Information erasure

Landauer's principle states that in erasing one bit of information, on average, at least k(B)T ln(2) energy is dissipated into the environment (where k(B) is Boltzmann's constant and T is the temperature of the environment at which one erases), Here, Landauer's principle is microscopically derived without direct reference to the second law of thermodynamics. This is done for a classical system with continuous space and time, with discrete space and time, and for a quantum system. The assumption made in all three cases is that during erasure the bit is in contact with a thermal reservoir.