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Frequently when trying to solve cosmology questions physicists turn to computer simulations of the universe (albeit massively simplified) in order to verify or disprove their hypotheses. This got me thinking.

My question is about the theoretical maximum possible complexity of these systems.

Let me give an example, if we imagine a tennis ball bouncing on a flat surface if we want to accurately simulate and measure the results of every single facet of the collision right down to the atomic and quantum effects you could actually find a tennis ball and drop it over your surface. In this case the universe is "simulating" the collision for you.

Would it be possible to simulate this same event just as accurately using a computer? Is there a theoretical reason why the computer would need to have more mass than the two colliding objects? (in this case a tennis ball and the planet!)

Now I have always assumed that the answer to this question is "yes you need a more massive computer to simulate any object with total physical accuracy" because if that were not the case there would be no reason why a computer less massive than the universe could not simulate the entire universe with total accuracy, which seems to me to be counterintuitive.

I hope I was clear with my question! I would like to find a hard and fast answer to this question with a reason preferably, been mulling it over for a little while now. Thanks for any help.

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    $\begingroup$ Simulating a tennis ball collision by treating it as a collection of atoms is way beyond any computational power we have or are likely ever to have. We have to take shortcuts like using fewer atoms or describing the overall statistical effect of that many atoms in a collision. $\endgroup$ – Brandon Enright Dec 13 '13 at 17:47
  • $\begingroup$ Of course, if I'm just being a jerk, I can certainly conceptualize systems that have arbitrarily large masses and that can be simulated by arbitrarily small computers -- take, for instance, a $0$ K ensemble of $N$ non-interacting fermions stored in a harmonic well. $\endgroup$ – Jerry Schirmer Dec 13 '13 at 17:49
  • $\begingroup$ But thats just boring Jerry! simulating no intereaction is not challenging! $\endgroup$ – James Dec 13 '13 at 17:56
  • $\begingroup$ The question is boring and of no physical consequence. You ask if we can compute something in the least intelligent way we might approach it and we obviously can't, but so what? $\endgroup$ – dmckee Dec 13 '13 at 17:58
  • $\begingroup$ No dmckee I wasnt asking if we could do it, I was asking if there was a physical principle at work meaning that it was impossible. Im sorry if my question offended you! $\endgroup$ – James Dec 13 '13 at 18:00
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The precise answer to your question can be found in section 2 of quant-ph/9908043, named "Entropy limits memory space ".

From that paper I can extract a heuristic summary to answer your question - why do we need massive computers to simulate massive things:

  1. before simulating anything involving information describing the universe to arbitrarily high accuracy, you need to store all that information.

  2. The amount of information you can store is limited by the number of degrees of freedom of your computer.

  3. This number of accessible states can be determined from the entropy of your computer.

  4. This entropy is determined by the mass of your computer.

  5. Hence the amount of things you can simulate and store in a theoretically "ultimate computer" is limited by the computer's mass.

Once again you can read about the details of any of these steps in the paper cited.

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    $\begingroup$ #5 says you can get the most advanced technology possible, the laws of thermodynamics yields an upper limit on the amount of information you can store in a computer of a given mass. What you gave is an comparison between two things neither of which saturates the upper limit given, and therefore doesn't say anything about the limit. $\endgroup$ – zzz Jul 25 '14 at 1:38
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    $\begingroup$ Yes computers are getting lighter, and storing more information, but this mass to information ratio is nowhere near the limit proved in the results cited. $\endgroup$ – zzz Jul 25 '14 at 1:45
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    $\begingroup$ There is a correlation between the maximum amount of information you can possibly store and your computer's mass. $\endgroup$ – zzz Jul 25 '14 at 1:46
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    $\begingroup$ In other words, your new, lighter, computer is getting better at reaching that maximum amount than your old, heavy computer, but it will never reach the ratio computed in the result cited. $\endgroup$ – zzz Jul 25 '14 at 1:48
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    $\begingroup$ You have never experienced using a computer which saturates the maximum ratio referred to by #5, and given by the literature cited, therefore #5 cannot possibly contradict experience. It may contradict intuition. $\endgroup$ – zzz Jul 25 '14 at 1:57
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The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa. --Heisenberg, uncertainty paper, 1927

On the macro level (stuff we can see and touch) it is easy to make these predictions, that is what mechanical engineers do. However, as you get to the atomic and sub-atomic levels, the predictions become more difficult not because of the complexity of the system, but because quantum level particles cannot accurately be predicted or tracked. Instead, you have to calculate the probabilities of where they are likely to go. Stephen Hawking's book "The Grand Design" goes into much more detail about how this works.

The Feynman sum-over-paths quantum theory takes this idea a step further and says things like electrons don't even follow a single path, but follow ALL possible paths at the same time!

"Thirty-one years ago, Dick Feynman told me about his ‘‘sum over histories’’ version of quantum mechanics. ‘‘The electron does anything it likes,’’ he said. ‘‘It just goes in any direction at any speed ,... however it likes, and then you add up the amplitudes and it gives you the wave- function.’’ I said to him, ‘‘You’re crazy.’’ But he wasn’t." --Freeman Dyson, 1980

"The electron is a free spirit. The electron knows nothing of the complicated postulates or partial differential equation of nonrelativistic quantum mechanics. Physicists have known for decades that the ‘‘wave theory’’ of quantum mechanics is neither simple nor fundamental. Out of the study of quantum electrodynamics ~ QED comes Nature’s simple, fundamental three-word command to the electron: ‘‘Ex- plore all paths.’’ The electron is so free-spirited that it re- fuses to choose which path to follow—so it tries them all."

From: Teaching Feynman’s sum-over-paths quantum theory Edwin F. Taylor, Stamatis Vokos, and John M. O’Meara Department of Physics, University of Washington, Seattle, Washington 98195-1560 Nora S. Thornber Department of Mathematics, Raritan Valley Community College, Somerville, New Jersey 08876-1265 (Received 30 July 1997; accepted 25 November 1997)

I hope this helps

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