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I am not a physicist, but I do have a sophomoric theory. I'd appreciate your indulgence.

The probability that all the matter in the universe will fly into exactly the same point at the same time is vanishingly small. So small that it's hardly worth considering. But, it is nonzero. Furthermore, it is decreasing with time, as the universe spreads out and entropy increases. In other words:

$$ p(\text{random crunch}) = f(H) $$ where $H$ is entropy.

Entropy cannot be infinite, if you assume (wrongly?) that matter will always have some tiny amount of Brownian motion.

If there were to be infinite entropy -- all matter an energy just kind of frozen, then we'd have $$ p(\text{random crunch} | H = \infty) = 0 $$

If you assume that $f$ is monotonic (or at least monotonic $\forall H>H_{now}$), then $p(\text{random crunch})>0$ for all $H$ in the feasible set.

Lastly, time can go to infinity, while entropy will asymptote. Therefore,

$$ \text{plim}_{time\to \infty} p(\text{random crunch}) = 1 $$

So: in the deep fastness of endless time, everything is guaranteed to begin again, for no particular reason.

Plausible?

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Your initial assumption is wrong. "The probability that all the matter in the universe will fly into exactly the same point at the same time is vanishingly small." If the universe is infinite. then if it were to collapse to a crunch, it would remain infinite. If the universe is finite, (assume for example it is the surface of a 4D sphere) it would not crunch to a single point. It would crunch to a condition called a singularity which is not actually reached, but is only approached. The math used to model such a physical crunch is incomplete.

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  • $\begingroup$ Why is the finitude (if that's a word) of the universe relevant? Like, who cares if the universe is infinite -- it just matters where all the stuff is, right? Also: I don't know the theory behind why a finite universe means that the singularity can't be achieved, but does the theory behind the big bang require that the singularity be achieved, or just approached? Like: could it just be super squished? Or does it need to be infinitely squished? $\endgroup$
    – generic_user
    Commented Apr 24, 2021 at 14:19
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    $\begingroup$ " Like, who cares if the universe is infinite..." I care. You apparently do not. Conclusion: some care and some don't. $\endgroup$
    – Buzz
    Commented Apr 24, 2021 at 15:03
  • $\begingroup$ "...does the theory behind the big bang require that the singularity be achieved, or just approached?" I suggest you take a look at en.wikipedia.org/wiki/BKL_singularity . $\endgroup$
    – Buzz
    Commented Apr 24, 2021 at 15:09
  • $\begingroup$ Yup, here's where you need years of study to follow stuff. Apparently Feynman -- widely considered a huge jerk -- said that there is no point in trying to explain physics to the uninitiated, because there is too much prerequisite knowledge. Anyway, I am glad that you care about the universe, and whether it's infinite or not. $\endgroup$
    – generic_user
    Commented Apr 25, 2021 at 16:07
  • $\begingroup$ The general cosmology understanding is that the math describing a cosmological model may have singularities, but the model itself does not include any singularities. I will say this another way to try to make this clearer. The singularity in the math is interpreted by the model as not being part of the universe being modeled. I hope you find this helpful. $\endgroup$
    – Buzz
    Commented Apr 25, 2021 at 17:18

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