systems on my computer and reverse all the dynamics at any time - which MUST then result in a decreasing entropy if the previous system had increasing entropy.
I assure you, James, that your method doesn't work. What you suggested has been known as Maxwell's daemon and the 20th century analyses have made it clear that no such proposed device may actually reduce the total entropy.

This blog has discussed Maxwell's daemon many times. See, for example,
Feynman on the arrow of time

Arrow of time understood for 100 years

Maxwell's daemon cannot do useful work
Recall that in his 1964 Messenger Lectures at Cornell, Feynman showed that Maxwell's daemon couldn't work because the "wheels with teeth" that were meant to undo the balance of a physical system ultimately worked so that they transmitted energy/heat from the warmer body to the cooler one, as expected, so the entropy goes up whether you like it or not.

Because James' comment shows that people, including frequent visitors of physics blogs, still haven't noticed that such devices cannot work, I decided to write one more blog entry of the sort and add some interesting related ideas that haven't been written on TRF yet.

Maxwell's daemon is a hypothetical agent or device that operates inside a thermodynamic system – in this case a vessel divided to two parts – and it does something intelligent in order to encourage processes that "naturally" occur in the opposite direction only.

For example, the daemon may open the door whenever a faster, hotter molecule is coming from the left, so that it gets to the right, and it may similarly encourage slower, cooler molecules to be concentrated in the left part. Or it may just try to concentrate all the gas molecules in one part. Or sort them in another way.

Whatever details we choose, the point is that the daemon is reducing the entropy of the gas (or another object). It is effectively able to increase temperature differences between the two parts of the vessel even though in Nature, temperatures tend to get more uniform as time goes by. Or the daemon may do something else that doesn't seem to happen naturally.

If such a daemon were possible, the advantage would be clear. We could construct the perpetual motion machine of the second kind because we could increase the temperature differences between the two parts of the vessel and use a part of the difference to do mechanical work. Well, in the case of the daemon that just spontaneously concentrates all the molecules in one part, we would seemingly construct the perpetual motion machine of the first kind because the pressure difference could be used to do mechanical work "immediately". However, if you look carefully, it would still be the perpetual motion machine of the second kind because the temperature of the gas would go down as you would be extracting work out of it.

Now, can the daemon exist? You may say that Maxwell's daemon is a metaphor for the government. So the people who are deluded and believe that the government may "social-engineer" things that work more effectively than the free market, may also believe that Maxwell's daemon that violates the second law may be produced and launched. Well, it cannot, those of us who understand the basic laws of thermodynamics and economics know.

But for a while, Maxwell's daemon may have been viewed as another "giant" who may perhaps "beat" the laws of thermodynamics. It wasn't quite clear who would win: the laws of thermodynamics, or Maxwell's proposed cleverness? Note that Maxwell designed the thought experiment as an example of a new effect that becomes possible when we replace the approximate laws of thermodynamics by their precise microscopic realization in terms of statistical physics. Such improvements of the foundations of physics often lead to new possibilities, so Maxwell's daemon could have been possible, couldn't it?

Leo Szilárd published the first article saying "No, the daemon won't work" in 1929. The paper said many nice things that were partly right. Equally importantly, the connection between the entropy and information appeared very clearly in that paper. He was able to say that doing something with the information of 1 bit was changing some entropy (or some part of it) by\[

S = k\ln 2.

\] The entropy equals Boltzmann's constant multiplied by the natural logarithm of two (nats). Also, he would figure out that some operations doing something with one bit created or moved energy or heat \(\Delta E = \pm kT\ln 2\) somewhere. I am being deliberately vague here because at this vague level, Szilárd was right.

(Did you know that Szilárd has mastered the method of getting grants by blackmailing that he would publish a paper on how to build your nuclear bomb? He was a proponent of bribing politicians to improve the world, too.)

The exact details about the moment when the entropy increases and guarantees that the total entropy can't go down were slightly wrong in his paper. He essentially believed that the entropy associated with the daemon (=the expenses of the government) had to increase when the information about the molecules was being accumulated.

These days, it seems that a different accounting of the entropy increase is much more convincing. The more precise explanation what's going on emerged as Landauer's principle in the 1961 paper by Rolf Landauer of IBM. See also a 1981 paper by Charles Bennett or newer lectures by John Preskill for some modern comments.

Landauer realized that the entropy measures the information about the microscopic arrangement that has been lost, that has become inaccessible. And this part of the information is actually not increasing when a computer accumulates the information about the molecules in the vessel. Instead, it is being lost when the computer erases its memory which it needs to do at some point before it accumulates new data – at least assuming that the computer's memory is finite.

(If the computer has an insanely high or "infinite" memory, one may assign its state an entropy in such a way that the second law will continue to hold even though the memory never has to be erased. At that moment, the validity of the second law may look vacuous or convention-dependent but that only occurs if we assume some unphysical assumptions.)

It's funny to look at the computers' minimal power consumption dictated by Landauer's principle. If your fast computer (well, one you may have in a few years) needs to erase one trillion bits per second, \(kT\ln 2\) per bit will tell you that the minimal consumption of such a computer at the room temperature is 2.85 nanowatts. Clearly, the actual microprocessors and memory chips still consume many orders of magnitude more energy than that – which is why it makes no practical sense to try to construct "reversible computers" that don't erase things and that could circumvent the Landauer bound. The experts say that such "reversible computation" is possible in principle. I have some modest doubts about it but I won't clarify them.

While the Landauer limit is much smaller a power consumption than what we seem to see in the real world, I feel that it may be necessary for the power consumption to be much higher than the Landauer limit if the computers are supposed to work flawlessly. For the bits of information inside the computer to behave as classical bits, I think it is necessary to copy them many times – in the sense of decoherence. Only when the information is copied many times, the classical-bit approximation becomes OK. For this reason, I would guess that the minimal consumption of reliable enough classical computers will always be greater than the Landauer minimum by at least an order of magnitude.

Nevertheless, we still have lots of room to reduce the power consumption. And one additional simple way to reduce \(kT\ln 2\) is to reduce the temperature \(T\). Highly cooled computers could consume less energy if the Landauer bound ever became relevant. Of course, this discussion isn't useful for your PC at home because the cooling systems needed to get this low would probably consume much more energy than your PC so you wouldn't get much.

In his belief that it's easy to "reverse any evolution" with the help of a computer and therefore to reduce entropy, James Gallagher makes a kind of isomorphic mistake as the proponents of Keynesianism, socialism, communism, and related crackpot theories in economics do. They overlook the expenses of the government and the inefficiency that the government itself brings to the system. Well, that's a pretty serious mistake.

Analogously, James thinks that the computer operates "for free" and doesn't create any entropy. But the point here is that the computer is a physical object, much like the government bureaucrats are people who still behave to maximize their utility function. When one does a proper analysis that includes the computer or the government offices, it becomes completely clear that Maxwell's daemon and the government simply cannot work to transform their utopia into reality.

James was thinking about a device that probably measures the positions and velocities – or the quantum state? – of all the molecules in the vessel and does a "global calculation" before it adjusts the motion of all the molecules in the vessel. This may look ambitious or different than Maxwell's daemon but if it is looked at rationally, it's easy to figure out that it's just another realization of Maxwell's daemon, and a highly inefficient one. To reconstruct the state of a kilogram of a classical gas, one would need something like \(10^{26}\) bits of information – well, a big multiple of that because the velocities would have to be known exponentially accurately. But to reserve the room in your PC memory for this huge inflow of data, you would have to erase a huge number of bits and the entropy would go up dramatically as a result.

Some people still keep on suggesting that Maxwell's daemon could work. A notable example is this 1998 paper by John Earman and John Norton in which they claim that the debunking of Maxwell's daemon is based on the circular reasoning. I find such texts deeply confusing. If you read the paper, they seem to deny the tight relationship between the entropy and information in general. This seems utterly indefensible to me.

In their "circularity" argumentation, they apparently criticize the "Maxwell's daemon won't work" arguments for their making assumptions about the behavior of the entropy of the daemon. Well, because Maxwell – and others – failed to present a precise model of the daemon's inner workings, the people who analyze this gadget must make some assumptions. In the particular models one may construct, it may be seen that the erasure of information does increase the entropy.

But even though it's impossible to "describe all the details" about the processes in a gadget – general Maxwell's daemon – that no one has clearly defined (it should be a task for the proponents of the daemon, shouldn't it?), it's still true that we have proofs that the entropy is increasing that don't seem to depend on the non-existence of computers, memory chips, and other devices at all. This simply means that these proofs have to apply to computers, too. A microprocessor is just another physical object. A memory chip is a physical object, too. And the fact that the entropy can't decrease in the thermodynamic limit may be proved quite generally, so it must apply to computers, too.

At least, without some convincing example showing a loophole, an example that hasn't been debunked yet, it seems silly to me not to consider the second law to be a general fact. Earman and Norton may try to criticize people with my opinion as people defending dogmas but it's a dogma analogous to \(2+2=4\) – it's demonstrably true although one may still be forced to work hard to debunk increasingly sophisticated attempts to prove that \(2+2\neq 4\).

To imagine that some "clever engines" may stand "above" the laws of physics is just a ludicrous religious belief, an elementary fallacy. It's a similar fallacy as the fallacy that the planned economies and their goverments may deny the general laws of economics. They cannot. People and even politicians are physical objects, too. Of course that if one envisions a divine daemon that doesn't have to be subject to the laws of physics, many things become possible. But in my opinion, the dreaming about such divinities doesn't belong to natural science. And because Earman's and Norton's argumentation is so analogous to the claims "science will never be able to prove the non-existence of supernatural beings and scientists are just bigots who have to assume the non-existence of supernatural beings and adjust their rules to defend their belief", I think that this argumentation isn't scientific, either.

Note that I have used this thermodynamics-economics analogy many times in this text. I actually believe that thermodynamics is the discipline of physics that is most analogous to economics. Some left-wing types who hate the free markets love to imagine that economics isn't a science in any sense. Well, it is a science and many of its statistical assertions, especially various inequalities, are closely analogous to various inequalities one may establish in thermodynamics.


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  2. I'm very busy today and have only skimmed the article.

    But just to make some things clear, the low entropy points in the deterministic dynamical phase space form a much smaller set than the high entropy ones - so it is not surprising that GENERICALLY (for "most" initial ensembles) you will evolve to a higher entropy state.

    However, GIVEN any one of these evolutions with increasing entropy there is an exactly reversed evolution with decreasing entropy - since the dynamics are deterministic I can exactly reverse them - hence a reverse entropy evolution is not really any more unusual than picking a low entropy starting point in the first place.

    HOWEVER, if I introduce true randomness into my evolution equation, eg it's good enough to use a computer's (deterministic!) random number generator to seed the evolution at each time step - then even me, as the programmer god of the simulation can not get a reversed entropy evolution unless I wait zillions of lifetimes of the (simulated) universe and get 'lucky' - ie I can't just reverse the dynamics this time - that will still result in an overwhelmingly likely evolution to higher entropy.

    And I don't need to add any psychologically convincing arguments here - unlike in the deterministic case.

  3. Dear James,

    But just to make some things clear, the low entropy points in the deterministic dynamical phase space form a much smaller set than the high entropy ones - so it is not surprising that GENERICALLY (for "most" initial ensembles) you will evolve to a higher entropy state.

    sorry, this is a completely sloppy "derivation" of the second law. Whether something forms a smaller set or a larger set doesn't immediately imply that it will evolve out of something else. The number (of elements in a set) is a completely different quantity than the probability (of an outcome), and so on, much like the body mass is a different thing than the balance of one's banking account or voltage in his power outlet. Your "reasoning" makes no sense.

  4. Ok I read the article above, it has nothing to do with anything I've been saying but it is interesting.

    Maxwell's Demon would work if the demon had true free-will - so that his actions weren't constrained by schrödinger evolution.

    Isn't that how we've made so many fucking low-entropy things on Earth? Do you think industrial machinery and vehicles spontaneously come into existence elsewhere in the universe without conscious life and free-will?

  5. Dear James, I have no idea why you think that the free will is related to violations of Schrödinger's equation – no object, whether animate or inanimate, may violate it – and independently of that, why the free will would help to make the impossible daemon possible.

    We make low-entropy things on Earth, like those in the fridge, for the price of increasing the entropy of the environment even more than the reduction of the entropy. It's always like that.

    We may do lots of other things because the Sun gives us tons of useful energy. It arrives in the visible and ultraviolet low-entropy photons – they have low entropy per unit energy because each photon has S of order 1 and the number of photons from the Sun is reduced by their having a low energy. On the other hand, we re-emit this energy as thermal radiation in the infrared which has much lower-energy photons that therefore carry much more entropy per unit energy. This allows us to reduce the energy of many things on Earth because the photons from the Sun are transformed to a lower-frequency form with a higher entropy, guaranteeing that the second law isn't violated even if some other objects decrease their entropy along the way.

  6. Yes I know that, and it explains the evolution of plant-life and simple animals - but it does not sensibly explain the creation of Shakespeare's plays, Mahler's symphonies, steam engines, aircraft carriers, apollo 11 etc etc.

    Clearly there is something in addition to schrödinger evolution (and random seeding) going on on earth.

  7. Lubos, Summing up Feynman you wrote (in Feynman link above): "We would say that some phenomena are emergent and many microscopic details become irrelevant along the way." I recall that Susskind suggested in one of his video lectures that this is the basic idea behind renormalization. It's why chemistry, biology, and, I presume, other disciplines like psychology and (as you say) economics have their own principles of analysis.

    I particularly agree with your observation that in economics the most important relationships (maybe the only ones) are "more than" and "less than". And since all economic policy decisions are political decisions the division of political science at the beginning of last century into the two disciplines of economics and political science was artificial and probably harmful. As for the goal of political science, I would define it as the discovery of policy that tends to increase the welfare of this and the succeeding generation. Of course policy doesn't imply more government necessarily: a reduction in government intervention is also policy.

  8. Dear Lubos, when you say: "... it would still be the perpetual motion
    machine of the second kind because the temperature of the gas would go
    down as you would be extracting work out of it." it remembered-me that a Black-Hole has negative Heat Capacity ( so it is such a beast you fear to find ... i.e. a realization of Maxwell's Daemon, right?

  9. Nice article, the thermodynamics - economics analogy is fun :-)

    This article reminds me of a funny story my supervisor has told us:

    When waiting at the central station of Hamburg for the next train, a crazy and characteristically dressed environventalist young man chatted him up, trying to convince him of his ideologies that everything that the modern society does is bad etc, everybody knows what I mean ...;-)

    And as the young man learned that my supervisor is a physicist, he claimed that the physicists are the worst guys among all of the devils and stated proudly that furthermore his is strictly against the second law of thermodynamics :-)))

    That made my supervisor rolling on the floor laughing, and everybody else too when he talks about this funny story :-D

  10. James, your argument with Lubos bears a lot of similarity to those of creationists, who say that the second law of thermodynamics disproves "Darwinian" evolution because life is complex. You're basically making exactly the same incorrect point.

    I'm almost expecting you to start talking about irreducible complexity, series A versus series B theory of time, et cetera

  11. Dear Numcracker, black holes have a negative heat capacity, yes, it is unusual, but it is something completely different - much less illegal - than Maxwell's daemon. Maxwell's daemon should lower the entropy of the system, whatever else it does. Having a negative heat capacity would mean that the entropy would decrease if the temperature spontaneously increased. However, this won't happen as the temperature spontaously decreases as black hole devours the material. And when it radiates, its temperature is getting hotter which, due to negative heat capacity, means its decreasing entropy, but it also emits the Hawking radiation that overcompensates it so that the total entropy grows.

    There is no Maxwell's daemon in Nature - it violates the basic rules of logic, really. This assertion is valid everywhere, including quantum gravity. One thing I constantly notice is that the laymen are always eager to throw away *any* knowledge within seconds and create nonsensical exceptions. There are no exceptions for many of the fundamental insights in physics, however.

  12. LOL, good for him, the green chap.

  13. Exactly, David, it's the same essence of the argument. With some "intelligence", a leader or creator of the world may circumvent similar laws of Nature.

    In reality, when one looks at this "intelligent person with a computer", how he's trying to reduce the entropy, there will be rather normal components and physical objects in it that will heat etc. From the viewpoint of thermodynamics, it will be completely clear that the entropy goes up, as it always does, and the complicated calculations will be seen as irrelevant distractions only.

    When current goes through wires, they heat up, the entropy goes up, and similarly for all other processes in a computer. When the computer manages to calculate something that may remove a few bits of entropy from a different system, it's damn obvious that the computer's own entropy will increase much more. There's actual heat, a macroscopic number of atoms, in the computer, and the computer is only trying to social-engineer the state of a rather small number of bits. It should be even intuitively clear that the information that the computer engineers is negligible relatively to the microscopic information about all the computer's atoms.

    Creationists also fail to see this hierarchy or inequality. The birth of a symmetric object means a decrease of entropy. However, most of the entropy isn't stored in the symmetry or asymmetry of an egg. Almost all the entropy is stored in the thermal motion of individual atoms.

  14. I'm about the layest of the lay men, and sorry if this is OFF TOPIC, but reading about the thermodynamical/information theoretical restraints of Maxwell's demon, it occurred to me, that is it possible to refute philosopher Nick Boström's famous "Simulation argument" that states that we almost certainly do not really exist but are living inside a computer simulation, using similar reasoning?

  15. It is so good to see that some people can laugh and have a good time when they observe such omnoxious idiots! :-)

    It is exhausting and not good for one's health to be put in a sour or angry frame of mind too often in this era of mega-facilitated dissemination of daft delusions. Hence the follow defiant self-preservation promoting smiley: $->. :-)

  16. Yeah, and this green chap was so dumb and naive that he is mostly harmless ;-)

  17. Oh come on, it's nothing like the silly arguments creationists use - they don't even believe the eyeball could have evolved.

    I saying something completely different -that the dynamics fundamentally change once conscious free-will emerges from the mechanical evolution.

    I'm not saying that conscious beings can do magic - you can think of it perhaps as a many-worlder might - free-will allows you to select the evolutionary path through the multiple universes. But I don't believe in many-worlds so I rather just think of it as conscious being being able to manipulate the fundamental randomness at the deepest level in nature.

    So the maxwell demon with free-will would be able to select one of the rare paths where the evolution leads to a low-entropy state.

    In real life this requires a lot of effort and knowledge of course, you need to understand how nature works and you need a lot of energy to create useful low entropy objects.

    I realise that you all think I'm more crackers than a christmas cracker but I really don't care :-)

  18. This response only makes sense, at least to me, from a dualist perspective; i.e. that mental states aren't subject to physics. Otherwise, symphonies, engines, etc... are the result of brain activity, and the brain is fueled by (among other things) glucose. So the output of intelligent activity is just as much the product of the conversion of solar energy as is anything else.