I was thinking about the director of this previous post and how she
could use a mechanical device to generate the fire alarm.
A long and narrow cylinder has a slowly moving particle A on one side and a particle B at rest on the other (*).
She sets everything up on Sunday and simply waits until she hears the 'click' of the two particles colliding, which then triggers the fire alarm.
Here is my problem: It would seem that the entropy of this closed system decreases until we hear the 'click'; The entropy due to the unknown location of particle B is
proportional to lnV, but the volume V decreases with time as A moves
from left to right.
Notice that she can make the cylinder (and thus the initial V) as large as she wants and she could use more than one particle at B so that the initial entropy kNlnV can be arbitrarily large(**); And if the velocity of particle A is very small this decrease can take a long time ...
added 1 day later: I think I (finally!) figured out where the problem is with this puzzle. See the comments if you want to know my solution (instead of finding out for yourself.)
(*) While the initial position and momentum of A are well known (the
particles are heavy enough that we don't have to worry about quantum effects), the position
of particle B is unknown (but we do know that it is at rest).
(**) Of course the effort to set up the device will increase entropy by an even larger quantity, but all this occurs already on Sunday.
added later: I am no longer sure about that. She might have simply picked a cylinder
of unknown length, but shorter than 5m. The (right) end of that cylinder and the particle B would be identical. Now she sets up particle A on the left side to move with a (constant) speed of 1m/day and when A hits the other end (=particle B) it triggers the alarm (at which point she then knows the length of the cylinder).
I don't see how the act of picking up a cylinder of unknown length increased the entropy on Sunday.