# Probability of spontaneous Boltzmann brain formation

I was reading through Scott Aaronson's notes (pdf), but I can't make sense of his discussion about Boltzmann brains on p61. Specifically the fact that it says:

But the problem is worse. Since in an eternal universe, you would have inﬁnitely many Boltzmann-brain doppelgängers, any observer with your memories and experiences seems inﬁnitely more likely to be a Boltzmann brain, than to have arisen via the “normal” processes of Darwinian evolution and so on starting from a Big Bang!

I don't understand what this last statement follows from and it seems very unintuitive.

A Boltzmann-doppelgänger brain arising by chance seems very improbable since that would require the formation of a brain with a full brain state, i.e. essentially all the individual atoms/molecules appearing in the exact (or at least close enough) configuration. In other words, a very high level of organization. Whereas the Big Bang scenario merely assumes a hydrogen/helium cloud and that the basic laws of nature are in place. This seems to be a much simpler object as a starting point and hence much more probable. This starting point, once it has occurred (maybe due to chance) then evolves (through the relatively simple laws of nature) into the state we have now, and gives rise to processes like natural selection etc. which eventually bring about intelligent life and the very high level of organization in the brain. Evolution through natural selection, while fundamentally also depending on stochastic events, seems much more probable since natural selection guides the process, allowing it to "tune in" on the target state, rather than there being a requirement of the target state just emerging in one step as with the Boltzmann-doppelgänger. Another argument is that the human brain, while seemingly a very complex structure, is probably the result (as part of fetus/child development) from a much simpler process, described in the human DNA. Like how a seemingly complex fractal can emerge from a simple equation. Hence, it seems a much easier way of producing a brain by chance is to produce this simpler mechanism by chance "and let it run" until it forms a brain, rather than demanding that a full brain itself appears by chance as in the Boltzmann case.

But given Scott Aaronson is very smart there must be something about this I'm missing. Because I'm reading the statement as saying that the Boltzmann-doppelgänger are more probable than the big bang, but to me it seems the converse is true.

• anyone who is claiming to calculate and compare the propabilities of forming a boltzmann-brain or a evolutionary brain in either scenario is a hypocrite and not a scientist. No one knows about the exact entropy fluctuation needed to form such stuff – image Mar 27 '15 at 12:42
• the lack of informations is too important, it's undecidable – user46925 Jul 14 '15 at 16:52
• The key thing is 'in an eternal universe': if you have enough time even very unlikely events become as probable as you want. – tfb Jul 30 '16 at 20:37
• Some detailed calculations are presented here – Count Iblis Jul 30 '16 at 22:12

You have to consider entropy to answer this question.

The idea of the Boltzamnn brain presupposes an expectation of a universe at thermal equilibrium, or maximally high entropy.

In order to create the initial state from which the natural processes you describe (evolution, development etc.) require to operate, the universe has to first start in a state of low entropy - either via an incredibly unlikely fluctuation into this state, or via some physical law which imposes an initial condition of low entropy on the universe.

In order to create a Boltzmann brain, the universe simply can fluctuate the Brain locally without consideration for the complex (low entropy) formations of galaxies, solar systems, ecosystems etc required to evolve you naturally.

Hence if we start with a universe at thermal equilibrium, then it is overwhelmingly more likely that you would be a Boltzmann brain and that everyone you observe is also one.

• everyone you observe is also one if you were a Boltzmann brain, there wouldn't be anyone else... – Christoph Jan 23 '15 at 11:14
• @Boo: Not sure I follow. I don't think it is in dispute that the universe started in a low-entropy state. Cosmology and the theory of evolution explains how such a low-entropy state can lead to the formation of brains etc. at high probability, much higher than brains materializing out of nowhere. But I think what you are arguing is that the initial low-entropy state is very improbable. But is it? By what prior? By the 2nd law of thermodynamics entropy only increases, so by that an initial low/zero entropy state seems very natural. – Morty Feb 1 '15 at 11:10
• "I don't think it is in dispute that the universe started in a low-entropy state" What does entropy means when talking about a single micro-state ? – agemO Feb 25 '15 at 12:27

# What Scott's saying

Okay, so the basic assumption I can phrase as this:

The universe is finite and has an end. During The End, cosmic inflation does not rip every particle apart into its own universe, and no cyclic model of cosmology predominates: The End of the universe is instead one big "soup" of particles in thermal equilibrium, all having some low average energy $$k_B T$$ in every degree of freedom, for all eternity.

Given these properties, a finite universe will evolve you approximately once or less, and will fluctuate you into existence approximately zero times, before The End. However, it will fluctuate you an infinite number of times after The End, because you have a finite probability of being fluctuated and the universe will have forever to keep trying.

If we're just trying to guess the universe conditional on our knowledge that we exist, we therefore should assume that we were fluctuated into existence, not evolved, because we only get evolved once but fluctuated infinite times.

# Where you're getting off on the wrong foot

You may be thinking that Scott is offering an alternative to the Big Bang: no, he's working within the formalism of the universe we know. You may be trying to explain the universe with a preference to the simplest possible explanation, rather than the most likely explanation: but his statement is not that the Big Bang isn't simple or fun, it's that as a consequence of one end to this Big-Bang universe, the most likely explanation is nothing like what we think our place in the universe is. So of course, if he's talking about what's most likely given what we think we know, and you're talking about what you think our place in the universe is, then you are his audience and he is making this point directly to you.

But you may have an almost-valid point: is it more probable for the thermal equilibrium to fluctuate-into-existence a whole Big Bang which follows normal evolutionary process? And the answer is a double-whammy. The first part of the answer is, if it did, then that's already sufficiently crazy that it upsets our current view. For, then we are still fluctuated into existence, and we just happen to have come from the rare fluctuation which produces a miniature Big Bang. The second whammy of the answer is, it's googols of orders of magnitude more likely that you're just a Boltzmann brain, so going with the more likely explanation, you're probably a Boltzmann brain.

Why the second? It's simple entropy. The universe started from a low-entropy state, and the lower the entropy of a state, the harder it is to form it by thermal fluctuations. You may feel like this assembly of particles constituting you is pretty rare, but once those particles were all in one point in spacetime along with all the particles of the Earth, Sun, Milky Way, and all the galaxies. You balk at putting all of these particles in precise positions? Well, your proposal is then to put all of those particles in one precise position. That's far, far less likely.

Although I am willing to accept that given infinite time, a thermodynamic soup would throw up my brain infinitely many times, I do not believe that is sufficient support for the theory that I am a Boltzmann Brain.

Because as far as I can tell, I am not just a brain in a thermodynamic soup. Such a brain would be short-lived, and would rarely have the consistent memories that I have (the state mentioned by the OP), and would not experience the sensory input which I experience.

I experience a consistent external universe through my senses, and each time I try to probe it, it comes back with consistent information. I take this as evidence that there is a real universe out there. (If you disagree, then feel free to test it for yourself!)

So whilst the probability of a brain like mine existing temporarily in thermodynamic soup is non-zero, the probability of one which has all of my memories and experiences (and furthermore experiences the consistent and responsive sensory feedback that I do) should be much lower than the probability of that universe actually existing, and my having actually evolved within it.

Put differently, whilst I agree that the probability of this universe spontaneously coming into existence as it is right now is astronomical, the probability of the Big Bang followed by the fairly modest process of evolution is much more reasonable, and significantly more reasonable that an emulation of my current state jumping out of thermodynamic soup (given that my current state also includes some comprehension of a consistent universe identical to ours).