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wandering sets, part 3 : Maxwell's demon

I gave 3 examples of things that dissipate in part 2: friction, electrical resistance, and hurricanes. I feel like I understand fairly well why we call these dissipative, although I've always felt or hoped that there is some unifying principle that sheds more light on the subject and explains why them and not other things. But there's a forth example that is far more interesting, and for that example I still don't feel like I really understand why exactly it's dissipative: computation.

Now, you might first think--maybe computation is dissipative because it involves the flow of electricity through circuits (whether those circuits be wires or be microchips), but that's beside the point. First, as I understand it, any kind of physical irreversible computational process must necessarily dissipate heat and increase entropy. So this applies not just to electrical circuits but to anything we could conceivably use to compute an answer to something, including for example, an abacus (of course the amount of computation that can be performed by an abacus is presumably so tiny that you wouldn't notice.) Second, it's not just the electrical resistance because supposedly, computers actually draw *more* electricity while they are involved in some intense computation, not when they are just idling. There are many circuits which are on while the computer is doing nothing, but it's not being on that creates the entropy I'm worried about... it's the entropy created specifically from irreversible computation, from switching those circuits on and off in just such a way that it computes a simple answer to a more complex question fed to it. Beforehand, there are many possible answers, but afterwards, there is only one... for example, 42. This reduces the available microstates of the system from many to one, and therefore represents a reduction of entropy (which remember, counts the number of available microstates). Because of the 2nd law, this cannot happen by itself without producing heat... it needs to produce heat in order to cancel out that entropy loss by a gain in entropy due to the heat, for exactly the same reason that the earth must dump heat into its environment if evolution is to result in more highly organized organisms. So even a perfectly efficient computer which caused no net entropy gain for the universe would still produce heat!

The only exception to the above process is if, instead of taking a large set of inputs and reducing them to one output, all of the inputs and outputs correspond exactly in a 1-to-1 fashion, in other words, you use all reversible logic gates to build the computer. An example of an irreversible logic gate is and AND gate. It takes 2 inputs and has 1 output, it outputs "Yes" if both the inputs are on, and "No" if either one of them is off. Another example is an OR gate, which outputs "Yes" if either input is on, and "No" if both are off. To build a reversible gate, you need 2 inputs and 2 outputs, so that if you ran the computation backwards, you could recover the question from the answer. For example, if you put 42 into the computer, it should be able to spit out what the ultimate question is, just as easily as going the other direction. This is the meaning of reversibility.

Maxwell's demon is a thought experiment that James Clerk Maxwell came up with which illustrates how weird this connection between entropy and information is. If there were a little demon who were watching the individual molecules in a box, and he had a switch that could slide in a divider instantly in the middle of the box, then he could sit there and watch for the moment when each gas particle (normally, bouncing around randomly in the box) was about to cross the boundary from one side of the box to the other. If he presses the switch at just the right time, he can deflect the gas particle back into the left side of the box without expending any energy. If he keeps doing this for hours and hours, eventually all of the gas particles will randomly wander into the left side of the box and get stuck there, because he will put in the partition just as they try to cross the boundary back over to the right. Because entropy is connected to volume (smaller volumes have a smaller # of microstates), the final state has less entropy than the initial state, due to having half the volume. And yet, no work was done and no heat was expended in the process! This seems to be a blatent violation of the 2nd law of thermodynamics. So what happened here?

Well, in the real world, demons don't exist. And humans do not have the supernatural powers that demons have that would enable them to see individual gas particles moving super fast around in a box. But what if we set up a computer that could play the role of the demon? In principle, a computer could detect a gas particle much faster than a human, maybe even as fast as Maxwell's hypothetical demon. But if it does this, it has to either store the information about where each of the gas particles are, or temporarily watch each gas particle for a moment and then forget about it. If it stores this information, then it needs to fill up an exponentially large memory storage system. If it wants to keep the storage from getting out of hand, then it has to erase some of this information at some point... and erasure of information is an irreversible process. Because it is irreversible, it must dissipate heat. I mostly understand this part of Maxwell's demon. The other part I've always been a little bit fuzzy on though... what happens if the computer chooses to just store and store more and more information in its memory? Then, it will be filling up its memory with more and more information about the trajectories of the billions and billions of particles in the box. But does this in itself represent an increase in entropy? Or is it just the erasure of such information which increases entropy? It seems to me that storing anything at a memory location which could have previously taken multiple values but is then set to a single value represents a decrease in entropy. It would seem that storing it decreases entropy and then erasing it undoes that increasing it again. But I must be thinking about things a bit wrong there. I admit, this is where my understanding has always grown a bit fuzzy.

In the next part, I hope to actually get to wandering sets, and by extension, Boltzman's brain paradox, Poincare recurrence, and Liouville's Theorem. But maybe that's ambitious. To be continued...


( 4 comments — Leave a comment )
Jul. 26th, 2013 12:35 am (UTC)
Wouldn't gas particles be bouncing back the other way, too? And the more of them that get trapped, the more of them will be bouncing around and be likely to cross back.
Jul. 26th, 2013 01:21 am (UTC)
Under normal conditions (without a demon) yes... every now and then, a particle from one side will cross over to the other, and this happens in both directions and is balanced. So in the long run, there remain roughly equal number of particles on either side of the box.

But the point of the demon, in this thought experiment, is he acts like a 1-way troll, who will only allow particles to pass in one direction. If he sees one coming in the direction he allows, he does nothing. If he sees one coming in the direction he doesn't allow, then he pushes a button that instantly drops a divider down in the middle of the box. As soon as the particle hits this temporary wall in the middle, it is deflected off and reverses its course going back to the side from whence it came. As soon as this happens, he pushes another button to instantly lift the divider back up. You have to imagine that all of this could happen at a ridiculously fast speed, before any other particles could cross the boundary in either direction.

The reason it is set up like this is that usually, if you want to compress a gas from an initial size down to half its size, you have to push one of the walls in, but this heats the gas as it is being compressed, because every time they bounce off of the moving wall, the motion accelerates the particles, making them move a little faster. When a lot of particles move around faster, it means they are at a higher temperature now. So you reduce the volume of the gas, but increase its temperature. The only way to do this without heating the gas is to not use any moving walls--you can only use stationary walls. But then you have to have reflexes as fast as the demon :-)
Jul. 26th, 2013 01:22 am (UTC)
Visual aid courtesy of wikipedia...


Edited at 2013-07-26 01:23 am (UTC)
Jul. 26th, 2013 01:27 am (UTC)
Actually the Wikipedia image may do more to confuse than help, as it depicts a slightly different version of Maxwell's demon than the simplified one I'm presenting. In their version, there are two different kinds of molecules, represented by blue and red, and the demon is sorting them. But this is unnecessary, the same paradox presents itself if you just have one kind and you want to move them all to one side.
( 4 comments — Leave a comment )


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