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Physics is defined as the study of an object {matter or energy} with its interaction with other objects:

Physics is the study of matter, energy, and the interaction between them.

On the other hand, Digital physics is based on computations and information.

Digital physics is a collection of theoretical perspectives based on the premise that the universe is, at heart, describable by information, and is therefore computable.

If we have a universal computer {a universal turing machine} and a program that can compute the evolution of the universe (Digital Physics) running on it, can this specific program simulate our normal physics laws like the classical mechanics laws or quantum mechanics laws ?

1- Can we simulate a universe with its fundemental laws {i.e physics} and change these laws by a universal computer accordingly ?

2- How a simulation of a universe (if any) with fundemental laws {like physics} would help to further understanding in physics ?

3- Is there any form of universal turing machine to apply these simulation ?

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    $\begingroup$ I don't see how to give a useful answer to this question without more explanation what you are asking. What are your thoughts? When you ask "...given a program that can compute the evolution of the universe, can this specific program simulate our normal physics laws like the classical mechanics laws or quantum mechanics laws?" it seems like the only possible answer is "If quantum mechanics is a complete theory of how the universe evolves, then of course the answer is yes, by definition" -- what more are you looking for? I can't tell what you are asking. $\endgroup$
    – D.W. ♦
    Oct 23, 2015 at 0:00
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    $\begingroup$ it is more or less an open question of physics whether physics is really computable. there are a lot of bordering-on-uncomputable aspects aka "fringe phenomena". almost all physics theory is about the computable aspects, but there seems to be endless (increasingly narrow) anomalies that dont fit into the models. although some anomalies/ question are quite large & play a large role eg dark matter/ energy, black holes, universe expansion etc... asked a related questions on Physics recently but it got deleted :( $\endgroup$
    – vzn
    Oct 23, 2015 at 15:39
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    $\begingroup$ Please note, a Universal Turing Machine is just one that can simulate any other Turing machine, it has no relation to "universe" in the physics sense, and it definitely can NOT solve all problems. $\endgroup$
    – Joey Eremondi
    Oct 23, 2015 at 19:31
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    $\begingroup$ wikipedia/ digital physics has a lot of citations. the area is controversial with Wolfram a leading theorist in the area. 't Hooft is a nobel prize winner writing extensively on it. also suggest drop by Computer Science Chat when you have the rep. $\endgroup$
    – vzn
    Oct 23, 2015 at 20:49
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    $\begingroup$ @Henryakpo when you say "the system itself is a universe," what do you mean? This feels like buzzwords with no actual meaning. There are undecidable problems, which no Turing machine, no matter how powerful, can solve, like the Halting problem. There is no way to encode "infinite-search" problems that guarantee a Turing machine will always halt. $\endgroup$
    – Joey Eremondi
    Oct 24, 2015 at 22:20

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The laws of physics are stated in the form of mathematics. Differential equations are pretty popular, for example classical mechanics can be described with differential equations.

You can program a computer to take some initial state of the universe and apply all the laws of physics to it to get a new state of the universe. This comes with a couple of caveats. First, you need to somehow discretize the time steps you compute. Differential equations operate under continuous time, but a numerical simulations does discrete steps. It's reasonably well understood how to do that without losing too much precision. This brings us to a second problem, precision. Physics is generally formulated using real (or complex) variables, but computers only deal with finite precision. Usually this is ok, because in reality we can also only distinguish things with finite precision. Lastly, our laws of physics are not yet complete. There are a number of phenomena where we don't really know what's supposed to happen, so the simulation won't represent the real universe perfectly, even if we start with a proper starting state (which we can also only guess) and use sufficient precision and sufficiently small time steps.

Note that it is completely impractical to simulate all laws of physics as we know them on real computers. Even simulating single molecules with all relevant laws is very hard even for supercomputers.

Nevertheless, physicists simulate things all the time, sometimes using bold approximations to cut down the required computation. It's really important in physics to do simulations. Wikipedia has on overview article on Computational Physics and Richard Feynman has written a classic paper called Simulating Physics with Computers.

It is of course also possible to tweak the laws of Physics a bit and do your simulation. You can change the equations, or the constants and see what happens to your simulated system. You can for example find that changing some constants just a little produces an inhospitable universe.

Now I don't think it's possible to change fundamental mathematical truths in your simulated universe, because Physics makes no prediction about mathematics. You can't change the laws of Physics such that 1+1 != 2.

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    $\begingroup$ more on scientific/ physics determinism see Laplace demon. more on precision/ sensitivity to initial conditions see butterfly effect $\endgroup$
    – vzn
    Oct 23, 2015 at 20:51
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    $\begingroup$ @adrian man you made my day with your answer + vzn's ... i would surely read the resources you both provided but just one thing -> what if laws of physics can be simulated or encoded as equations or constants on a computer, how useful that would be ? surely 1+1 is always 2 but what if there exists a universe when only the law (x)=(-x) is applied alone or along with other rules ? iam sure there exists a logical algorithm that would make (x)=(-x) by somehow. $\endgroup$
    – Henry akpo
    Oct 23, 2015 at 21:16
  • $\begingroup$ @Henryakpo Mathematicians sometimes look at what happens when you change basic axioms. It's still maths though. For example you get different kinds of geometry if you change your assumptions a bit. There has been an interesting period in mathematics when we tried to find which axioms are truly necessary. $\endgroup$
    – adrianN
    Oct 26, 2015 at 6:46
  • $\begingroup$ Ya, i just read about it ... well i can say both axioms are helpful in different ways but more interestingly when you use both of them at once, i am talking based on my experience of both axioms with cellular automata, but defining which axioms are necessary ... necessary for what ?? $\endgroup$
    – Henry akpo
    Oct 26, 2015 at 14:55
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It is currently believed that physics requires real numbers. Digital computers cannot work with real numbers. We spend a great deal of time in simulations trying to cope with the difference between real numbers and IEE-754 floating point representations of those numbers. We also try to avoid operating in the known environments where those differences are resolvable (which occurs in many chaotic systems). We also abuse statistics, and seek to argue that a sampling of universes can describe a statistical variable in the real world.

One option would be to do symbolic manipulation, if you could identify a set of symbols which fully identified state of the universe. However, the ability to model our universe that way is highly questionable, due to self-referential issues. Read about Godel's incompleteness theorem if you're interested in questions about predicting the state of our universe, not just an arbitrarily specified universe.

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  • $\begingroup$ Well, i got you idea clearly => so if that simulated universe just contains integers (all positive) there is no way to implement physics law rights ? what if we transferred our current physics laws that involves floating points to just integers and implemented them, what would it involve, how hard it can be ? surely i will read godel's imcompleteness but i would be very interested in knowing things in physics that involve decimal points or represented as floating points. $\endgroup$
    – Henry akpo
    Oct 24, 2015 at 21:54
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    $\begingroup$ Floating points (like IEE-754 floats) are still not real numbers. There is actually a mapping from every floating point operation you can do to a set of integer operations. Floats do a better job of approximating real numbers than normal integers do, because they capture traits of those real numbers that we consider important. However, there are plenty of cases where the details needed to simulate our universe are not well captured in floating point arithmetic (at least to the best of our scientific knowledge to date... we may always learn something new tomorrow) $\endgroup$
    – Cort Ammon
    Oct 24, 2015 at 22:22
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    $\begingroup$ An easy example is catastrophic cancellation of floating point numbers. There are things you can do to work around that issue, but it is a well understood way to demonstrate limits of floating point numbers. Chaotic systems, however, are much more resistant to being modeled in floating point numbers, so much so that I would make the claim that dynamic chaotic systems cannot be fully captured in floating point numbers. Edward Lorentz found this with his weather simulator, and we've been dealing with the results ever since. $\endgroup$
    – Cort Ammon
    Oct 24, 2015 at 22:24
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This question is too broad to be useful I think. The general answer is, of course, no because all computers we have operate in a discrete manner while laws of physics seem to be pretty damn continuous (even with quantum mechanics in the way). So you'll always have rounding errors, steps which have nothing in-between and insufficiently random behavior.

I think the best example is just simple bouncing balls problem. If you have ever implemented it you know that, as the program goes on, balls get stuck in each other, as in this image I've drawn for you:

image description

This is one of many possible cases where your program breaks laws of physics and what you have to do is to either calculate collision before it happens or calculate what should've happened once laws of physics have been broken.

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  • $\begingroup$ so the problem is about the representation of decimal numbers and how they will appear, but i don't understand why physics cannot only be represented as integers ??? would you please provide something from physics that require writing it as decimal ? The images explains it clearly, but i am wondering what if we implement rules of physics that only involve integers without any rounding process or objects that require them, or represent the law of physics as a whole as integers (without decimals) would that be possible ? $\endgroup$
    – Henry akpo
    Oct 24, 2015 at 21:44
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    $\begingroup$ But physics doesn't happen in integers, which is whole problem. Man, just look at a circle. Π is a number that not only isn't integer, but can't even be represented by float of any size apart of indefinite. Which means that some calculations, such as circle circumference will never be precise unless you have indefinite amount of memory at hand. So the idea of simulating physics for real breaks pretty soon. $\endgroup$
    – Tomáš Zato
    Oct 24, 2015 at 23:43

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