1/r attractive force by cellular automaton

Question

Does there exist a cellular automaton (in 2D) which simulates a $1/r$ force between particles?

More specifically, I would like to know whether it is possible, with strictly local update rules, to have two objects (defined within the model) attract each other with a $1/r$ force, where $r$ is the distance separating the objects. This would in particular entail an acceleration of the object (particles) as they get closer together.

More generally, can long range attractive forces between objects (blobs) be simulated in a cellular automaton setting with strictly local rules?

Answer 1

If by "simulate" you mean something like "generate a picture of what the dynamics would be under such a force," then the answer to your question is yes: there exist universal cellular automata (including Conway's original Game of Life rule set).

If, however, you're asking about whether our universe can be explained in terms of strictly local update rules, then your question is still open. Konrad Zuse was one of the first to explore this question explicitly in terms of CA; see Wolfram, Schmidhuber, or t'Hooft for more recent work.

Answer 2

this is a very significant research question & there is a more general question here that is studied by some. the deeper question is "to what degree can CA(-like) rules reproduce the laws of physics". the larger question is a very important open question with large amounts of speculation and research on the subject, but unfortunately conventional scientific/physics wisdom considers it a more fringe area of modern physics. my understanding is that your specific question is basically open also.

regarding your question in a more general way, here are links on many closely related themes, having researched this thread/area recently:

• research into the game of Life (which has been proven Turing complete by Conway and others) is highly relevant. "gliders" would seem to exhibit laws of attraction to some degree, but the topic and analysis can be subtle. suppose two glider guns are pointing to each other, are the gliders "attracting" each other?

• 't Hooft, Nobel prize winning physicist has investigated in several papers the general question/theme of whether local discrete laws can reproduce QM dynamics or other low-level laws of physics eg in this paper, Relating the quantum mechanics of discrete systems to standard canonical quantum mechanics

• an example of opinion on 't Hoofts directions (being considered fringe), see ’t Hooft on Cellular Automata and String Theory by Woit, a theoretical physicist/String theorist expert/skeptic

• Fredkin speculated long ago on "Digital physics" and some of this has been expanded on by Wolfram eg in New Kind of Science.

• a key angle: 2d/3d solitons seem to be able to be generated from purely local "rules" ie local differential equations, and therefore it seems solid/likely that CAs exist that replicate those same differential equations, although this appears to be yet to be demonstrated. solitons are known to have many strong resemblances to particle/atomic interations including attraction/repulsion aspects/properties. see eg Solitons & cellular automata

• recent breakthrough analytical/theoretical work by Brady shows that a soliton-like system called sonons has strong analogues to basic physics such as particle, electromagnetic/quantum analogies. The irrotational motion of a compressible inviscid fluid.

• a new site dedicated to the subject of classical fluid particle physics with references to Bradys work, tying it in with physics phenomena, eg summary to classical fluid dynamics theory