# We’re All Just Boltzmann Brains, Living in a Boltzmann Universe: A Crash Course in Statistical Mechanics

By Mike Sha

Let’s take a quick trip through the groundbreaking theory of statistical mechanics and use it to smash our current understanding of the universe to pieces. Around the late 1800s, Ludwig Boltzmann (Figure 2) started thinking about order, disorder, and atoms, a line of thought that led him to become the father of diesel engines, refrigerators, electric generators, and anything involving heat1.  Boltzmann showed that starting from the simple assumption that all possible configurations of particles (microstates) are equally likely, the logical consequences are beautiful laws of physics. With this assumption, we can answer perennial questions like, such as why do hot things transfer heat to cold things? In principle, we could count the number of ways to arrange the atoms in a physical system (symbolically denoted as Ω). The typical explanation of “what is temperature?” is a little more involved than “random movement of particles”. In fact, there is a precise mathematical definition (Figure 1) which results in the same definition as measuring the expansion of mercury in a glass tube. Boltzmann’s answer to the heat transfer question above is that there are more configurations for two systems of medium temperature than there are for one hot system and one cold system.

In the equation for temperature above in Figure 1, U is the energy of the system and S is entropy, a measure of disorder. In fact, the definition of S (entropy) is on Boltzmann’s gravestone, where Boltzmann used W instead of the modern Ω (Figure 2).

Considering this equation, temperature is then: how much does changing the disorder of a system change its energy? This begs the question: Why is entropy important?

The second law of thermodynamics states that total entropy always tends to increase with time. To wrap your head around this concept, let’s picture a bookshelf. There are only so many ways you can arrange your books so that it looks neat. If I rip out some pages, pour water on the books, set the bookshelf on fire, or perform many other destructive actions, I can make a bookshelf look disordered. The second law is simply a consequence of the statement that all configurations are equally likely: each way the bookshelf configuration can evolve to look neat is equally likely as each of the tons of ways the bookshelf can be destroyed. Directly counting these configurations gives you the exponential of the entropy (which is on Boltzmann’s gravestone). Notably, there is also a maximum entropy: when the bookshelf is ashes and dust and cannot become more disordered.

Let’s take this discussion to the logical extreme: in the beginning, the universe was super small. This means that it was super ordered! There are only so many ways you can pack all the particles in a tiny space, like a crystal. Then, the universe became really big. Entropy on average always increases, and entropy is a form of energy. Going back to the definition of temperature, if you rearrange the equation a bit, you’ll see that dU=TdS: the change in internal energy is equal to the temperature times the change in entropy. This includes the caveat that the entropic energy in the system always increases, so other changes to U (energy), such as mechanical work, chemical bonds, and nuclear binding energies, must on average decrease by conservation of energy. So, the universe starts off with no energy in entropy and all the energy packed into particles, and as the universe gets bigger, more energy can be dumped from those other sources (chemical bonds, kinetic energy, gravitational energy) into entropy. From this bird’s eye view of the timeline of our universe, we get the prediction that the universe will eventually undergo “heat death” when no more energy can be dumped into entropy as the universe reaches maximum disorder.

Let’s fast forward past the heat death of the universe. There’s nothing now. Not even black holes. For 10^(1050) years.

What was that? A flash of light? A random collision of particles in the endless void? It turns out that quantum mechanics lets conservation of energy be violated occasionally, though not indefinitely. Imagine in the far future, suddenly an entire brain pops into existence that has the exact same configuration as your brain2. It has all your memories up to this point in your life. Of course, it dies immediately in the vacuum of space, but at these timescales, let’s say the brain is magically (statistically) kept alive. It’s way more likely that this brain appeared in isolation than that the entire universe appeared as it does today, out of the void. Stop and think about that. Our universe, statistically, is extremely unlikely.  Moreover, calculations show it is even more likely for these brains to appear than for the universe to have started out as that tiny nugget 13.7 billion years ago, in an infinite universe. The final prediction is that we expect the number of these Boltzmann brains to outnumber human brains by an infinite number. Thus, it’s infinitely more likely for us to be in the universe’s brains-popping-into-existence era than for us to be so close to the beginning of the universe. You are literally more likely to be a Boltzmann brain drifting through the void than you are to be a human being.

So why do humans exist now, so soon after the Big Bang?

It’s more than a philosophical question, it’s a scientific question because all of these assumptions lead to calculable, observable results. On the same order of timescales as the appearance of Boltzmann brains, entire observable universes will quantum fluctuate into existence, and infinitely many of them (if dark energy is not expected to rip apart the universe, which it currently isn’t3). Following our beliefs about the universe to their ultimate logical conclusions is difficult, but our trust in logic requires that someone thinks about these issues.

I hope your worldview has changed a little after reading this article:

Further reading includes Wikipedia’s timeline of the far future.

References:

1. Reif, F. (1965). Fundamentals of Statistical and Thermal Physics. McGraw.
2. https://en.wikipedia.org/wiki/Boltzmann_brain
3. Vikhlinin, A.; Kravtsov, A.V.; Burenin, R.A.; et al. (2009). “Chandra Cluster Cosmology Project III: Cosmological Parameter Constraints”. The Astrophysical Journal. 692 (2): 1060–1074.