boltzmann brains i

i found a paper on google

Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. [His student] Paul Ehrenfest, carrying on the work, died similarly in 1933.

— David Goodstein, States of Matter

One interested in discussing the Boltzmann brain might be reassured to learn that the modern form of the thought experiment was not exactly proposed by Boltzmann per se, and rather formed much more recently—and that consequently it could not have played a role in the explanation of his death. The topic creeps me out anyways. It’s not like I expect metaphysical conclusions to actually cause much issue, they’ll presumably add up to normality¹¹ Until the Singularity, whereupon all bets are off., but nonetheless the quantity of open questions in metaphysics makes me rather uncomfortable.

To reassure ourselves, we will read an article published by Rovelli and Wolpert to the arχiv last year which purports to tell you Why you do not need to worry about the standard argument that you are a Boltzmann brain.

For those unfamiliar with the Boltzmann brain concept:

Boltzmann brain, or b-brain for short, is the name given to a phenomenon that is in principle possible in statistical mechanics. Imagine a large statistical system formed by a mixture of particles of different kinds that remains in thermal equilibrium for an arbitrarily long span of time. According to statistical mechanics, there are fluctuations at thermal equilibrium, and in principle all configurations can be reached by such fluctuations with enough time available. Consider one of these random fluctuations giving rise –just by chance– precisely to a brain like yours, containing all the memories, the information and the perceptions that you have right now, fluctuating for a brief moment into existence. That would be a b-brain. That brain will have exactly your worldview, your memories, your perceptions. It would know and feel precisely what you know and the way you feel right now. Now, how do you know that you are actually not such a b-brain?

Looking at this account of the thought experiment… I won’t say I would like to up the ante slightly, but I do feel obliged to. My point is simple: who says it has to be a whole entire brain? You in your special little seat of consciousness²² to whatever extent that’s even a thing don’t have simultaneous access to your whole entire brain state. And if you’re functionalist enough to believe in uploads, and such, you’re not married to the specific substrate—there are many ways to implement the pattern that you are.

If I want to find a Boltzmann brain, I do not need to look around the far future heat death quantum fluctuations until I find an entire human brain. I might be able to just look at like, a hot plasma, until I find some particles bouncing around in the right way³³ What counts as “the right way”? Super unclear.. I do not feel that, a priori, I can totally rule out the idea that these things already pop up all the time in the modern day. There may be some good argument that they don’t, but it hasn’t yet popped fully-formed into my head. So that’s one of the questions I will have going through Rovelli and Wolpert.

Another worry I have about Boltzmann brains is the simple “what about Tegmark IV or other notions of a Greater Reality”? A big enough multiverse gets pretty scary, if I’m not prepared to simply take for granted that we probably pretty much have an entire Universe embedded into Greater Reality somehow. Maybe most of the places you find an April in Greater Reality are rough approximations of Universes, good enough that April doesn’t notice the issues, currently, but which are about to start falling apart at the seams in a few minutes… I’d love to be able to just fully trust inductive reasoning, but maybe something something anthropics vacuum decay mangled worlds. I’m not sure I expect Rovelli and Wolpert to help me here⁴.

Our intrepid historian-philosophers of physics introduce the objects 𝓛 and 𝓓, denoting the “set of all mechanical Laws of physics that we know” and “the set of all present Data that we have,” respectively. Now, I’m a little hesitant to say that I know any of the true Laws of physics or that I have very much Data beyond my raw sensory experience, but I am willing to set these cautions aside for now.

They proceed to say:

The set of laws 𝓛 admits a large family of solutions. Let us restrict ourselves to these solutions to those compatible with 𝓓, in the sense that 𝓓 has non-infinitesimal likelihood under those solutions. Ignoring the other solutions, for simplicity, let us assign equal probability to all these solutions that we restrict ourselves to.

I hesitate a little to accept the non-infinitesimal part—what if 𝓛 has solutions parameterized by a real-number value? But Claude told me that in this case they probably mean something like probability density when the say “likelihood”, so sure. What I find harder to get over is the “let us assign equal probability to all these solutions” part. Like, no! You can only do that if you have finitely many solutions! And even if you do, you probably need a complexity penalty or something! But as good Bayesians we will pick our favorite prior and move on.

Rovelli and Wolpert next say that 𝓛 and their (concerning) equiprobability assumption imply something they call Boltzmann’s Η-theorem⁵⁵ That’s a capital η, not a capital h., “which says that entropy increases both into the past and into the future from any given time at which entropy is known to have some specific low value. This theorem is time-symmetric.”

Now, when physicists discuss theorems this always scares me—they seem to use the word in a subtly different way from how mathematicians do, and it is very alien to me. But in the modern day, there is a solution: tell Claude to phrase the theorem like a mathematician would.

Let f(t,x,v) denote the one-particle distribution function for a dilute gas, where t ≥ 0 is time, x ∈ R³ is position, and v ∈ R³ is velocity. Assume f evolves according to the Boltzmann equation:

$$ \frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}} f = Q(f, f) $$

where Q(f,f) is the collision operator (assumed to be the Boltzmann collision integral for elastic collisions).

Define the H-functional by:

$$ H(t) = \int_{\mathbb{R}^3} \int_{\mathbb{R}^3} f(t, \mathbf{x}, \mathbf{v}) \log f(t, \mathbf{x}, \mathbf{v}) \, d\mathbf{v} \, d\mathbf{x} $$

(assuming the integrals converge appropriately).

Theorem (Boltzmann’s H-Theorem): Under suitable regularity and decay conditions on f, we have:

$$ \frac{dH}{dt} \leq 0 $$

for all t ≥ 0. Moreover, dH/dt = 0 if and only if f is a local Maxwellian distribution, i.e.,

$$ f(\mathbf{v}) = \rho \left(\frac{m}{2\pi k_B T}\right)^{3/2} \exp\left(-\frac{m|\mathbf{v} - \mathbf{u}|^2}{2k_B T}\right) $$

for some density ρ(x), bulk velocity u(x), and temperature T(x).

Physical Interpretation

The H-functional is related to entropy S by S = -k_BH (up to constants), so the theorem asserts monotonic entropy increase toward equilibrium.​​​​​​​​​​​​​​​​

Now, I don’t know about you, but personally I’m a little hesitant to grant this theorem quite the philosophical weight that Rovelli and Wolpert seem to assign to it! I am not so confident that 𝓛 fits whatever criteria we need, here. Like, “under suitable regularity and decay conditions on f"? We’re talking about the set of all known physical Laws, here. Was the equiprobability thing supposed to establish those conditions on f? And how am I to make sense of this in a differential geometry context—I assume someone has probably ever tried to do so? But whatever, we carry on.

There is a common idea that the Big Bang was a special point that had low entropy, and this explains why the universe isn’t just in thermodynamic equilibrium. But Rovelli and Wolpert contest this:

Note though that the most probable situation, given the current observations, is that we happen to be at that special point, not in its future. In other words, the most probable situation is that we are just an entropy fluctuation. And this is to say that we are a b-brain.

Needless to say, I do not yet feel convinced that I don’t have to worry that I am a Boltzmann brain. Before they begin digging us out of this deep hole they have dug us into, Rovelli and Wolpert note that even ignoring the time-reversibility point⁶⁶ Though they then require that “𝓛 admit thermalization in the distant future and the universe settles into a long living equilibrium state (Boltzmann’s “thermal death”)”. and supposing the standard story that there was a low-entropy point sometime in the past⁷⁷ Sometimes I wonder if the past can simply be defined to be the low-entropy direction, thus resolving our metaphysical quandaries. I’m not sure you can actually make that work., there will eventually be the more classic⁸⁸ Or less classic? This account might show up more often in modern pop-sci, but I think the first argument they walk through corresponds a little more closely to Boltzmann’s thoughts—for whatever that’s worth. far-future Boltzmann brains.

Why aren’t we Boltzmann brains?

Only fools think that they are now awake and that they really know what is going on, playing the prince and then playing the servant. What fools! The Master and you are both living in a dream. When I say a dream, I am also dreaming. This very saying is a deception.

— 庄子 (Zhuangzi)

Here, Rovelli and Wolpert (henceforth R&W) introduce their first expression:

which means that P(𝓑 | 𝓓, 𝓛), where 𝓑 is the event that you’re a Boltzmann brain, is close to one. Boldly, given the title of their article, R&W concede that 𝓑 is nearly⁹⁹ I try to avoid the phrasing “almost certain” unless I mean “with probability one (given a certain model)”, though I make no promises I will always remember to do so. certain given (i) the physical laws 𝓛 and (ii) the observed data 𝓓.

They proceed to say:

The problem we intend to point out is not in the logic of the argument: it is in the premises 𝓛 and 𝓓. And it is not either than [sic] (i) and (ii) are individually wrong. It is that we infer 𝓛 from 𝓓 — but that inference in turn invokes those very laws 𝓛 we wish to infer. (Specifically, it relies on our using the second law of thermodynamics Wolpert2023.) So we have circular reasoning.

Viewed differently, Both [sic] (i) and (ii) are right. And yet, the argument is incorrect. The problem can be seen as arising because (i) and (ii) are incomplete, and likelihood derived from incomplete premises can be strongly misleading.

I think I agree that there is some circular reasoning, and that this is concerning. But I’m a little unsure of this whole “rejecting neither 𝓛 nor 𝓓, but rejecting the conclusion 𝓑 that follows from 𝓛 and 𝓓” thing. It seems a little suspicious.

To help make their argument more clear, R&W tell a little story.

Suppose that the information i is a video of a closed camera circuit, from which we see that John was killed at 6pm in his house, and at 6pm Bob, upset because John had kissed his girlfriend, was in the same house, and forensic evidence shows that the gun, with Bob [sic] fingerprints, is the weapon that killed John. This seems to be strong evidence against Bob. But imagine that there is also a second video, where we see that at 6pm a robber had entered John’s house, taken the gun from Bob’s coat hanging in the entrance, and shot John while he was discussing in a civilized manner the jealously issue with his dear friend Bob. Then the evidence on the basis of i appears to be misleading, in the light of the larger evidence I. A shadowy secret service could have Bob convicted by carefully hiding the second video recorded by the closed circuit camera.

Why did John kiss his “dear friend” Bob’s girlfriend, given that Bob and his girlfriend are presumably monogamous? Why did the robber randomly shoot John? Why would the “shadowy secret service” want to frame Bob? Can we hear more about this fascinating shadowy secret service character that was introduced in the final sentence?

Answering none of these very important questions about character motives¹⁰¹⁰ Granted, I never myself provided a justification for the motives of Shifty Sam., R&W put the point that incomplete information can be misleading “in a bit more formal terms” using two expressions:

The meanings of these expressions are left as an exercise for the reader.

Anyways, after this story, R&W present us with a new equation:

where 𝓡 is the event that our Records or memories are reliable, or basically that induction works.

Their point appears to be that it does not make any sense to take 𝓛 seriously unless you’ve granted 𝓡. But 𝓡 seems to strongly suggest that we aren’t Boltzmann brains! So the argument based on 𝓓 and 𝓛 should also include 𝓡, and thereby conclude that probably not 𝓑.

In the next section of their article, R&W claim that Last Thursdayism is sort of like a Boltzmann brain—actually, they say 500 years ago, but what’s the difference. They then argue that believing the Big Bang to be a special moment of low entropy is sort of like a Boltzmann brain hypothesis. Which… I mean, okay, sure, I guess.

One of the reasons that the Past hypothesis — this particular variant of the b-brain hypothesis — is widely accepted is that it is consistent with lots of cosmological data. In contrast, none of that cosmological data supports the original version of the b-brain hypothesis, or the 500 years-ago variant of the b-brain hypothesis. The conclusion is a simple one: No, you don’t have to worry about the usual argument that you are a b-brain, and yes, the usual argument for the second law is sound.

I suppose I mostly accept their conclusion that the usual argument is a little self-contradictory. But… well, I’m left with a feeling that maybe just saying “well, maybe the Boltzmann brain argument still holds to some degree, but the Big Bang is what seems most likely to be a low entropy point where a Boltzmann brain sort of event happened” is… a liiiiiittle suspicious? If the laws of physics strongly suggest that we’re a Boltzmann brain, but we have no reason to accept the laws of physics if we are, in fact, a Boltzmann brain… then our current 𝓛 seems deficient, in that it hasn’t reached any sort of reflective equilibrium.

I’m not totally sure if, in fact, the laws of physics do fail to be in reflective equilibrium in this sense. I am unqualified to say whether the argument that physics implies we’re probably Boltzmann brains is a good one. But if it is, I think that is probably suggestive of a need for better physics, rather than something that lets us keep the same physics while just ignoring it.

Come back another day for my consideration of unusual arguments that we’re a Boltzmann brains, which Rovelli and Wolpert do not address.