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The best explanation for this paradox and one that should be especially convincing to computer scientists is that the kolmogorov complexity of any brain aware enough to have consistent memories is higher than the universe which contains it. A Boltzmann brain might not be impossible but it is far,far less likely than arbitrary universes occurring as fluctuations. It makes sense if you think about it. It's also consistent with the fact that 17th century mathematics could make so much headway unravelling the laws of physics and we're still fumbling trying to understand bacteria.

Suppose you were playing with procedural generation. Which would be harder: a procedural universe, a procedural universe with a generated AI able to interact with it or an isolated AI which has learned to act optimally in some arbitrary but specific universe? It is not clear that the last program, meant to represent a Boltzmann Brain with a memory of interactions with the universe is any less difficult than the one with the complex generated AI. The isolated AI might be even harder since a dumb algorithm could, given enough time and feedback, produce something fairly intelligent.

So it would take more bits to describe a brain with your specific memories than to specify a universe which could end up with someone similar to you and easiest of all is the program which generates all possible universes. Schmidhuber goes into this here: http://www.idsia.ch/~juergen/computeruniverse.html.

Even if the universe is not some automaton, the argument is still valid. Shorter programs are more likely and a program specifying you with memories of being embedded in some specific universe requires more bits than evolving a universe which has the laws required to support and eventually evolve someone like you.

I approve of you giving this some thought, but entropy doesn't care about Kolmogorov complexity at all. A box full of helium where exactly half of the helium is on one side and half is on the other is not appreciably more likely than one where there are 17 more helium atoms on one side the box than on the other despite requiring less bits to describe.

The regions of the Canonical Ensemble corresponding to a "A Brain" are much smaller than those corresponding to "A brain that's inside a body". And the incompressibility of phase space means that it's therefor proportionally less likely in the steady state. Just be very glad that the universe we live in is such that we seem due for a Big Crunch or Big Rip before getting anywhere near steady state.

EDIT: A little bit of help for the intuition. In a long run steady state universe pretty much all matter is going to be inside black holes. Sometimes you get a spontanious particle/anti-particle pair forming near the radius and one happens to escape. The odds of this happening enough times to accumulate a brain's worth of mass outside a black hole are staggeringly unlikely. Each additional brain's worth of mass decreases the odds exponentially. So I hope you can see that an entire solar system's worth of mass forming outside a black hole is so unimaginably unlikely that it dwarfs the unimaginable unlikely-hood of some mass happening to assemble into a brain.

>A box full of helium where exactly half of the helium is on one side and half is on the other is not appreciably more likely than one where there are 17 more helium atoms on one side the box than on the other despite requiring less bits to describe.

Plus, how does one decide which information is counted ("which of two distinct volumes the atoms occupy") and which information is not ("full position & velocity of every particle constrained only by uncertainty")?

Because if we don't make that [artificial] distinction, each box has exactly the same amount of information.

> It's also consistent with the fact that 17th century mathematics could make so much headway unravelling the laws of physics and we're still fumbling trying to understand bacteria.

Keep in mind that's pretty relative. 17th century physics was low-energy, newtonian, and euclidian. No insight into the microcosmos whatsoever. No insight into the high-energy macro-structures. No science of complexity.

In other words, a very narrow slice of the whole range of existence.

Also worth considering is Dust Theory, from the inventive mind of Greg Egan, which adds a simulation layer and time/space independence to the Boltzmann paradox.

http://schwitzsplinters.blogspot.com/2009/01/dust-hypothesis...

And the FAQ for those who have read Permutation City:

http://gregegan.customer.netspace.net.au/PERMUTATION/FAQ/FAQ...

This is pretty silly. The ``paradox'' only arises because of a confused notion an apparent incongruity between the second law of thermodynamics and the subjective perception that complex "things" in the universe (rivers,trees,people) are highly organized. The second law makes a precise statement about the evolution of closed thermodynamics systems. The latter observation is only a vague sentiment without an exact statement, and is at any rate concerned with open systems to which the second law of thermodynamics does not apply. So there is no paradox.

Addendum: So I made my way through to some of the references, where one can find all these papers by respected cosmologists (I am not a cosmologist, but I have heard of these people and get the sense they that are the real deal). Anyway, it would seem that the term "Boltzmann brain" refers to a technical problem associated with the growth rate of the creation of metastable states from fluctuations during inflation (an actual particle physicist should feel free to chime in). This article appears to hijack the associated buzzword to advance a philosophical discussion around the ``paradox''. It is worth pointing out that none of these cosmology articles are referenced inline in the text. Who are these people and why are they writing this?

(I am not a cosmologist or particle physicist, but my thesis did require me to delve into both, and was defended in front of someone who was a particle physicist)

It doesn't require confusions about the second law.

It does require some very specific conditions which may not obtain in the actual universe. It requires, for example, that the universe can eventually reach thermodynamic equilibrium, that the universe in this state can persist over incredibly long time periods, that at least some types of particles are stable enough to exist over such a time range, etc.

And then starts considering whether situations that we normally associate with states where free energy and the ability to do work exist could arise occasionally as a result of random collisions of particles, possible quantum effects, etc.

The key is that the second law is statistical in nature. It describes a tendency, wherein macroscopic entropy-decreasing events in a closed system are merely incredibly improbable. Or, more simply: entropy-decreasing events can occur, they are just so improbable/rare compared to entropy-increasing events that we should expect not to observe them.

In that context, the Boltzmann brain and other strange things are at least not outright impossible.

I think I completely agree with you. As you point out, one expects that there is a small but finite probability to have Boltzmann brain-like things pop up in the universe. And if it did happen it would violate the second law, which is making a statistical statement.

So my problem with the article (perhaps your comment should help to clarify) is that it attempts to start a discussion, involving some kind of anthropic principle/natural selection to resolve a paradox that doesn't exist in the first place. The body of the article states:

>The usual resolution of the Boltzmann brain paradox is that we and our environment are the products of a long process of natural selection, which can produce complex and improbable outcomes without violating the laws of thermodynamics.

>The Boltzmann brain paradox is that any observers (self-aware brains with memories like we have, which includes our brains) are therefore far more likely to be Boltzmann brains than evolved brains, thereby at the same time also refuting the selection-bias argument. If our current level of organization, having many self-aware entities, is a result of a random fluctuation, it is much less likely than a level of organization which only creates stand-alone self-aware entities.

The authors of this article somehow turn a correct argument of the expectation value of the number of Boltzmann brains in an ensemble of universes to a claim that most brains in these universes must be Boltzmann brains. They do this without mentioning at all how to calculate the probability of non-Boltzmann brains (assuming the distinction is even valid). TO deal with the feeling of existential anxiety that they are left with at the worry they might be Boltzmann brains they then invoke a natural selection+ anthropic principle argument.

As far as I can tell has very little to do with the concept as it was introduced by the cosmologists.

I suspect, but cannot be sure, that a lot of the handwaving about the second law is due to the common misconception that evolution somehow violates the second law.

It's unfortunate but common, especially among people who refuse to accept evolution, so I'm not surprised to see it being mentioned.

Here's a Sean Carroll article on the subject: http://www.preposterousuniverse.com/blog/2006/08/01/boltzman...

here's another one, linking the higgs field to the cosmological entropy problem: http://www.preposterousuniverse.com/blog/2013/08/22/the-higg...

(that second one is a bit whacky...)

Peter Woit also has some material on this :)

http://www.math.columbia.edu/~woit/wordpress/?s=boltzmann+br...

as does Luboš Motl: Boltzmann brains: popular misconceptions http://motls.blogspot.de/2008/08/boltzmann-brains-trivial-mi...

Taking the argument further, it seems more likely that you are a Boltzmann half-brain, with the other hemisphere's input arriving randomly.
It must be noted that we evolved the way we did because nature have selection functions interwoven in it. Self awareness (consciousness) is still a tricky subject to grasp. People think of Infinite Loop Apple's office name as play on programming. But I am pretty sure it's play on thinking.
He was definitely ahead of his time... "Mathematicians Solve 140-Year-Old Boltzmann Equation"(2010) http://www.sciencedaily.com/releases/2010/05/100513162755.ht...
I have some observations on this.

1. The universe we observe is interpreted by a brain(or conscious entity)

2. There is no proof if the universe is real or fake(Any proof for its existence if all brains are destroyed?)

3. Existence of the universe seem to be a property of the conscious mind

4. There is no probability theory in this realm. Even Randomness is an attribute of existence which itself is under question

5. It is impossible to prove if there is only one conscious mind or there are several(what if all men are fake)

6. Since the physical brain is part of the universe which again depends on its own conscious perception, we have a total breakdown. One possibility is resort to Godel and suggest that consciousness exists outside the physical/energy plane.

7. Therefore what are we talking about (super mario questions his own origins on an 8 pixel screen)

~random

Even with a knowledge of psychology, I'm sure few people realize nightmare of such vision.

Please, consider that many geniuses feel deprived and stressed (http://en.wikipedia.org/wiki/Gifted_At-Risk). It would be unwise to assume that these problems would not arise for a brain without access to culture and society. What is a Boltzmann Brain, if not an isolated genius?

To be such an isolated and self-aware brain might be a scary experience, if we were ever to record it.

Or impoverished one, if it lacked emotions.

Loosely related: Some time ago I wrote a fictional story "History of Everything" about a "overlord of chaos" which is roughly similar to the Boltzmann brain, at least in its basic idea:

http://profv.de/history/

However, I never thought that this strange idea could ever be part of a serious scientific discussion, let alone that such a discussion already happened more than a century ago.

As a first time introduction, my brain instantly related the Boltzmann brain concept to Spontaneous Generation (http://en.wikipedia.org/wiki/Spontaneous_generation). Oddly, Ludwig Boltzmann lived during the time the Spontaneous Generation theory was discredited after being a widely accepted theory for 2 millennia.