First, a little technical precisation: Poincaré′s recurrence theorem applies to classical systems whose phase space is finite. So, if you believe that the universe is finite, then you have recurrence; otherwise no recurrence.
I think that your conclusion is correct under the hypothesis that the universe exists from an infinite time, and that our current situation of low entropy is the result of a random fluctuation.
The symmetry is broken by the initial condition. If at t=0 the entropy is very low, then it is almost sure that it will rise. The expert consensus is that there has been a special event (the big bang), that you can modelize as an initial condition of extremely high entropy.
It seems unlikely that the big bang was the result of a fluctuation from a previously unordered universe: you can estimate the probability of a random fluctuation resulting in the Earth’s existence. If I recall correctly Penrose did it explicitly, but you do not need the math to understand that (as already pointed out) “a solar system appears from thermal equilibrium” is extremely more likely that “an entire universe appears from thermal equilibrium”. Therefore, under you hypothesis, we should have expected with probability of almost 1, to be in the only solar system in existence in the observable universe.
I wish to stress that these are pleasant philosophical speculations, but I do not wish to assign an high confidence to anything said here: even if they work well for our purposes, I feel a bit nervous in extrapolating mathematical models it to the entire universe.
In order to apply Poincarè recurrence it is the set of available points of the phase space that must be “compact” and this is likely the case if we assume that the total energy of the universe is finite.
The energy constrains the moments, but not the positions. If there is infinite space, the phase space is also infinite, even at constant energy.
Take two balls which start both at x=0, one with velocity v(0) = 1 and the other with velocity v(0) = −1, in an infinite line. They will continue to go away forever, no recurrence.
First, a little technical precisation: Poincaré′s recurrence theorem applies to classical systems whose phase space is finite. So, if you believe that the universe is finite, then you have recurrence; otherwise no recurrence.
I think that your conclusion is correct under the hypothesis that the universe exists from an infinite time, and that our current situation of low entropy is the result of a random fluctuation.
The symmetry is broken by the initial condition. If at t=0 the entropy is very low, then it is almost sure that it will rise. The expert consensus is that there has been a special event (the big bang), that you can modelize as an initial condition of extremely high entropy.
It seems unlikely that the big bang was the result of a fluctuation from a previously unordered universe: you can estimate the probability of a random fluctuation resulting in the Earth’s existence. If I recall correctly Penrose did it explicitly, but you do not need the math to understand that (as already pointed out) “a solar system appears from thermal equilibrium” is extremely more likely that “an entire universe appears from thermal equilibrium”. Therefore, under you hypothesis, we should have expected with probability of almost 1, to be in the only solar system in existence in the observable universe.
I wish to stress that these are pleasant philosophical speculations, but I do not wish to assign an high confidence to anything said here: even if they work well for our purposes, I feel a bit nervous in extrapolating mathematical models it to the entire universe.
In order to apply Poincarè recurrence it is the set of available points of the phase space that must be “compact” and this is likely the case if we assume that the total energy of the universe is finite.
The energy constrains the moments, but not the positions. If there is infinite space, the phase space is also infinite, even at constant energy.
Take two balls which start both at x=0, one with velocity v(0) = 1 and the other with velocity v(0) = −1, in an infinite line. They will continue to go away forever, no recurrence.
Good point but gravity could be enough to keep the available positions in a bounded set
While of course it could, current measurements suggest that it is not.