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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.

Urb
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James
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2 Answers2

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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.

Urb
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zzz
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-2

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 wavefunction.’’ 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: ‘‘Explore all paths.’’ The electron is so free-spirited that it refuses 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)

Urb
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Jeff Goetz
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