One example in which the problem of a demon-gas system can be easily examined analytically is a ``tube demon,'' which operates at one end of a closed tube. See Figure 5.
The demon observes a small section at the of length of a tube of length D. If there are no molecules in this region, the demon inserts a partition closing off the section. There is now a pressure difference between the length , which is empty, and the rest of the tube. The gas has been compressed into a smaller volume, because the section closed off by the demon is no longer accessible. The demon can now get work out of the gas by expanding it into the full volume of the tube.
The only information gathered by the demon here is whether or not the section is empty of particles. If the times at which the demon looks are separated by sufficient time for the gas to have reached equilibrium, successive observations will have uncorrelated values. For simplicity, the length can be chosen so that the two observational outcomes are equally likely. Doing this enables us to approximate the algorithmic entropy by the number of bits gathered. The entropy changes in the gas can be easily calculated, so the ability of the demon to reduce the entropy of the demon-gas system can examined.
If the tube has a length of D, then there is a probability of that any given molecule is not a section of length . If there are N molecules, there is therefore a chance that no molecules are in the section. If the bits in the demon's memory are to be uncorrelated, then it must be equally likely that the section be empty or otherwise. The length should therefore be given by the equation
The difference in the entropy of the gas due to the insertion of the wall can be determined using standard thermodynamics.
The demon therefore reduces the entropy of the gas by one bit each time it places a wall in the chamber. Unfortunately for the demon, this happens only half the times it looks. The demon must use on average two bits of its own memory for every bit of the gas's entropy it decreases. The entropy increase of the demon in therefore
which is twice as large as the drop in the entropy of the gas, so the second law continues to hold. The demon's entropy rises at such a large rate in comparison to the drop in the entropy of the gas because the demon does not make full use of the observations it makes. When the demon observes a particle within the length , it does nothing to take advantage of this information. It is more difficult to design a demon which would take advantage of this, because the number and energy of the molecules within the end are unknown by the demon.