The Second Law and the Past Hypothesis

The Second Law tells us that entropy always increases. So while it is far from maximum today, it must have been even smaller in the past. And indeed our best cosmological observations tell us that the deep past was in a very low entropy state.

But what is entropy? The relevant parameter here, Boltzmann entropy, is asso­ciated with a state of knowledge of the “macrostate” of the system (the big-picture properties), not the actual system itself, which is in some particular “microstate”. From what we know about the system (its macrostate-features), we can compute a measure W of the number of different microstates that are compatible with our knowl­edge. The entropy of the macrostate is engraved on Ludwig Boltzmann’s tombstone: S = k log W, where k is fittingly known as Boltzmann’s constant.

Note that the entropy is actually associated with a macrostate (a state of inexact knowledge), not a microstate. If we knew the actual state, there would be only one compatible microstate (itself!), and the entropy would be k log(1) = 0. It is only logically possible to talk about assigning entropy to a microstate if there is some clear rule as to what types of macrostate should be considered in the first place.^[38] Entropy is a measure of how uncertain you are about which microstate the system is really in. The more possible underlying states, the higher the entropy.

Because entropy is only definable in terms of states of knowledge, rather than the one microstate that actually exists, it follows that the Second Law of Thermodynamics cannot be fundamental in its own right.

3.1 Dynamical Explanations?

Looking to dynamics to explain the Second Law initially seems like a hopeless task, because known dynamical laws are time-symmetric, and the Second Law is time-asymmetric (it does not look the same in reverse). True, we have a method for calculating quantum probabilities that also does not look the same in reverse (we never computationally un-collapse a quantum wavefunction), but all the predictions of quantum theory are perfectly time-symmetric. This fact can be shown in the conventional quantum formalism [4], but is more clearly evident if one looks at the manifestly time-symmetric path integral version of quantum theory [5]. These time- symmetric rules evidently cannot be used to explain the time-asymmetric increase in entropy.

But this argument is not ironclad, because one could argue that there might be unknown dynamical laws at work, with a true time-asymmetry (For example, maybe quantum wavefunctions really are collapsing into the future but not into the past, in some deep-level time-asymmetric theory). One could then argue that the Second Law might be some empirical manifestation of this time-asymmetry, resulting from new dynamics still unknown to modern physics.

This position also falls apart—not because we know anything about yet-to-be- discovered dynamical laws, but because we can replicate the entropy-increasing behavior of the Second Law using computer simulations. In these simulations, we use only time-symmetric dynamics, with no possibility of hidden dynamics that we don’t know about. We get to write the simulation programs, after all.

A careful analysis of these simulations [6] makes it clear that the time-asymmetry results from the boundary conditions on the problem, not from the dynamics them­selves. When one starts with a low-entropy state, dynamics almost always takes that state to a higher entropy state, no matter which direction in time the simulation is run.

3.2 Random Explanations?

When trying to explain the macrostate of the early universe without using a dynamical explanation, it might seem that one option would be to resort to randomness, to the equal a priori probability postulate. If all Big Bang microstates are equally probable, this logic goes, then the Big Bang was overwhelmingly likely to be in a high-entropy macrostate (Just as any random drop of water is far more likely to be in the Pacific Ocean than in your sink). Randomness predicts high-entropy.

And yet, we know (from our best observations) that the early universe was clearly a low entropy macrostate! Here, the explanation-from-randomness has failed entirely. This is considered by many physicists to be a great and enduring mystery. Alterna­tively, if one takes the view that random explanations can’t possibly explain anything fundamental, then this mismatch is hardly evidence of anything.

One option at this point is just to hypothesize that the Big Bang macrostate was low entropy and take that as a given. Given this “Past Hypothesis”, one can easily prove the Second Law. But this is even less informative than a random explanation, the equivalent of the annoying: “Because I said so!”. What’s more, one can only assign the “low entropy” status to a macrostate, which is a state of knowledge—and any such rule about our knowledge of the early universe could hardly be a fundamental rule. We want to know why the early universe had such a smooth distribution of matter—we want to know the explanation, and a random explanation doesn’t seem to work.

Another option at this point is to drop back to a different sort of dynamical explanation—using dynamics to explain the Big Bang as a consequence of something in the even-more-distant past, as in the popular “cosmological inflation” models. But as you might imagine, this just shunts the same mystery about the improbable initial state to a different point.

So what might explain the success of the Second Law? The first person to tackle this problem was Boltzmann himself, after he realized that his “proof” of the Second Law had mistakenly included a time-asymmetric assumption. Boltzmann’s instinct then was the same as many physicists today: to forge ahead with “random explana­tions” all the same!

3.3 Random Anthropic Explanations?

With his statistical understanding of the Second Law, Boltzmann knew that it wasn’t an absolute rule. Dynamical processes—with some very low probability—can evolve the actual microstate of the universe into a macrostate with lower entropy. If you wait long enough, he reasoned, anything would eventually happen, no matter how improbable. And high-entropy states can’t support life and consciousness, so we don’t notice the universe until a rare low-entropy moment happens. This is an additional “anthropic explanation” of why we find ourselves in an improbable macrostate: even­tually something like our universe would randomly happen, and we find ourselves here because we can’t exist elsewhere.

Before we broach the serious problems with this account, it’s worth taking a step back to see what such a “random anthropic explanation” amounts to. The only input requirements are randomness and an infinite amount of time (along with dynamical processes that have a non-zero chance of exploring every point in possibility space).

This type of story suffers from precisely the same flaws as “random explanations” in general. They can’t answer “Why this but not that?”, and indeed have to posit “This and that.” (And how could it be otherwise, with no other starting point or principle?) Such reasoning is the antithesis of a fundamental explanation. It’s easy to come up with plenty of more-probable options in such a Boltzmann universe—say, a single planet orbiting a single star in a high-entropy background, randomly created at this very moment (The most probable is the “Boltzmann Brain” scenario, where you are some disembodied brain experiencing one blip of consciousness, before lapsing back into macro-equilbrium).

Boltzmann’s proposal was abandoned, but this general logical thrust—that some­how dynamics and randomness can explain the Second Law—lives on in many other approaches. One recent proposal from Barbour and colleagues [8, 9] notes that essentially any group of gravitationally interacting particles will pass through a “Janus Point” where the coarse-grained macrostate is at lowest entropy. If the entropy of the universe is unbounded, the argument goes, entropy will increase in both time directions from this special point (which would look like the Big Bang, when rescaled). The Second Law would be due to us being on one side of the Janus Point, for any random history of the universe.

It is easy to see that all the critiques to Boltzmann’s proposal apply here as well. In random anthropic reasoning, absolutely anything that can happen, will happen. Furthermore, if a very-coarse-grained Second Law is really coming about from such logic, then it could easily be reversed by the same logic at a finer graining. Taken as a subsystem, the Milky Way and the Andromeda Galaxy are heading for a collision, with its “subsystem Janus point” clearly in the future, not the past—and yet our local Second Law is in disagreement with this argument.

Another group of modern Boltzmannians are using a version of cosmological infla­tion, with a multitude of universes, to try to resolve the improbable-initial-state prob­lem [10, 11]. But almost all of these utilize some type of time-asymmetric/Second- Law-style reasoning. The only hint of a plausible time-neutral solution here would be some variant of a proposal from Carroll and Chen [12]. But even if some serious technical problems [13] are overcome, such an account falls directly into the essential difficulty with random explanations: it would “explain” an infinite number of very different universes, and hence would not really explain anything.^[39]

It is my view that these approaches aren’t merely unpromising or difficult [14, 15], but rather that they’re essentially misguided. Dynamics plus randomness may be popular, but it doesn’t actually work without adding in the Second Law from the start. To explain the Second Law from something fundamental, we need to understand the smooth matter distribution near the Big Bang, and from a thermodynamic perspective this distribution is essentially non-random. Looking to randomness to account for such a situation would be like looking to statistical letter-frequency tables to explain the popularity of George R.R. Martin’s novels.

But what other options do we have? Projecting further into the past would only deepen the explanatory mystery. Dynamical explanations can only explain one state in terms of another, lacking a logical starting point. And once we understand that randomness should be off the table, there’s really only one other type of physi­cal explanation available. The only reasonable path forward is to think in terms of boundaries.

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