## Turing machine and Landauer limit

My inactivity period was due to a lack of real news around the World. But I was not inactive at all. My friend Alfonso Farina presented to me another question that occupied my mind for the last weeks: What is the energy cost for computation? The first name that comes to mind in such a case is Rolf Landauer that, on 1961, wrote a fundamental paper on this question. The main conclusion drawn by Landauer was that at each operation on a bit there is an entropy cost of $K\ln 2$ being $K$ the Boltzmann constant. This means that, it you are operating at a temperature $T$ there will be heat emission for $KT\ln 2$ and this is the Landauer limit. This idea stems from the fact that information is not some abstract entity living in hyperuranium but just to stay in the real world it needs a physical support. And wherever there is a physical support thermodynamics and its second principle is there at work. Otherwise, we can use information to evade the second principle and build our preferred perpetual motion. As Charles Bennett proved, Maxwell demon cannot work due to Landauer limit (for a review see here).

Recently, a group of researchers was able to show, by a smart experimental setup, that Landauer’s principle is indeed true (see here). This makes mandatory to show theoretically that Landauer’s principle is indeed a theorem and not just a conjecture.

To accomplish this task, we would need a conceptual tool that can map computation theory to physics. This tool exists since a long time and was devised by Alan Turing: The Turing machine. A Turing machine is a thought computational device aimed to show that there exist mathematical functions that cannot have a finite time computation, a question asked by Hilbert on 1928 (see here). A Turing machine can compute whatever a real machine can (this is the content of the Church-Turing thesis). There exist some different kinds of Turing machines but all are able to perform the same computations. The main difference relies on the complexity of the computation itself rather than its realization. This conceptual tool is now an everyday tool in computation theory to perform demonstrations of fundamental results. So, if we are able to remap a Turing machine on a physical system and determine its entropy we can move the Landauer’s principle from a conjecture to a theorem status.

In my paper that appeared today on arXiv (see here) I was able to show that such a map exists. But how can we visualize it? So, consider a Turing machine with two symbols and a probabilistic rule to move it. The probabilistic rule is just coded on another tape that can be consulted to take the next move. This represents a two-symbol probabilistic Turing machine. In physics we have such a system and is very well-known: The Ising model. As stated above, a probabilistic Turing machine can perform any kind of computations a deterministic Turing machine can. What is changing is the complexity of the computation itself (see here). Indeed, a sequence of symbols of the tape in the Turing machine is exactly a configuration of a one-dimensional Ising model. This model has no critical temperature and any configuration is a plausible outcome of a computation of a Turing machine or its input. What we need is a proper time evolution that sets in the equilibrium state, representing the end of computation.

Time evolution of the one-dimensional Ising model has been formulated by Roy Glauber on 1963. Glauber model has a master equation that converge to an equilibrium as time evolves with a Boltzmann distribution. The entropy of the model at the end of its evolution is well-known and has the limit value for the entropy $K\ln 2$ as it should when a single particle is considered but this is just a lower limit. So, we can conclude that the operations of our Turing machine will involve a quantity of emitted heat in agreement with Landauer’s principle and this is now a theorem. What is interesting to note is that the emitted heat at room temperature for a petabit of data is just about a millionth of Joule, a very small amount. This makes managing information convenient yet and cybercrime still easy to perform.

Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process IBM Journal of Research and Development, 5 (3), 183-191 DOI: 10.1147/rd.53.0183

Bennett, C. (2003). Notes on Landauer’s principle, reversible computation, and Maxwell’s Demon Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 34 (3), 501-510 DOI: 10.1016/S1355-2198(03)00039-X

Bérut, A., Arakelyan, A., Petrosyan, A., Ciliberto, S., Dillenschneider, R., & Lutz, E. (2012). Experimental verification of Landauer’s principle linking information and thermodynamics Nature, 483 (7388), 187-189 DOI: 10.1038/nature10872

Marco Frasca (2012). Probabilistic Turing Machine and Landauer Limit arXiv arXiv: 1206.0207v1