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Suppose there are two gases in a recipient with a movable piston in the middle, separating the recipient into two halves. If both halves have different pressure, work can be extracted from the system by letting the pressure difference move the piston in one direction or the other.

But, if the two halves are connected to each other by a hole in the middle of the piston, such that they are in thermodynamical equilibrium with each other, no work can be extracted, by the Second Law of Thermodynamics.

Suppose, after some time, we close the hole. At this moment, it is likely that one side will have slightly more pressure than the other. We can then let the piston move and extract work from this (extremely small) pressure difference.

This process can then be repeated after both pressures return to equilibrium again, apparently extracting work from a gas in equilibrium. However, this goes against the Second Law of Thermodynamics, so what is wrong in this process?

João Marcos Cardoso da Silva
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    How do you know which side has the higher pressure? If you put a work-extracting mechanism on both sides to be sure, it will also fluctuate and push the wall back and forth. More information is needed to determine whether this Maxwell’s-demon-type problem has an information-related resolution or a Brownian ratchet resolution. But one of these provides the resolution. – Chemomechanics Jun 18 '22 at 17:34
  • By "recipient" do you mean a cylinder? And if so, is the cylinder insulated? – Bob D Jun 18 '22 at 17:40
  • @BobD you can assume that the recipient is an insulated cylinder with a piston in the middle. – João Marcos Cardoso da Silva Jun 18 '22 at 18:09
  • @Chemomechanics thanks a lot! After reading your links I believe that this setting is an example of a Brownian ratchet. The way I understand it, that'd mean that if the temperatures are the same, the piston will randomly move back and forth instead of go in a single direction. However, I don't understand why that makes it impossible to extract energy from this movement. For instance, if the piston is electrically charged, couldn't we extract energy from the electromagnetic radiation emitted when it accelerates, regardless of direction? – João Marcos Cardoso da Silva Jun 18 '22 at 18:26
  • Ultimately, any detector at the same temperature will be emitting its own electromagnetic radiation, resulting in no net collection. You could use a cooler detector/rectifier, however—this is just a type of heat engine. See here and here. – Chemomechanics Jun 18 '22 at 19:56
  • @Chemomechanics thanks a lot once again! I'll try to read the papers you've linked – João Marcos Cardoso da Silva Jun 18 '22 at 20:03
  • This is one of the many versions of Maxwell's Demon. I've linked what seems to me the most upvoted answer on the subject, but searching this site for Maxwell's demon will fond many others. – John Rennie Jun 19 '22 at 05:58

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