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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?

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    $\begingroup$ How do you encode the distance and the object? If the rules are strictly local, i.e., around one object, how would you know which way an object should be attracted? $\endgroup$
    – John K.
    Commented May 10, 2013 at 13:09
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    $\begingroup$ Indeed, this is precisely what makes the problem non-trivial. I would naively expect that if solution exists, then it would be of the following form: a 2d lattice which can be populated by "particles", superposed with an 'ether' which would send out 'signals' in all directions when a particle is present and do nothing otherwise. When a signal reaches another "particle" then it tells the particle to move in the direction of the sent out signal. Somehow the signals should also have some memory otherwise there would be an excess accumulation of them for distant particles... $\endgroup$
    – MJK
    Commented May 10, 2013 at 15:00
  • $\begingroup$ But whether this actually acts as a long-range force, moreover one depending on the distance, is not clear to me. I was wondering whether this question has already been considered? $\endgroup$
    – MJK
    Commented May 10, 2013 at 15:01
  • $\begingroup$ imho extremely deep/significant open research question crosscutting key disciplines such as TCS,QM,(particle) physics,emergent behavior, etc. suggest migrate/promote this to cstheory.se $\endgroup$
    – vzn
    Commented Jun 14, 2013 at 17:30
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    $\begingroup$ re MJKs idea about attraction via "signals". another basic physical model for particle attraction is overall density of a field. so imagine you have a large pool with a density gradient, and constant density particles in this pool. the particles will move/drift from regions of higher density to lower density. ie "float" in a way. this may be a unified theory of both attraction and gravitation that even the standard model hasnt really unified yet and is largely a key open question in physics. $\endgroup$
    – vzn
    Commented Jun 14, 2013 at 19:03

2 Answers 2

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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.

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  • $\begingroup$ +1 For a really beautiful answer. It demonstrates that what the OP is asking is definitely possible, without giving any hint of an indication as to how such a thing might be accomplished. Well, the hint's there, but following it to completion would be about the most tedious thing I could possibly imagine. $\endgroup$
    – Patrick87
    Commented Jun 14, 2013 at 15:16
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    $\begingroup$ I suppose Paul Gordan would say, "Das ist nicht Mathematik. Das ist Theologie!", but even theology has its merits! $\endgroup$
    – rphv
    Commented Jun 14, 2013 at 17:23
  • $\begingroup$ +1 clever idea but think this should be sketched out a little more. guess it assumes that "attraction", particles, etc are based on algorithms that generate or simulate them. $\endgroup$
    – vzn
    Commented Jun 14, 2013 at 17:33
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    $\begingroup$ I suppose it hinges on the definition of what it means "to simulate." I suspect the OP was looking for a CA rule set such that "live" cells (or some configuration thereof) are "attracted" to each other with a 1/r force. I suspect this is possible, but tedious to construct and largely irrelevant. I stand by my original answer since an analogous observation could be applied to any computer simulation - after all, strings of 1's and 0's in a processor pipeline don't "look" much like n-bodies interacting in a gravitational field, yet we accept that as a "simulation." $\endgroup$
    – rphv
    Commented Jun 14, 2013 at 17:57
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    $\begingroup$ @rphv: Turing completeness doesn't mean that you can find a CA so that information travels faster than the speed of light, even though Turing machines can easily simulate CA where information travels faster than the speed of light. Here, if the OP wants a CA such that objects are attracted with $1/r$ force which runs at some constant slowdown, Turing completeness does not give you that. If you want a machine which, every so often, displays a picture of what the dynamics would be under such a force, Turing completeness will give you that. $\endgroup$
    – Peter Shor
    Commented Jun 15, 2013 at 11:26
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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:

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