Landauer's Principle

In 1961, Rolf Landauer published a landmark paper titled "Irreversibility and Heat Generation in the Computing Process" in which he observed that irreversible computational processes emit heat. Landauer's principle is significant because it provides a direct link between physics and information theory, and, as of this writing, the full importance of this connection is not fully understood.¹

Thirteen years prior to Landauer's discovery, Claude Shannon launched the field of information theory with his seminal paper A Mathematical Theory of Communication. In this paper, Shannon defined the notion of information entropy, which quantifies the information content of any particular message.² Interestingly, information entropy H was so named NOT because of a direct link to the thermodynamic quantity S, but because the formulas for S and H look exactly the same:

Of course, p_i represents a fundamentally different³ quantity in each equation, and the constant k takes on different values and units, but the similarity is striking nonetheless.⁴

What, then, IS an "irreversible computational process?" Consider the humble Boolean AND gate, which is a fundamental component of computers. This circuit accepts two bits as input and yields one bit of output:

An AND gate computes the logical conjunction of its two input bits: it produces a '1' if and only if BOTH inputs are '1'. This means that there are three other possible input pairs that generate the output '0': (0,1), (1,0), and (0,0). Thus, the list of inputs and outputs⁵ for an AND gate looks like this (the '∧' symbol means AND):

(1∧1) → 1
(1∧0) → 0
(0∧1) → 0
(0∧0) → 0

Now, imagine that an observer is given access to the output bit and ONE of the input bits. Here, the important thing to notice is that ½ of the time it's impossible to determine the OTHER input bit just by looking at one of the input bits and the output bit. To illustrate: if the known input bit is '0' and the output bit is also '0', then the unknown input bit could be EITHER a '1' or a '0'.

In a sense, this implies that one of the two input bits⁶ is "lost" as it passes through the AND gate. What Landauer observed is that this "lost bit" is not really "lost" at all - it turns into heat. Specifically, that bit becomes (kT ln 2) joules worth of heat, and in this case the k is the SAME k that's in the equation for S - namely, Boltzmann's constant.

Most people intuitively understand that information is powerful. Is it possible that this is also literally - physically - true?!

[1] By me.
[2] Such messages are usually expressed as a string of bits - 1's and 0's.
[3] We think.
[4] See this Wikipedia page for more details.
[5] Called a "truth table," this list completely defines the behavior of an AND gate.
[6] ...or perhaps more accurately, ½ of each of the input bits...