# Is there a way to perform Ising model simulations with a Game of Life approach?

This question may turn out to be trivial or nonsensical as I do not have more than undergraduate understanding of both the Ising model and computational physics simulations.

I just wanted to post this to see if this "shower thought" that I just had has already been explored and studied.

The question is pretty straightforward.

The lattice setup of Conway's Game of life is identical to the one of the Ising model, the difference being that the former evolves through Conway's "rule" while the latter evolves through classical statistical mechanics.

The ising model though is subject to a thermodinamical constraint concerning temperature which can cause fluctuations in the lattice that could not be foreseeen with the use of the Conway's rule.

I was thinking that maybe a location and temperrature-dependent rule for the lattice might be used to recreate the behaviour of an Ising lattice.

Has this approach ever been explored? If yes could you link some reference because i'd be extremely interested to read some of those. If not is there any specifical reason why not?

EDIT: I understand now both from the comments and from the answers that my question wasn't probably well-posed, so I'll make an attempt to be clearer. Is there a way to model Conway's rule in a location and temperature dependent fashion such that the lattice exhibits the same macroscopical behaviour as in an Ising Model (e.g. low temperature ferromagnetism)? If the answer to the latter question is "yes" then can this teach us anything about the physical model?

• i doubt anyone would try exploring the Ising model with GoL model. For one, the Ising model has a physical background to the mathematical model... – Kyle Kanos Dec 28 '19 at 11:58
• I am not sure I understand your question, as Conway's game of life is deterministic. If you meant to ask whether there are probabilistic cellular automata that converge to the Gibbs measure associated to the Ising model, then the answer is yes (to some extent); see, for instance, this paper. – Yvan Velenik Dec 28 '19 at 12:02
• As far as I can understand after you last EDIT, you are looking for a deterministic automata which is able to model the behavior of Ising model. Am I right? If so, I wouldn't call it a modification of Conway's rule, although this is matter of tastes. In any case, I'll add something on this theme to my answer. – GiorgioP Dec 29 '19 at 9:00

Both, the usual 2D Ising model Metropolis Monte Carlo (MMC) simulation and Conway's Game of Life (CGL) can be considered as cellular automata, although of a different kind.

MMC is stochastic and ergodic (at any finite temperature, every configuration has a finite probability of been reached whatever is the starting point). CGL is deterministic and non-ergodic: many initial configurations end up into some final attractors like fixed points or limit cycles (configurations periodically repeating in a cyclic way).

Of course, nothing prevents to change the rules of both. The real question is what the change is for? Both automata were designed as models for different phenomena. Changing the rules usually means changing the model. Some times it could be useful to study a class of models by systematically playing with some parameter to explore different capabilities of a class of models. The important issue is that in any case on starts with a clear question and tries to find an answer. The other way around, playing with answers and then looking for the correct questions it is usually not the most efficient way of making progress in science.

NOTE added after the EDIT of the question.

I felt the question too bound to a small modification of Conways' rules. If instead the real question is if it is possible to have a deterministic automata, even far from CGL rules, able to model Ising model, the answer is affirmative.

There have been different proposals for such a class of automata. From the point of view of the Statistical Mechanics, controlling temperature or energy is not an issue while the ral problem is to find a dynamical system on the lattice which is able to show th dsired average behaviors.

One interesting cellular automata was proposed in the eighties by Creutz ( Creutz, M. (1986). Deterministic ising dynamics. Annals of physics, 167(1), 62-72.) (the link allows access to the paper).

Creutz's model is based on an enlarged state variable at each site. The usual two values spin variable is augmented by a four value additional variable which acts like a kinetic energy term. Moreover, the lattice has to be updated globally using each time half of the sites, in order to circumvent an issue noticed by Vichniac, G. Y. (1984). Simulating physics with cellular automata. Physica D: Nonlinear Phenomena, 10(1-2), 96-116. Creutz's model is energy conserving but modifying it to control the temperature shouldn't be a problem.

Alternative models are mentioned in Creutz's paper and other could be found just looking for papers citing Creutz.

• okay I understand now both from the comments and from your answer that my question wasn't probably well-posed, so i'll make an attempt to be clearer. Is there a way to model Conway's rule in a location and temperature dependent fashion such that the lattice exhibits the same macroscopical behaviour as in an Ising Model (e.g. low temperature ferromagnetism)? If the answer to the latter question is "yes" then can this teach us anything about the physical model? – Defcon97 Dec 28 '19 at 18:01
• @Defcon97 I do not see a reasonable way to force CGL to behave as an Ising model. It is not matter of temperatue or of stochastic vs deterministic automata. There are more fundamntal problems. One of them is that Ising model has a built-in symmetry such that, at zero external field, each spin configuration has the same probability as the one where all spin have been flipped. CGL has not such a symmetry. If you start with a lattice with all empty cells it will remain empty forever. if you start with a lattice with all the sites occupied, at th next step, all cells die. With ... (contine) – GiorgioP Dec 28 '19 at 18:10
• ... probability 1 the lattice will finish as an empty lattice. – GiorgioP Dec 28 '19 at 18:12
• @GiorgioP : Of course, you cannot simulate the Ising model using the rules of CGL, but I understood the question as being about finding suitable sets of rules to mimic this model (in particular, these rules should incorporate the required symmetry). The latter is possible if you consider stochastic rules (these prevent the type of absorbing configurations you mention in your comment). As per your answer, I agree with it, of course. – Yvan Velenik Dec 29 '19 at 8:37
• Yes I was looking exactly for something similar to Creutz's paper. @GiorgioP thank you very much – Defcon97 Dec 29 '19 at 11:02