Physics Asked by Defcon97 on February 16, 2021

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?

If you're looking for a deterministic cellular automaton that exhibits similar dynamics to the Ising model then you might enjoy Norman Margolus' paper Crystalline Computation. It's an overview of his work on reversible cellular automata, and in particular, it describes one that mimics the statistics of the Ising model.

Reversible cellular automata are a fascinating topic. Roughly, the idea is that since information can neither be created or destroyed, they tend to exhibit an analog of the second law of thermodynamics, becoming increasingly disordered over time. (When simulated on a finite grid they must eventually return to their ordered initial state, but it may take far longer than the current age of the universe for them to do so.)

They also often have interesting conservation laws. The one above conserves a quantity analogous to the energy in the Ising model. The total energy depends on the initial conditions, which causes it to behave as if it has a temperature despite being deterministic.

One can construct other reversible cellular automata, known as lattice gases, that conserve analogs of kinetic energy and momentum. These can be used to simulate turbulent fluid flow.

Reversible cellular automata are generally constructed in a slightly different way from Conway's game of life, however, because otherwise it's difficult to make sure that the rules will indeed be reversible. The cellular automaton shown above operates on a checkerboard scheme, updating the 'odd' cells on odd time steps and the 'even' cells on even time steps. The rule set is very simple though, arguably simpler than the Game of Life. You can see the paper linked above for more information.

Answered by Nathaniel on February 16, 2021

The usual 2D Ising model Metropolis Monte Carlo (MMC) simulation and Conway's Game of Life (CGL) can be considered 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 cyclically).

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. It could sometimes be useful to study a class of models by systematically playing with some parameters to explore the different capabilities of a class of models. In any case, the important issue 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 is usually not the most efficient way of making science progress.

**NOTE added after the EDIT of the question.**

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

There have been different proposals for such a class of automata. From the Statistical Mechanics point of view, controlling temperature or energy is not an issue. The real problem is finding a dynamic system on the lattice that can show the desired average behaviors.

One interesting cellular automaton 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 as a kinetic energy term. Moreover, the lattice has to be updated globally using each time half of the sites, 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 others could be found just looking for papers citing Creutz.

Answered by GiorgioP on February 16, 2021

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