On Boltzmann brain formation As a popular thought conundrum it is said that if our universe can randomly fluctuate into a self-conscious formation (called Boltzmann brain), then given enough time it will do so and hence the paradox is claimed to be that for a random given observer there's way higher chance to be a Boltzmann brain rather than a normal evolved consciousness.
My thought is: what if the chance to randomly fluctuate into simple cellular life that then evolves intelligence is just way higher than the probability to create an entire brain from a single enormous random fluctuation? If this is the case, then for a given observer the chance to be a normally evolved brain is way higher than a chance to be a Boltzmann brain and hence even though formation of Boltzmann brain can be possible, it can just have negligibly low probability for a given observer to emerge randomly than to randomly evolve intelligence.
This is a conundrum that's used to disprove serious cosmological theories that allow Boltzmann brain formation, so there clearly has to be something that I don't understand about it.
 A: I think you have to look at the whole picture to tease out the relative odds.
With an extremely low entropy Big Bang in the past, and the Second Law increasing the entropy ever since, and treating the universe as an isolated system (this part may be controversial), the Fluctuation Theorem says "as the time or system size increases, the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially." https://en.wikipedia.org/wiki/Fluctuation_theorem
So fluctuations occur exponentially more, and are exponentially larger, the further we are from equilibrium/heat death. You would therefore expect the vast majority of fluctuations to occur at earlier times, and they would be exponentially larger too. This helps reduce (not eliminate though) the seemingly infinite amount of BB observers in the far future once natural/evolutionary/AI life is no longer possible in near heat death conditions.
Tiny fluctuations are exponentially more common than larger ones, so yes it seems much more likely for a microscopic fluctuation -> life -> observer than for a BB observer. Additionally, a BB with the memories of the past like we posses is even less likely to pop into existence.
Does this mean we are the result of an tiny initial fluctuation that went against the second law that then lead to evolution, rather than progressively increasing entropy with no reverse arrow? I guess, but that isn't surprising. At small scales and far from equilibrium fluctuations are exponentially more common according to the FT, so it is very likely life/evolution made some use of a fluctuation to get it us here. That isn't surprising.
Lastly, if the universe is infinite, how do we compare the possible infinite evolved observers vs infinte BB observers? That is basically a problem of measure https://en.wikipedia.org/wiki/Measure_problem_(cosmology) and it is not known what assumptions to make to solve it. (It doesn't just deal with multiverses as the Wiki article suggests.)
So with all that said, I think you are correct that a BB is exceedingly less likely than the type of observers we are.
A: Your idea about a possible resolution doesn’t work, because of conservation of information. First, let me define some terms.
By "brain like yours" I will mean the exact current configuration of your brain. By "good"/"bad" states of the universe, I will mean states of the universe with an evolved/Boltzmann brain like yours.
Your hypothesis can be re-stated as follows:
Hypothesis: consider all the possible initial states of the universe. Maybe there are more initial states that lead to good states than those that lead to bad states?
But conservation of information, which is a well-known principle in quantum mechanics, dictates that the number of initial states that lead to good states is precisely equal to the number of current good states. And the same for bad states. So your hypothesis now becomes:
Final version of the hypothesis: maybe there are more good current states than there are bad current states?
But we know that that's not true. A good current state would involve a brain like yours surrounded by some sort of biosphere, in other words a brain surrounded by some kind of ordered environment. But a bad current state would involve a brain surrounded by a completely random environment. So of course there are many more bad current states than good current states.
