Are You a Boltzmann Brain?

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Matt O'Dowd (1:03)

This episode is supported by The Great Courses Plus. Statistically speaking, you are a disembodied brain. You just came into being randomly out of the chaotic void. Your memory of your entire life also just came into being through a chance arrangement of particles. That includes your memory of that last statement, and that one, and that one.

You, my friend, are a Boltzmann brain. Statistically speaking.

Ludwig Boltzmann revolutionized physics in the latter half of the 19th century. His most incredible insight was the kinetic theory of gases. He showed that the laws of thermodynamics can be explained by thinking of gas as a collection of microscopic particles in constant random motion.

Take entropy, for example. Prior to Boltzmann, entropy was only understood as a measure of the proportion of energy in a system that can be used for useful work. For example, the hot, dense gas in a recently fired engine piston only expands because it's surrounded by a colder, less dense environment. The energy in that burning gasoline does the useful work of driving your car down the street. But if you put the compressed piston in an environment that's full of equally hot, dense gas, nothing happens. The gas in the piston still contains the same amount of energy, but it's useless energy. Why? Because in the latter case, the gas inside the piston is in thermal equilibrium with the gas outside the piston. Everything the same temperature and pressure, perfectly mixed. Entropy is a measure of how far from equilibrium a system is. The lower the entropy, the further from equilibrium. Any part of the universe left to its own devices always tends to flow back to equilibrium. That's the second law of thermodynamics right there. Entropy must always increase in a closed system. And it's that flow back to equilibrium, that increase in entropy, that can be harnessed to do work.

But why does the universe have to return to equilibrium? Why does entropy always increase? Boltzmann figured this out. Entropy is just a measure of the specialness or the degree of order in the current arrangement of positions and velocities of a system's particles. It's a proportion of possible states that are indistinguishable from the current state. Instead of an engine, imagine a room full of air. The air molecules are moving randomly. And over time, they pass through all possible arrangements that they could have. In most of those random arrangements, the room is filled pretty evenly. And we can't tell those arrangements apart from each other. That's a high entropy situation, because a high proportion of all possible states look just like this one.

But in a tiny fraction of those possible distributions, all of the air molecules end up randomly bunched together in one corner. Or, I don't know, produce a density wave playing the ballad of serenity over and over.

If you count up all the possible arrangements of particles, only a tiny proportion do weird, highly ordered stuff like that. So they would be a low entropy situation. Entropy increases because particle positions and velocities get randomized over time. Boltzmann's interpretation of entropy leads to a prediction that seems innocuous but has some astounding implications. His statistical interpretation doesn't prohibit entropy from decreasing. In fact, it allows it. For example, tiny localized dips in entropy happen all the time when you get a chance convergence of a few particles in one corner of the room. The larger the random dip in entropy, the less probable it is. But improbable isn't impossible. There's an incredibly tiny chance that all of the particles in a room of gas will happen to all end up in one corner of the room due to their random motion. It would take vastly longer than the age of the universe for it to happen. So in practice, we never observe the second law of thermodynamics being broken on macroscopic scales. But given infinite time, any non-impossible arrangement will happen. Imagine an infinitely large room. A universe that's in perfect thermal equilibrium for infinite time. All sorts of dips in entropy will happen. Particles will occasionally converge into a dense environment like a black hole or a galaxy.