# Does basic probability guarantee a cyclic bang/crunch cycle?

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?