The cost of compressible dynamics, time-reversibility, Maxwell's demon, and the second law

A tantalizing version of Maxwell's demon is presented which appears to operate reversibly. A container of hard core disks is separated into two chambers of equal volume by a membrane that selects which disk can penetrate depending on the disk's angle of incidence. It is shown that the second law of thermodynamics requires the incompressibility of microscopic dynamics or an appropriate energy cost for compressible microscopic dynamics.