• Thomas Kweon

Boltzmann Brains: How You Are Most Definitely an Almost Impossible Entity

They say that if you leave a monkey with a typewriter, with enough time, it’ll reproduce all works of Shakespeare by aimlessly swinging at it. Sure, it may take a long time (assuming our virtual monkey is well fed through cosmological time scales!), but it is a possibility that cannot be ruled out due to its simplicity. Random movements, a series of input, and pure chance.

Now, let’s take that to the extreme. Instead of a monkey, assume a soupy mess of the remnants of a ‘dead’ universe. A point in time where every star will burn out and the last blackhole will give out. A plain field of space evenly filled by a seemingly uninteresting jumble of randomly moving particles. As the final product, we don’t want to type up Shakespeare’s theatre pieces; we want to recreate his brain in its entirety, alive and conscious.

How likely is it that Shakespeare’s brain will materialize out of a perfectly disordered array of particles?

Sometime around the 19th century, Ludwig Boltzmann (1) devised a thought experiment in which random brains with the capacity to think can fluctuate into existence. When I first heard of Boltzmann Brains, it blew my mind. It was such an exaggerated example of statistical certainty that seemed radically reasonable. While I did not believe in random brains (even replicas of real people!) popping into existence for a spark of consciousness, it frustrated me that no direct disproof could be offered against their possibility, much like the monkey above.

So how likely really is it that these brains form? Is it even possible? What are the implications?

Entropy and the second law of thermodynamics

Okay, let’s not rush to spawning a random brain out of nowhere. Understanding entropy and the second law of thermodynamics will help you understand what is to come after.

Let’s talk about entropy. Although we won’t be discussing it in detail and how it is measured by scientists, conceptually, it can be thought of as the measure of ‘disorder’ in a system. Simply put, the more chaotic and random something is, the higher the entropy, and the more ordered and structured something is, the lower the entropy. For example, let's take our system to be a sealed room as represented by the box below (2).

Figure 1: low entropy start

Looking at figure 1, there is low entropy when perfume particles are released into the room because they are highly ordered (densely packed into one corner of the room). Their subsequent behavior is likely to be predictable, as we can expect them to spread out evenly through the room as time passes. This is in agreement with the 2nd law of thermodynamics: the total entropy of a system (on a macroscopic level) increases over time. Therefore, as you’d expect, the particles will shift into any of the following configurations:

Figure 2: high entropy configurations

At maximum entropy, the particles are evenly spaced out. All particles behave completely randomly, as they shift through the volume of the container. In most cases of observation, the particles will be evenly spread out and retain high levels of entropy.

But a system will not stay at maximum entropy forever and instead experiences small ‘dips’ in entropy as a result of the particles’ random movement. For example, when the particles move around randomly, at one instance, they will look something like this:

Figure 3: partial low entropy configuration

Notice the region encircled by the dotted line. The spaces between the particles are much tighter and as a result, the entropy of the system is temporarily lowered.

This is the very idea that Boltzmann played on. He claimed that when given infinite time, the conditions would coincide, and that any non-impossible configuration would be attainable, even including this highly unlikely behavior of the perfume particles packing themselves into one of the corners before spreading out to high entropy again.

Figure 4: Boltzmann fluctuation

This phenomenon of reaching lower entropy through random occurrence is referred to as ‘Boltzmann fluctuations’.

Understanding what a Boltzmann Brain is.

Now let’s scale things up a notch! Instead of a small room, the universe is now our system. Here is a visual analogy that may help:

The white A4 paper below represents the universe. We’ll have different building blocks (atoms, electrons, the basic building blocks of matter), simplified and represented by the labeled jig-saw pieces, floating through empty space. On most occasions, they will behave randomly and just stay adrift.

Figure 5: high entropy condition

But with aforementioned occurrence of a Boltzmann fluctuation, some of the pieces may happen to assemble by colliding into each other in the right orientations (but with very low probability).

Figure 6: partial low entropy condition

With infinite time and infinite collisions, the puzzle may eventually complete itself!

Figure 7: low entropy condition (brain)

And as you may have guessed, the puzzle we’re trying to solve is a functional human brain capable of reasoning and thinking. However, because it results from random fluctuations, its reasoning capabilities, intelligence, and memory will all be random.

(Note: The analogy is an extremely simplified version of the discussion. Many more particles would be required (order of magnitudes of at least 10^23 from Avogadro’s number) to exhibit Boltzmann fluctuation behaviors to complete a brain. However, with infinite time, not only is this not impossible, it’s unavoidable.)

Key criterion: TIME!

A key aspect of the Boltzmann Brain (for simplicity this will now be abbreviated as "BB") problem lies in the amount of time given in this universe. In order to understand the magnitude of time required to assemble a BB from random fluctuations, it is necessary to be able to predict the eventual fate of the universe. We need to know whether space will expand endlessly without bounds or if we’ll collapse down into another singularity like a reverse big bang called the “big crunch.”

For simplicity, consider the cosmic scale factor R to be representative of the radius of the universe (*disclaimer: the real definition would be more accurately put as ‘the parameter that describes how the size of the universe is changing with respect to its size at the current time’ (3) ). Studying how this changes over time gives us an idea of what to expect as a result of the expansion.

Before that, let’s go over how scientists theorize and study this stuff.

We know that anything with mass tends to attract other things with mass due to gravity. This attraction becomes stronger the more mass there is. Now, think of all matter in the universe as a part of a larger spherical ‘cloud.’

The ρc (critical density) can be defined as the minimum density (minimum amount of mass in a given volume) required for a spherical cloud to keep itself bound with gravity. Any density less than the ρc will result in the cloud of gas slipping away and losing its form to obey the second law of thermodynamics (entropy generally always will increase, and the cloud of gas will diffuse out into empty space). Any density higher than the ρc will mean that gravity will dominate over the cloud and compress it.

So if we measure the density of the current universe, we can predict whether it will keep on expanding outwards or compress due to gravity!

Figure 8: Evolution of the scale factor (4)

Density parameters, 𝛀, are numbers that show the relationship between the measured density of the universe and the critical density. More specifically, 𝛀 represented the ratio of a measured/predicted density to the critical density. These parameters help indicate the different possible outcomes of the universe illustrated in Figure 8.

For average matter, the density parameter 𝛀m is expressed as:

For dark energy (will be elaborated on), 𝛀Δ is expressed as:

There are 3 different outcomes depending on the density parameter that inform the likelihood of the production of a Boltzmann Brain.

Case 1 (Closed model): This corresponds to the yellow curve, where the radius of the universe expands then collapses. If the parameter 𝛀m is bigger than 1, it would mean that the currently measured density of the universe is higher than the critical density, resulting in an eventual collapse of the universe. In this case, the time given in our universe is finite, at just over 20 billion years (5) (and we will explore whether this is enough time to form a Boltzmann Brain). This is called the closed model of the universe.

Case 2 (Open model): This corresponds to the blue and red curves. If the parameter 𝛀m is smaller than 1, it would mean that there isn’t sufficient matter for gravity to act as a binder to slow expansion. In this case, the universe would continue to expand without a limit. While this would mean a theoretically infinite amount of time for Boltzmann Brains to form, it would also imply a limitless stretching of spacetime which would continue to dilute the density of the universe. This is called the open model of the universe.

The two models above are not entirely consistent with observations as they fail to acknowledge the gravitational influences of dark matter - as observed in rotation curves - as well as the expansionary effects induced by dark energy. As this is not entirely relevant to our discussion, to sum up briefly, the two cases above fail to explain experimental data because studies point towards an undetectable mass that is binding galaxies together (dark matter) as well as an unknown source of energy that is accelerating the expansion of the universe (dark energy). If you’re interested, have a read of NASA’s attempt at explaining dark matter and energy, as even they say “more is unknown than is known” (6).

Based on these theories, Einstein and de Sitter (a renowned Dutch physicist) created their own model of the universe known as the de-Sitter model.

Case 3 (De sitter): This corresponds to the green curve. 𝛀Δ + 𝛀m equals to around 1. Observations from the Planck satellite observatory (7) indicate that 𝛀m ≈ 0.32 and 𝛀Δ ≈ 0.68. Since this would mean that the measured density is equal to the critical density, the expansion slows to halt at infinity and stays there without collapsing. Therefore, the lifetime of the universe is infinite.

So why are time and volume important to the formation of BBs?

The real life plausibility of the formation of Boltzmann Brains has been long debated by physicists. The 3 scenarios above give rise to whether, and accordingly how, a BB may form under the given circumstances.

For the content below, I refer to the work of a renowned theoretical physicist, Sean Carroll at the California Institute of Technology, and his paper “Why Boltzmann Brains are bad,” 2017. By theorizing the rate of fluctuation of a Boltzmann Brain constituted by Avogadro’s number of molecules (6 x 10^23), he explores the number of Brains forming in a given timeframe (this is a little too math heavy to be explained fully in detail. If interested, have a read of his paper!).

Also, he compares his estimated number of BBs to his estimated number of ‘ordinary observers’ (OOs). This is our conventional regard of a ‘person,’ born from the aftermath of the big bang (real people like us, or as you may think). He does not limit this to the number of people that have lived on Earth, but holds a larger estimate by assuming a certain probability of intelligent life to form extraterrestrially.

His work is summarized below:

NBB(A) << NOO (A,B) << NBB(B) (8)

NBB(A) : number of BBs in universe A (short lived)

NBB(B) : number of BBs in universe B (long lived)

NOO(A,B): number of OOs (ordinary observers) in both universes

When A (on the left) is defined as a universe with finite time (about the current age of our universe) and B (on the right) is defined as a universe with almost infinite time, Sean Carroll, through the use of order of magnitude calculations, shows that the number of BBs in a short lived universe is close to 0, while the number of BBs in a long lived universe approximates to around e^(10^122) (9). That's e (the base of the natural exponent, approximately 2.7) to the power of ten to the power of 122 - a number with more than a googol of zeros!

NOO is much larger than the number of BBs in a short lived universe but much, much smaller than the number of BBs in a long lived universe (so small that it is practically negligible compared to the number of BBs in a long lived universe).

This means that in a long lived universe, of all the total brains that exist - like yours and mine - the vast majority are Boltzmann Brains.

How Boltzmann Brains would form under each model

Case 1 (closed model): As shown by Sean Carroll, the probability of a BB forming in such a short time is close to 0. Therefore, no problems are raised regarding BBs, or any other similar macroscopic fluctuations.

Case 2 (open model): Since time is infinite, there is always a nonzero probability of a BB forming. However, this condition will make it less likely for BBs to form than the de Sitter model because of the endless stretching of spacetime, making it more difficult for particles to arrange correctly to create BBs. The average distances between molecules will increase and with finite matter, the formation of a BB becomes increasingly difficult.

Case 3 (de Sitter): With infinite time and a finite volume of space in which particles may roam, this is the ideal condition for forming BBs. However, Sean Carroll states that “In a modern cosmological scenario, the temperature of empty de Sitter space is extremely low, much lower than the mass of the lightest massive particles”(10). Furthermore, hadrons (consider them to be your “average” particles), including regular protons and neutrons, will cease to exist. Instead, he states that “The eventual empty universe is therefore dominated by massless particles – photons (the particles of light) and gravitons (suspected particles responsible for gravity)” (11).

However, since it is impossible to create a brain out of massless particles, he claims that the process in creating a Boltzmann Brain will require what is known as "pair production and annihilation processes." When photons gain sufficient energy (by interacting with other particles) they can very briefly split themselves into 2 parts: the average matter (e.g. a proton) and a counterpart of average matter known as antimatter (e.g. an antiproton), in a process called pair production.

Their regular tendency is to interact with each other right after production and reform back into a single photon in pair annihilation. However, with small probability, the newly produced matter and antimatter pair may drift far away from one another, creating the materials needed to produce a brain.

This process would have to be repeated multiple times, assembling the macroscopic brain molecule by molecule. Consequently, this makes the creation of a BB quite difficult but, given the amount of time, still possible.

You may be a Boltzmann Brain

The philosophical part of the argument comes in when we assume that these brains have consciousness just like our own. Although we still do not understand what really creates ‘consciousness,’ we will regard it as the chemical and electrical processes in our brains. With enough fluctuations, not only will instantaneous copies of BBs form, but also ones with the replicated processes that give us thoughts and sensations that provide it with the illusion of ‘life.’

As we discussed in the sections above, according to current predictions, our universe is very likely in a state of producing endless numbers of Boltzmann Brains, and therefore the theoretical number of BBs vastly outnumbers the number of OOs produced. Then how do we know whether or not we are disembodied brains with the fluctuated delusion of thought and sensation as opposed to being an actual intelligent entity capable of thought?

Because statistically, you are most definitely a Boltzmann Brain.

You cannot prove with empirical evidence that you are not a Boltzmann Brain.

Here is my own paraphrased, simplified version of Sean Carroll’s fictional dialogue between 2 hypothetical physicists, S and W, that highlights how the argument is unfalsifiable by observation (12):

W: I am worried that our best cosmological model predicts that the majority of ‘observers’ are Boltzmann Brains.

S: There may be many BBs, but I am not one of them.

W: How do you know?

S: I don’t feel like a disembodied brain. I have arms and legs, and the world around me seems normal.

W: But a Boltzmann Brain is only a single example of what can fluctuate into existence. There may be infinitely many copies of Boltzmann Observers (BOs) just like you - with arms, legs, sitting in an office. Also, it may all be an illusion in a disembodied brain.

S: But I can see that I’m not! For example I can go and get my radio telescope and observe the cosmic background radiation (an evidence for the big bang billions of years ago) and prove that I am a result of the big bang, which wouldn’t be likely if I was a BO.

W: But how do you know that the radiation you are observing entered your telescope as a result of the big bang? It is overwhelmingly likely that such photons too - may have fluctuated into your telescope at the right time of your observation, without any connection to a big bang.

S: Fine okay. Then if that photon is a result of a random fluctuation, it is very unlikely that it would persist over time. So I’ll just wait a while and check my radio telescope again. If you’re right, I shouldn’t be able to observe such photons…. Nope I still see it! I’m not a BO!

W: You still can’t conclude that. In a randomly fluctuating - infinite time - universe, just as most observers are disembodied brains, and almost all observers are Boltzmann Observers with arms and legs, it’s also the case that for all observers with arms, legs... etc. to have witnessed the cosmic background radiation and waited a few seconds to observe it again, they are all random fluctuations as well. No matter what macroscopic constraint you add, it remains overwhelmingly likely that you are a random fluctuation.

S: But everything I know and feel and think about the world is what I would expect if I were an ordinary observer who has arisen in the aftermath of a big bang, and nothing that I perceive is what I would expect if I were a random fluctuation.

W: And in a randomly-fluctuating universe, the overwhelming majority of people who would say exactly that are, as a matter of fact, random fluctuations.

Pretty interesting, huh?

Hold up - A little too early for existential crisis

Hold your existential crisis - there is a weakness in the argument for BBs as explored by Sean Carroll as well, as he calls the BB argument ‘cognitively unstable.’

Essentially he argues that you cannot both believe that you are a Boltzmann Brain and trust your reasoning that you are a Boltzmann Brain. Let me explain why: If you conclude that you are indeed a Boltzmann Brain, then you also have to admit to the fact that your reasoning, thoughts and knowledge are all random fluctuations and not a result of ‘true intelligence’ from an ordinary observer. Therefore, reasonings from such an entity should not be trusted. Essentially, it’s a paradox in which the logic undermines itself.

Even if random fluctuations were to replicate the thoughts and reasonings of a truly intelligent ordinary observer, the number of such intelligent BBs would be overwhelmed by a larger number of random, unintelligent BBs and therefore the probability that we are one of them is infinitely small.


So which side of the argument are you on? Do you still think that you are indeed a Boltzmann brain according to the statistics or are you with Sean Carroll in that the argument is ‘cognitively unstable?’

While I personally hope that Sean Carroll is correct and I am an ordinary observer, these thoughts fill me with excitement that far out there, somewhere out on the edge of space, another Shakespeare may be writing yet another tragedy.


(1) Ananthaswamy, Anil, NewScientist.com.

(2) Following diagrams and visualizations were constructed by the author of this article

(3) Vaibhav Sharma, PhD Physics, Cornell University (2014)

(4) Tsokos Physics for the IB Diploma, Option D Astrophysics, 2015, pg 42.

(5) Tsokos, pg 42.

(6) NASA science, dark matter, dark energy.

(7) Tsokos, pg 43.

(8) Sean Carroll, Why Boltzmann Brains…, 2017, pg 8.

(9) Sean Carroll, pg 9

(10) Sean Carroll, pg 9

(11) Sean Carroll, pg 9

(12) Sean Carroll, pg 16

Works Cited

Ananthaswamy, Anil. "Universes That Spawn 'cosmic Brains' Should Go on the Scrapheap." New Scientist, www.newscientist.com/article/mg23331133-200-universes-that-spawn-cosmic-brains-should-go-on-the-scrapheap/.

Carroll, Sean M. Why Boltzmann Brains are Bad. 2017. California Institute of Technology, PhD dissertation. arxiv.org/pdf/1702.00850.pdf.

"Dark Energy, Dark Matter." Science Mission Directorate | Science, 21 Apr. 2020, science.nasa.gov/astrophysics/focus-areas/what-is-dark-energy.

Tsokos, K. A. Physics for the IB Diploma. Cambridge UP, 2005.

"What is the Scale Factor in Cosmology?" Quora - A Place to Share Knowledge and Better Understand the World, www.quora.com/What-is-the-scale-factor-in-cosmology.

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