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For a long time it was taken for granted that storing, manipulating
and transmitting information would always necessarily dissipate some
nonzero quantity of energy. Engineering prowess might lower the
energy cost somewhat, but there was a threshold level, a lower limit
we could never cross. A device that could compute without loss of
energy was seen as the information equivalent of a perpetual-motion machine.
John von Neumann, in a 1949 lecture, set the minimum price of
"an elementary act of information" at kT ln 2. In
this formula k is Boltzmann's constant, which is the
conversion factor for expressing temperature in energy units; its
numerical value is 1.4 x 10-23 joules per kelvin.
T is the absolute temperature, and ln 2 is the natural
logarithm of 2, a number that appears here because it corresponds to
one bit of information—the amount of information needed to
distinguish between two equally likely alternatives. At room
temperature (300 kelvins), kT ln 2 works out to about 3 x
10-21 joule, or 3 zeptojoules. This is a minuscule amount
of energy; Ralph C. Merkle of the Georgia Institute of Technology
estimates that it is the average kinetic energy of a single air
molecule at room temperature.
Von Neumann's pronouncement was based on a thermodynamic argument.
Consider a computation that answers a single
yes/no question, where the two possible outcomes
appear equally likely at the outset. Once the question has been
settled, we know one bit more than we did beforehand, and so the
computational process reduces the uncertainty or entropy of the
computing system by one bit. But the second law of thermodynamics
says that total entropy can never decrease, and so the reduction
inside the computer must be compensated by an entropy increase
elsewhere. Specifically, the computer must stir up at least one
bit's worth of disorder in its surroundings by expelling an amount
of heat equal to kT ln 2. Von Neumann—along with
everyone else at the time—assumed that every
"elementary act of information" has the effect of settling
at least one yes/no question, and thus it seemed
that each step in the computer's operation inevitably dissipates at
least three zeptojoules of energy.
Von Neumann's ideas on the thermodynamics of computation were widely
accepted but never formally proved. In the early 1960s Rolf Landauer
set out to supply such a proof and found that he couldn't. He
discovered that only a certain subclass of computational events have
an unavoidable three-zeptojoule cost. Ironically, these expensive
operations are not those that produce information but rather those
that destroy it, such as erasing a bit from a memory cell.
Landauer's work on the cost of forgetting was counter-intuitive, and
initially it got a frosty reception. Now the idea has been
thoroughly assimilated, and it's hard to see what the controversy
was all about. Erasing a memory cell amounts to ignoring its present
contents—which may in fact be unknown—and resetting the
cell to some standardized state (usually 0). Thus an indeterminate
bit becomes a fully specified one, and the entropy of the machine is
diminished accordingly. For this reason the corresponding amount of
heat energy (kT ln 2) has to be rejected into the
environment. The consequences are even clearer if you think about
erasing the entire memory of a computer, so that the system goes
from a random state to a highly ordered one; this is a process of
refrigeration, and so it obviously gives off heat.