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Say you had a very powerful computer and wanted to run a completely lossless simulation of a universe approximately the same size as our own: $10^{80}$ particles.

Each particle in the simulation has properties like velocity, mass, charge, etc. Assuming that your program didn't use any tricks (like compressing this simulated universe by storing groups of 1000 particles as if they were one), does the pigeonhole principle mean that you would need a computer made out of at least $n$ particles to losslessly simulate a universe of $n$ particles?

I say this because I don't see how it's possible to store all of the physical properties of a particle on a piece of hardware without using at least one actual, physical particle.

Am I right about this? Does this mean it would be impossible to ever hope we could create a high resolution, lossless simulation with a number of particles similar to the actual number of particles in our universe?

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No, the pigeonhole principle doesn't immediately imply that. It might be true, but you'd need some additional reasoning to prove it.

For instance, you seem to be assuming that the way the simulation works is by separately simulating each simulated particle using one or more real particles, and that a single real particle cannot simulate more than one simulated particle. It's not obvious that either of these need to be true. For instance, perhaps you can simulate K simulated particles using L real particles, using a complex algorithm that simulates them jointly without any clear mapping from simulated particles to real particles. It's also not clear why we'd need to have $K \le L$; perhaps there is some algorithm where $L=K/2$ (say). Some additional reasoning would be needed to rule out these possibilities -- the pigeonhole principle alone isn't enough.

It does seem like a plausible-ish sort of conclusion, under reasonable assumptions about your model of computation. But I don't think it's that easy to prove.

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    $\begingroup$ Would this be a "lossless simulation", according to the rules of the OP? $\endgroup$
    – shgr1092
    Jan 7, 2020 at 14:56
  • $\begingroup$ But if it were possible to simulate K particles with L real particles and L was less than K, you could simulate a larger simulator from within the simulator: meaning that you could recursively, infinitely improve your processing power merely by uploading the right software (namely, simulations simulating more simulators... and so on) $\endgroup$
    – Greg Lane
    Jan 7, 2020 at 16:47
  • $\begingroup$ @GregLane, not necessarily; suppose that was only possible when $L\ge 10^{70}$, say. Doesn't sound awfully plausible, but it seems like a more involved argument is necessary to rule that out. $\endgroup$
    – D.W. ♦
    Jan 7, 2020 at 17:46
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Imagine that you’ve built such a simulation. Now, using only a small fraction of all resources, you can reproduce whatever is possible in the universe. You run your simulation and faraway aliens are free to do whatever they want.

But it means you can run the same simulation inside simulation, again using only a fraction of simulated resources. You can simulate the simulation and using the same space simulate aliens that do something irrelevant to your simulation.

Nesting the simulations further, you can achieve a simulation that uses no resources at all. And this is a bit suspicious.

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    $\begingroup$ I don't see why this is necessarily the case: a solution in the same vein as D.W. may not infinitely recurse. $\endgroup$
    – shgr1092
    Jan 7, 2020 at 14:53
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    $\begingroup$ @D.W. I don't understand why this is an obstacle. If you can take $10^{70}$ particles and build a simulator of $10^{80}$ particles, surely you can reproduce the same process inside the simulator? This then can be repeated at deeper level. And each level will give you free space to simulate additional $10^{80} - 10^{70}$ particles. $\endgroup$
    – Dmitri Urbanowicz
    Jan 8, 2020 at 15:58

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