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I would like to understand where the waste heat is generated in the Maxwell's demon problem. To this end I've come up with the simplest scenario I can think of. If my scenario is workable I am hoping someone will be kind enough to apply the right math to it to show what is going on and why it does not work to provide a "free lunch."

My scenario is as follows:

Take a 1 dimensional container of length l. Start with an evenly distributed population of particles; slow red ones and fast blue ones, traveling at +v or -v, and +2v or -2v respectively.

The demon sits at the boundary at l/2.

He lets slow red particles at +v and fast blue particles at -2v through and keeps the opening between the two halves of the container closed otherwise.

The idea of using a one dimensional container is to keep the math as simple as possible. In order for this to work the particles have to be able to go right through one another without interacting.

I am imagining that if one can work out what happens in this simplest case then it should be possible to extend the scenario to a 2-dimensional container in which the line forming the barrier between the two halves of the container is made up of discrete copies of the Maxwell's Demon in my 1-dimensional scenario.

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The dimensionality is not relevant. The commonly-believed "solution" of the Maxwell's demon puzzle utilizes the fact that the demon must store information as measurements are made. To bring the demon back to its initial state, that memory must be erased. According to Landauer's theorem, erasure of each bit of information generates at least $k \rm{ln}(2)$ J/K of entropy in the environment via a heat process. Erasure is dissipative. You can read about this in various places, including Leff & Rex, "Maxwell's Demon 2: Entropy, Classical and Quantum Information, Computing (Institute of Physics, Bristol, 2003) and von Baeyer, "Maxwell's Demon: Why Warmth Disperses and Time Passes"(Random House, 1998). C.H. Bennett also had a nice article in Scientific American magazine: C. H. Bennett, "Demons, engines and the second law," Sci. Am. Vol. 257, 108-16 (1987).

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