Can a *perfectly* accurate physical simulation be faster than the real process?
Question: assuming one day we know the exact laws of the physical universe, and assuming we have enough computing power, is it possible to simulate a part of the universe (even a tiny one like a molecule) with perfect accuracy (i.e, indistinguishable form the real thing for EVERY experiment we can imagine) but faster than the real process?
The question may seem stupid but it's more complicated that it appears at first thought. Think about this: assuming a classical, newtonian universe, and a perfectly isolated particle, we can easily compute at any time, say, the position by using a simple mathematical formula. Then the "simulation" does not need approximating the behavior of the system using any sort of time step: the computation time may be much less than the time required by the real particle to actually reach the position. But as soon as we introduce real physics, things get complicated: perhaps there are no analytical formulas that describe the observables of the particle; or perhaps these formulas are so complicated that it takes more time to compute the result than to actually simulate it (something that could be done with arbitrary precision, but this is not what we want because we want *perfect* accuracy - and by that of course I don't mean necessarily a deterministic result, but at least a perfect description of the probability distribution from which our universe is but an observation among others).
Obviously, we could build a physical replica of the system we want to simulate, and consider it a "computer": then the time it takes to simulate is exactly the time it takes for the "real" system to evolve. This gives an upper bound for the minimum time it takes to run a perfect "simulation" (if one does not impose conditions on the type of "computer"). But what about a lower bound? can it run faster than the real thing?
22.7.2013
Or does a perfect simulation require at least exactly the same duration (and/or physical space, energy, etc)?
The question may seem stupid but it's more complicated that it appears at first thought. Think about this: assuming a classical, newtonian universe, and a perfectly isolated particle, we can easily compute at any time, say, the position by using a simple mathematical formula. Then the "simulation" does not need approximating the behavior of the system using any sort of time step: the computation time may be much less than the time required by the real particle to actually reach the position. But as soon as we introduce real physics, things get complicated: perhaps there are no analytical formulas that describe the observables of the particle; or perhaps these formulas are so complicated that it takes more time to compute the result than to actually simulate it (something that could be done with arbitrary precision, but this is not what we want because we want *perfect* accuracy - and by that of course I don't mean necessarily a deterministic result, but at least a perfect description of the probability distribution from which our universe is but an observation among others).
Obviously, we could build a physical replica of the system we want to simulate, and consider it a "computer": then the time it takes to simulate is exactly the time it takes for the "real" system to evolve. This gives an upper bound for the minimum time it takes to run a perfect "simulation" (if one does not impose conditions on the type of "computer"). But what about a lower bound? can it run faster than the real thing?
22.7.2013
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