{"id":7114,"date":"2013-05-14T15:55:27","date_gmt":"2013-05-14T22:55:27","guid":{"rendered":"http:\/\/dabacon.org\/pontiff\/?p=7114"},"modified":"2013-05-14T15:55:27","modified_gmt":"2013-05-14T22:55:27","slug":"resolution-of-tooms-rule-paradox","status":"publish","type":"post","link":"https:\/\/dabacon.org\/pontiff\/2013\/05\/14\/resolution-of-tooms-rule-paradox\/","title":{"rendered":"Resolution of Toom's rule paradox"},"content":{"rendered":"

A few days ago our Ghost Pontiff Dave Bacon wondered<\/a> how Toom’s noisy but highly fault-tolerant 2-state classical cellular automaton\u00a0 can get away with violating the Gibbs phase rule, according to which a finite-dimensional locally interacting system, at generic points in its phase diagram, can have only only one thermodynamically stable phase.\u00a0 The Gibbs rule is well illustrated by the low-temperature ferromagnetic phases of the classical Ising model in two or more dimensions:\u00a0 both phases are stable at zero magnetic field, but an arbitrarily small field breaks the degeneracy between their free energies, making one phase metastable with respect to nucleation and growth of islands of the other.\u00a0 In the Toom model, by contrast, the two analogous phases are absolutely stable over a finite area of the phase diagram, despite biased noise that would seem to favor one phase over the other.\u00a0 Of course Toom’s rule is not microscopically reversible,\u00a0 so it is not bound by laws of equilibrium thermodynamics.
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\nNevertheless, as Dave points out, the distribution of histories<\/em> of any locally interacting d<\/em>-dimensional system, whether microscopically reversible or not, can be viewed\u00a0 as an equilibrium Gibbs distribution of a d<\/em>+1 dimensional system, whose local Hamiltonian is chosen so that the d<\/em> dimensional system’s transition probabilities are given by Boltzmann exponentials of interaction energies between consecutive time slices.\u00a0 So it might seem, looking at it from the d+1 dimensional viewpoint, that the Toom model ought to obey the Gibbs phase rule too.
\nThe resolution of this paradox, described in my 1985 paper with Geoff Grinstein<\/a>,\u00a0 lies in the fact that the d<\/em> to d<\/em>+1 dimensional mapping is not surjective.\u00a0 Rather it is subject to the normalization constraint that for every configuration X<\/em>(t<\/em>) at time t, <\/em>the sum over configurations X<\/em>(t<\/em>+1) at time t<\/em>+1 of transition probabilities P(X<\/em>(t<\/em>+1)|X<\/em>(t<\/em>)) is exactly 1.\u00a0\u00a0\u00a0 This in turn forces the d<\/em>+1 dimensional free energy to be identically zero, regardless of how the d<\/em> dimensional system’s transition probabilities are varied.\u00a0 The Toom model is able to evade the Gibbs phase rule because<\/p>\n