The Boltzmann brain idea as I understand it: suppose the universe has an infinite lifetime. Once heat death is achieved, there are no more large-scale structures to the universe -- everything is just particles random floating around.
But, just by chance these particles will sometimes randomly form themselves into structures. One possible structure is a human brain, in exactly the state as your brain right now, e.g. with neurons firing in a way appropriate to the memories + perceptions you have right now.
Given that heat-death state lasts indefinitely, we would expect to be one of these randomly-formed brain structures, as opposed to being a human that exists because of being born, etc.
Usually, when I see this argument, it's asserted that a Boltzmann brain would vanish almost instantaneously. My question is, why? If atoms randomly arrange themselves into something like a brain, why wouldn't this structure persist like an actual brain? (This is the position I understand this blog post to be taking).
I think there's something I don't understand about entropy, because from this perspective I don't even see why there would be a heat death in the first place. Given infinite time, wouldn't the uniform sea of randomly-moving atoms just by chance happen to arrange itself into a universe (that is, a self-perpetuating state) and continue from there?