Hamiltonian memory: An erasable classical bit

Erasing a bit of information requires probability concentration in phase space, which by Liouville's theorem is impossible in pure Hamiltonian dynamics. It therefore requires dissipative dynamics, leading to the Landauer limit: kBTlog2 of heat dissipation per erasure of one bit. We show that when a conserved quantity confines the dynamic to a single shell with zero thickness, it is possible to concentrate the probability on this shell using Hamiltonian dynamic, and therefore to implement an erasable bit with no thermodynamic cost. This implies that there is no thermodynamic cost associated with bit erasure in the microcanonical ensemble, where the energy of the system is precisely known. However, any uncertainty in the energy results back in the Landauer bound.