A few days ago our Ghost Pontiff Dave Bacon wondered how Toom’s noisy but highly fault-tolerant 2-state classical cellular automaton 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. The Gibbs rule is well illustrated by the low-temperature ferromagnetic phases of the classical Ising model in two or more dimensions: 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. 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. Of course Toom’s rule is not microscopically reversible, so it is not bound by laws of equilibrium thermodynamics.

Nevertheless, as Dave points out, the distribution of *histories* of any locally interacting *d*-dimensional system, whether microscopically reversible or not, can be viewed as an equilibrium Gibbs distribution of a *d*+1 dimensional system, whose local Hamiltonian is chosen so that the *d* dimensional system’s transition probabilities are given by Boltzmann exponentials of interaction energies between consecutive time slices. 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.

The resolution of this paradox, described in my 1985 paper with Geoff Grinstein, lies in the fact that the *d* to *d*+1 dimensional mapping is not surjective. Rather it is subject to the normalization constraint that for every configuration *X*(*t*) at time* t, *the sum over configurations *X*(*t*+1) at time *t*+1 of transition probabilities P(*X*(*t*+1)|*X*(*t*)) is exactly 1. This in turn forces the *d*+1 dimensional free energy to be identically zero, regardless of how the *d* dimensional system’s transition probabilities are varied. The Toom model is able to evade the Gibbs phase rule because

- being irreversible, its
*d*dimensional free energy is ill-defined, and - the normalization constraint allows two phases to have exactly equal
*d*+1 dimensional free energy despite noise locally favoring one phase or the other.

Just outside the Toom model’s bistable region is a region of metastability (roughly within the dashed lines in the above phase diagram) which can be given an interesting interpretation in terms of the *d*+1 dimensional free energy. According to this interpretation, a metastable phase’s free energy is no longer zero, but rather -ln(1-Γ)≈Γ, where Γ is the nucleation rate for transitions leading out of the metastable phase. This reflects the fact that the transition probabilities no longer sum to one, if one excludes transitions causing breakdown of the metastable phase. Such transitions, whether the underlying *d*-dimensional model is reversible (e.g. Ising) or not (e.g. Toom), involve critical fluctuations forming an island of the favored phase just big enough to avoid being collapsed by surface tension. Such critical fluctuations occur at a rate

Γ≈ exp(-const/*s*^(*d*-1))

where s>0 is the distance in parameter space from the bistable region (or in the Ising example, the bistable line). This expression, from classical homogeneous nucleation theory, makes the *d*+1 dimensional free energy a smooth but non-analytic function of *s*, identically zero wherever a phase is stable, but lifting off very smoothly from zero as one enters the region of metastability.

Below, we compare the *d* and *d*+1 dimensional free energies of the Ising model with the d+1 dimensional free energy of the Toom model on sections through the bistable line or region of the phase diagram.

We have been speaking so far only of classical models. In the world of quantum phase transitions another kind of *d* to *d*+1 dimensional mapping is much more familiar, the quantum Monte Carlo method, nicely described in these lecture notes, whereby a* d* dimensional zero-temperature quantum system is mapped to a *d+*1 dimensional finite-temperature classical Monte Carlo problem. Here the extra dimension, representing imaginary time, is used to perform a path integral, and unlike the classical-to-classical mapping considered above, the mapping is bijective, so that features of the* d*+1 dimensional classical system can be directly identified with corresponding ones of the *d* dimensional quantum one.

“the d to d+1 dimensional mapping is not surjective”

I think you meant injective (1-1) rather than surjective (onto)

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I think I meant surjective. For every d dimensional probabilistic rule there is a d+1 dimensional local Hamiltonian model corresponding to its time evolution. But because of the normalization constraint, most d+1 dimensional local Hamiltonian models do not corresponds to time evolutions of any d dimensional rule. These d+1 dimensional models, therefore, lie outside the range of the d to d+1 mapping.

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OK I’m still missing it (in my defense, while I’d read your 1985 paper a while ago, I now had to go begging on twitter for a copy of the paper. Ah the joys of paywalls! Also I spend my time writing code, so my physics brain cells are atrophied.)

Maybe if I try it this way you can help me figure out where I’m confused:

1) There exists a local Hamiltonian in d+1 such that in equilibrium the distribution is equal to the distribution produced by a history of the Toom’s rule in d dimensions.

2) We know that Toom’s rule has this nice property of being robust to noise plus a bias.

3) If I define an order parameter = sum of the spins at one time slice, shouldn’t this order parameter behave according to what we know about Toom’s rule?

Is the problem that my order parameter is not extensive in d+1 dimensions?

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All of what you say is true. The problem is not with the order parameter, which behaves as you say, but with the d and d+1 dimensional free energies, which behave as indicated in my last diagram. For reversible models like the Ising model, the d dimensional free energy is sensitive to the magnetic field, giving rise to the first order phase transition at zero field where the two phases’ free energy curves cross. For irreversible models like the Toom model, the free energy is undefined in d dimensions. In d+1 dimensions the free energy is defined for both kinds of models, but it does not behave like a normal d dimensional free energy. In particular it can be identically zero over large regions of the phase diagram. Each phase’s d+1 dimensional free energy is identically zero wherever that phase is stable, and lifts off very gently and nonanalytically as one passes into the metastable zone. In the Toom model, the two phases both have identically zero free energies over the whole green wedge-shaped region of the phase diagram. This violation of the Gibbs phase rule can happen because the d dimensional free energy, which would have to obey the Gibbs rule, is undefined for irreversible processes such as the Toom rule.

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Thanks Charlie, I think I’m beginning to understand. Fascinating.

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