Why would a Boltzmann brain be transient? 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?
 A: The Boltzmann brain paradox arises due to smaller fluctuations being more probable than larger ones. So, if you contemplate how our universe started out with low entropy initial conditions then it's difficult to explain this in terms of a generic high entropy state. Fluctuation yielding the early universe that in turn would have given rise to you, would be far more unlikely than just you appearing out of thin air. 
So, the reason why a Boltzmann brain would vanish almost instantly is because the longer it is to survive the more support structures it would need, a larger part of the local environment would have to be compatible with a longer existence, and all that makes it less likely for the fluctuation yielding such an outcome to have occurred. 
One has to note that this reasoning also applies to what a Boltzmann brain would be aware of, it's far more likely to have random information stored in its brain that doesn't make sense to itself than information that is more structured and internally consistent. This means that we can be sure that we are not Boltzmann brains. This is then the core of the paradox: we exist in a state that according to anthropic reasoning should not be typical.
A: A Boltzmann brain is not that different from what might be called Boltzmann cheese. Given enough time a set of atoms or particles might arrange themselves by statistical fluctuations into big wheel of cheese. If that happens there is no reason to think the cheese would then rapidly be demolished unless it formed in a star, or falling into a black hole or in similar circumstances that would demolish it, say on my table where I start to eat it. The Boltzmann brain would appear to require far fewer special circumstances for its occurrence than an entire universe. The number of microstates, or Hilbert space of states for a brain is less than an entire cosmos. So the difficulty that is raised is that it would statistically be more plausible for a Boltzmann brain to spontaneously occur and from there dream all that I or the rest of us observe. It is similar to the philosopher's question of how do we know that we are not just a brain in a vat with input stimuli that gives us the conscious appearance of an exterior world that actually does not exist.
There is though one key difference between the spontaneous generation of a cosmos and the appearance of a Boltzmann brain. We have some reason to think that a cosmology emerges from a high energy vacuum, called a false vacuum, that by various means tunnels or transitions into a lower energy vacuum. This idea originated with Coleman and de Luccia as a bubble nucleation and was found to work well with inflationary cosmology. This gives an open direction for the generation of a cosmology, and as an open spacetime is not subject to the constraints of closed thermodynamics. The Boltzmann brain occurs by pure statistical fluctuations with no “gradient” or direction given by energy potential differences, say from a high energy vacuum of complete symmetry (false vacuum) to a physical vacuum at low energy. A comparison between the spontaneous quantum generation of a cosmology and the thermodynamic occurrence of a Boltzmann brain appears to be different categorical problems.
Sean Carroll is concerned about Boltzmann brains (BB). I read quite some time ago an estimate on the occurrence of a BB, and as I recall the time is comparable to or longer than the estimated stability of the de Sitter vacuum of $10^{10^{70}}$ years.  In a multiverse situation with something like $10^{1000}$ cosmologies on the landscape, and F-theory is on the order $10^{10^5}$, there is maybe no way that we can prove that BBs do not exist. On the other hand the purpose of science is not to cover ground like that and to disprove that something can't exist in some absolute sense. On the whole I would say the BB problem is one that is not exactly worth being overly concerned about.
A: I would not say it decays instantaneously, but it will decays quickly, at least the conscious part. There is no body that supports it, so it will lack the oxygen to think in a brief moment. The rest will be bombarded by  radiation and other particles until it disintegrates again, the timescale for this to occur will be a function of the temperature. 
A: I just want to add a second option that was proposed by some author whose name I cannot remember (it might be wikipedia, I did not check). Perhaps the laws of physics are such that it is very difficult to form a spontaneous and isolated boltzmann brain. According to this idea the only way to form a brain is to first spontaneously form a universe, like ours, in which complex structures develop. The fact that an isolated brain would have much less entropy does not necessarily imply that it will be more likely to form than an entire universe, because not all arbitrary physical structures would form spontaneously over the eons, but only those compatible with evolution of the physical laws.
