Time in Physics by Renato Renner & Sandra Stupar

Author:Renato Renner & Sandra Stupar
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

or, equivalently,

If we introduce

independence of a and b is I K (a, b) ≈ 0.

In the same spirit, we can define conditional independence: We say that a and b are independent given c if

or, equivalently,

or

3 Complexity as Randomness 1: Work Extraction

3.1 The Converse of Landauer’s Principle

In our search for an intrinsic notion of randomness—independent of probabilities or the existence of alternatives—expressed through the properties of the object in question, we must realize, first of all, that such a notion is impossible for single bits, since neither of the two possible values, 0 nor 1, is in any way more an argument for the “randomness” of that bit than not. The situation, however, changes for long strings of bits: No one would call the one-million-bit string 000⋯0 random (even though, of course, it is not impossible that this string originates from a random process such as a million consecutive tosses of a fair coin). In the spirit of Rolf Landauer’s [30] famous slogan “information is physical,” we may want to test our intuition physically: If the N bits in a string encode the position, being in the left (0) as opposed the right (1) half of some container, of the molecules of a gas, then the 0-string means that the gas is all concentrated in one half and, hence, allows for extracting work from the environmental heat; the amount is NkTln2 if k is Boltzmann’s constant and T is the temperature of the environment. This fact has also been called the converse of Landauer’s principle. Note that any other system which can be transformed by a reversible process into that maximally asymmetric gas has the same work value; an example is a physical representation of the first N bits of the binary expansion of π of the same length—although this string may look much more “random” at first sight. This reversible process is, according to the Church-Turing thesis,6 imagined to be carried out by a Turing machine in such a way that every step is logically reversible (such as, e.g., a Toffoli gate) and can be uncomputed by the same device; the process is then also possible in principle in a thermodynamically reversible way: No heat is dissipated [23]. It is clear that most N-bit strings cannot have any work value provided there is no perpetuum mobile of the second kind.

For a given string S, its length minus the work value of a physical representation (divided by kT) may be regarded as an intrinsic measure for the randomness of S. We address the question what in general the fuel value is of (a physical representation of) S. Since (the reversible extraction of) the string 0 N from S is equivalent to (the gain of) free energy of NkTln2, we have a first answer: Work extraction is data compression.

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