Saturday, October 23, 2010

Dreaming Boltzmann Brains

In a previous post, I discussed the proposition that all the world is a coherent dream.  This is basically the ur-example of pointless philosophical exercises, but I promised that it has applications in cosmology, to be explained now.

Let's take a closer look at the Second Law of Thermodynamics.  The Second Law says that the universe tends towards disorder as time progresses.  More precisely, it tends towards a state of higher entropy, where "entropy" is a precise measure of disorder.

In any basic statistical mechanics class, they'll tell you that the Second Law has perfectly logical justifications.

First we need to understand that every physical system has a large number of possible configurations, called "microstates."  Some of those microstates look more or less the same.  For example, it doesn't matter if a particular molecule on my nose is moving to the left or to the right, my nose still looks the same to me.  On the other hand, there are some microstates which look different.  If the right half of my nose has significantly more energy than the left half, I can tell that one side of my nose is cold and the other is warm.

A "macrostate" is a collection of microstates that all look more or less the same.  Some macrostates consist of more microstates than others.  Entropy is a measure of how many microstates there are in the current macrostate.

The Second Law rests on the Ergodic Hypothesis, which says that all microstates are equally likely in the long run.  Therefore, in the long run, we're more likely to be in macrostates consisting of more microstates.  In other words, in the long run, we're more likely to be in a state of high entropy.  That's why entropy increases as time progresses.

But there's a problem with this "perfectly logical" justification of the Second Law.  The Ergodic Hypothesis and the Second Law make very different predictions.

According to the Second Law, the universe starts at some low entropy state, and entropy increases throughout time.  Eventually, the universe reaches the maximum entropy state (called the "heat death" of the universe).  This does not agree with the Ergodic Hypothesis, which says that the universe would be in the maximum entropy state to begin with.

But hold on!  The Ergodic Hypothesis doesn't say that the universe has to be in the maximum entropy state.  It just says that all microstates are, in the long run, equally likely.  At least some of those microstates have lower entropy.  Therefore, the universe will occasionally fluctuate into lower entropy states.  Perhaps such a fluctuation accounts for the observations we see now.

Such a fluctuation seems extremely unlikely, but we could argue, by the Anthropic Principle, that we just weren't around to see the vast majority of the timeline in which there was no large fluctuation.

Let's try out our new hypothesis, which is the Ergodic Hypothesis plus fluctuations.  The first thing we want to figure out is how large the fluctuation was.

 The above plot is a comparison of three different fluctuation sizes.  The fluctuation could have been big enough to account for the entire universe.  Or it could have been just barely big enough to account for our present observations.

The probability of a fluctuation decreases exponentially as the size of the fluctuation gets larger. So the most likely explanation for our current observations is that the fluctuation was just big enough, and no bigger.  Any signs that the universe used to have lower entropy are illusory.  Taking this conclusion to the extreme, I must be a brain that spontaneously assembled itself in such a way that it just thinks that it lives in an ordered universe with things like blogs and the internet.  This brain is known as a Boltzmann Brain.

We can draw a comparison between Boltzmann Brains and the dream hypothesis.  If the world were just a dream, then most likely, it is a dream which is just coherent enough to explain our observations.  If the world were just a statistical fluctuation, then most likely, it is a fluctuation just large enough to explain our observations.

The problem is that this makes very strange predictions which are perpetually falsified.  The dream hypothesis predicts that as we see more of the world, it will no longer be coherent.  The Boltzmann Brain hypothesis predicts that only part of the universe is ordered: the part we currently occupy.  As we look out to the sky, to a part of the universe that we have never seen before, we would predict it to be in a maximum entropy state.  But everywhere we look, we see stars that have not yet burnt out.

When we consider the dream hypothesis, something is not right about it.  So we reject the dream hypothesis.  When we consider the Ergodic Hypothesis, something is also not right.  But we need the Ergodic Hypothesis!  It explains the Second Law of Thermodynamics.

Cosmologists understand that the Second Law arises from two principles.  First, all accessible microstates are about equally likely.  Second, the universe has an initial condition which is highly ordered.  But why?  Is a large fluctuation not as unlikely as it appears?  Perhaps there is no maximum entropy state, and new baby universes are formed repeatedly throughout history?  It's an open question in cosmology.


Larry Hamelin said...

Correct me if I'm wrong, but it's interesting that we seem to be able to actually calculate levels of entropy in the past by observing the present, even though the thermodynamic laws are themselves time-symmetric (i.e. there are more higher entropy states than lower entropy states that could have had a small fluctuation to result in any given present state).

It seems that past lower entropy states leave some "clues" in the present.

But of course I'm not a physicist, so I may be completely misguided here.

Anonymous said...

The issue of Boltzmann Brain is resolved: