sitive entropy. Therefore, the erasure of information must increase the entropy of the surrounding environment. Such a revolutionary concept was initially suggested by J. von Neumann4 and then formalized in 1961 by R. Landauer. The key conclusion takes the form of Landauer's principle (or the Landauer bound or limit, or the von Neumann-Landauer bound): The energy dissipated into the environment by the erasure of a single bit of information is at least equal to k T log 2. The environment entropy increases by at least KB log 2. Landauer's principle, thus, provides a lower bound on the energy dissipation and entropy increase associated with information erasure. It is not coincidental that this minimum is proportional to log 2 (natural logarithm). Without pretending to provide a formal demonstration of this result, consider the following. The memory to be erased can be viewed as a binary source X = {1, 0} having a uniform distribution, i.e., p(1) = p(0) = 1/2 (the probability that the memory bit is either 1 or 0). As we have seen in Chapter 3, the self-information associated with any of the two events x E X is I(x) = - log.[p(x)] = - log.(1/2) = loge 2, as expressed in nats. The source's entropy H(X) corresponds to the average information (I(x)) = p(1)log.[p(1)]-p(0)logep[(0)] = 0.5log.2+0.5log.2=log.2, 3 For a given memory address, the initial state is the bit value 0 or 1. Once cleared, the bit value is reset to 0 (for instance). From this final state, it is not possible to decide whether the initial state was the bit value 0 or 1, and SO on for all previously recorded states, which illustrates the irreversibility of the erasure process. 4 See http://en.wikipedia.org/wiki/Landauer%27s_principle 15.1 Maxwell's demon and Landauer's principle with H(X) = KB(I(X)) = KB log, 2 in the units of physics. According to Landauer's prin- ciple, clearing the memory, thus, corresponds to the conversion into energy of the mem- ory's average information, which corresponds to the heat Q = kB (I) = kB log 2. The cleared memory can b