this is a serious question i actually have in case any physicists or ex-physicists wanna chime in
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Certain dynamical systems (e.g. a bound state) will provably return arbitrary close to their initial conditions - en.wikipedia.org/wiki/Poincar%C. So I think this is the best answer.
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i think it has something to do with this
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The response is that it's not just the equations of motion that matter but also the initial conditions. Part of what explains the "arrow of time" is the low entropy initial state of the universe.
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Drescher's *Good and Real* helped clear this one up for me. The big bang caused spacetime to expand. We see space expanding in three directions, but we can't see time expanding in the other direction. That causality is separate from ours.
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people have mentioned entropy/initial conditions but just recently i saw a pop sci video that discussed a related question -- why is there only one time dimension -- from the point of view of special relatively as applied to particle decay
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it went something like this (wish i could find it again)
conservation of momentum requires the sun of the 4-vectors of particle decay products to equal the 4-vector of the original particle and with one time dimension this implies that
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I think wiki gives the right answer, that Boltzmann's H-theorem which is irreversible uses an extra assumption of molecular chaos.
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There are systems in which one can show the propagation of chaos results and does not need additional assumptions. My guess is that there when we start using reversible dynamics on infinitesimal particles, some weird approximation error creeps in.