How many years will it take before there's a 50% chance that all particles would have assembled into another Big Bang?

Question

Going off of the Boltzmann brain thought experiment, approximately how many years from now would there be a 50% chance that all the particles in the Universe would randomly assemble themselves into another state of very low entropy similar to what was present at the beginning of the Big Bang?

Answer 1

Consider the phase space of the universe as a box with hypervolume $V$ and divide its phase space into volume $h^6$ boxes. That gives us $N=V/h^6$ boxes. If we put in $k$ particles we get $S=\binom{N}{k}$ possible states. One can bound this as $N^k/k^k\leq S \leq N^k/k!$. So if we assume random states occur each $\tau$ seconds, the time until a particular state shows up is somewhere between $\tau N^k/k^k$ and $\tau N^k/k!$.

Now, the observable universe is about $V=10^{80}$ cubic meters and has about $k=10^{84}$ particles (mostly photons). So $N=10^{160}/(10^{-36})^6=10^{376}$ (where I assumed the phase space volume is the 3D volume squared). So the time is between $\tau (10^{376})^{10^{84}}/(10^{84})^{10^{84}}$ and $\tau (10^{376})^{10^{84}}/(10^{84})!$. So the timescale is roughly $10^{2.92\times 10^{86}}$: at this point we can drop the $\tau$ since to a good approximation measuring the time in Planck seconds or Hindu kalpas give the same answer - double exponentials are big.

The big bang might not correspond to a single state, so the actual recurrence time is shorter. But it is still doubly exponential.

The fundamental problem is that space is expanding and that particles are lost over the cosmological horizon. This also causes a very weak horizon radiation of finite temperature that in principle can add new particles. This hardly replenishes the universe enough.