[Symmetry]

A mere trillion years is a tiny fraction of the time it would take for a brain to form like this spontaneously. This concept arose when people generally thought the universe was in a generally steady state and had existed forever. Even though generally say that entropy always increases if you wait long enough you might see reversals if you wait periods of time proportional to something like e^N where N is the number of particles in the system. So it's weird that we're in this low entropy solar system, but you'd expect something like that to happen every once in a while across infinite time. But since a brain has many less particles than a solar system [citation needed] you should expect brains to form from spontanious entropy reversals much more often than galaxies do. There's a principle called the "incompressability of phase space" that means that a low entropy solar system with a low entropy brain is necessarily much less likely than just the low entropy brain, because entropy reversal across an entire solar system is just so mind boggling unlikely.

Of course nowadays the cosmic redshift, etc, make us think that the universe is not eternal but began a short few billion years ago and will end in a big crunch or big rip long before we expect a single Boltzman brain to arise through the random walk of particles.

[raattgift]

You have the analysis right (incompressibility of phase space), but maybe not the application.

Bbs are arguments against the early low-entropy state of the universe being a fluctuation out of thermal equilibrium, and of a future universe fluctuating out of the approximately de Sitter state of the far future.

Expanding steady state was an effort to capture the increasing evidence (including redshift relations) in favour of a Lemaître-style dense early universe, and to avoid several problems with ~static universes.

Since a steady-state cosmology has neither an early low-entropy configuration nor a late homogeneous equilibrium state (steady-state means homogeneity & isotropy in time as well as space: the "perfect cosmological principle"), I'm not sure how a BB argument arises in such a model. In an expanding steady-state model, is there some mechanism for making BBs other than to have them appear with the other components of new gas which under self-gravitation fragments into systems with negative heat capacity?

[sdwr]

"Because the earth existing is more unlikely than winning the lottery, we should expect to see winning lottery tickets floating in space"

[bee_rider]

We should expect that there will be at some point winning lottery ticket floating in space.

I don’t think we should expect to see them, though, because they are probably very far away in time and/or space.

Also the concept of a winning lottery ticket would seem to require the existence of a lottery game, which would seem to require some sort of society to play it. We are probably not the minimal working example of a society that is able to invent a lottery, but I bet that society is closer to us in complexity than it is to a scrap of paper.

[qingcharles]

This reads like a paragraph from H2G2.

[immibis]

Yes - if the observable universe is just a low entropy random fluctuation.

[sdwr]

The point of the Boltzmann brain is that it's spontaneous (which is also what makes it impossible). If a billion years of slowly ratcheting up complexity counts as a "random fluctuation", then every brain is a Boltzmann brain.