# Entropy, time reversibility, and the uncertainty principle

I had a coworker bring up time reversibility during a lunchtime conversation the other day and how physical systems would behave. Sparing the unimportant details of the conversation, his position was that: if you were to break an egg it would unbreak by running the clock backward.

I've never studied this, but my knee-jerk reaction was that it this was incorrect, as it would violate the 2nd law of thermodynamics. My argument was that, although the equations of motions and dynamical laws are symmetric with time-propagations, there are situations in quantum that are not. My counter scenario was this:

Consider a universe which only contains a single hydrogen atom. The atom is initially in an excited state. The bound electron decays and spits out a photon at the characteristic energy. The photon travels for some distance, at which point you freeze time and reverse it. As I understand it, this would effectively be performing a time measurement on the photon, and because of the uncertainty principle, the photon traveling back is no longer guaranteed to be at the same energy. Effectively changing the probability of absorption back into the hydrogen atom, thereby not guaranteeing you can "rebuild the egg" (by egg I mean hydrogen atom).

I've poked around a bit, and it seems like my position is not the correct one, and that entropy would, in fact, decrease with time-reversal.

But I was wondering if anyone would clarify in the context of my scenario?

• I believe that the statistically-based conclusions embodied by the 2nd Law of Thermodynamics are, in spite of its obvious importance, technically secondary to the so-called "fundamental" laws of physics, which deal with fields of fundamental (or irreducible) particles, and perhaps only to those particles which have been found in nature. (Some, which I believe include the neutral K-mesons often involved in discussions of time and symmetry, are only known to occur in the partially-artificial environment of colliders.) Sep 6, 2019 at 17:04

The uncertainty principle isn't quite relevant here. Quantum physics is unitary, which means you can un-do everything by running it backward — provided you include the whole system (not just the egg) in the quantum model, including whatever physical entities are used to make measurements. The uncertainty principle implies that that certain measurements can't happen simultaneously, but it doesn't mean that the state of the physical system is "uncertain."

So even in quantum physics, your coworker is right: the egg would un-break by running the clock backward.

Despite how it looks at first, this does not actually violate the 2nd law of thermodynamics. The 2nd law isn't a statement about what is or isn't possible. It's a statement about what is overwhelmingly likely or overwhelmingly unlikely. When dealing with such huge numbers as the number of molecules in a typical egg, "overwhelmingly unlikely" might as well mean "impossible".

More explicitly, this is what the 2nd law says about un-breaking eggs: If you consider all the different possible microscopically-different states in which egg-material is scattered across the floor, practically none of them will have all of the molecules with just the right posititions and velocities to end up as an intact egg jumping back up into your hand. The key word here is practically. The very fact that you can break an egg implies that some of those states in which egg-material is scattered across the floor do have just the right molecular arrangement to end up as an intact egg jumping back up into your hand.

Why don't we ever see that happening? Because if $$n$$ is the number of microscopically-distinct splattered-egg states that have this property, and if $$N$$ is the total number of microscopically-distinct splattered-egg states including ones that don't have this property, then $$n <<<<<<<<<<<<<<< N. \tag{1}$$ (I probably didn't write nearly enough "$$<$$"s here!) In words, the ratio $$n/N$$ is so insanely close to zero that no matter how hard we try, we will never be able to arrange the splattered-egg molecules in just the right way to see it un-break.

If we take quantum physics into account (as we must, because we're talking about arrangements of molecules), then the situation is even more extreme: when you break an egg, its contents become entangled with the floor, the air, and so on. In effect, the floor and air (etc) have "measured" the contents of the splattered egg by virtue of being influenced by them in a microscopically-complicated way. So not only would you have to get just the right arrangmenet of molecules in the egg material, you'd have to get just the right microscopic configuration of the whole room in order to end up with an intact egg jumping back up into your hand.

So yes, if we could run the clock backward, the egg would un-break (even in quantum physics), but that's because when we run the clock backward, we're starting with a very, very, very special molecular arrangement of splattered-egg material (and floor, and air, etc). But then why is it so easy for us to break an egg? Well, that's because any unbroken egg is itself a very, very, very special molecular arrangement of egg-material! The overwhelming majority of the choices for the positions and velocities of those molecules would not represent an intact egg at all. This, of course, begs the question of how we got such a special configuration of egg-molecules (namely an intact egg) in the first place. And if we pursue that question far enough, we run into unsolved mysteries about the quantum measurement problem and the beginning of the universe.

So I guess you could say that the fact that we never see eggs un-break has its roots in some of the deepest mysteries of the universe.

In quantum theory, the application of Born's rule doesn't represent the occurrence of measurement. Measurement is a physical process in which one part of the system (the egg in this case) influences other parts (floor, air, etc) in such a way that we have no hope of ever un-doing those effects in practice. The occurrence of such a measurement (which is a physical process) is a prerequisite for applying Born's rule (which isn't a physical process). The important point here is that we aren't required to apply Born's rule as soon as the measurement occurs. We can, but we're not required to, because — by the definition of "measurement" — the predicted (or retrodicted) distribution of possible outcomes doesn't depend on how long we wait. The fact that we can defer Born's rule indefinitely is the reason I said that the uncertainty principle isn't relevant here.

Of course, we don't (and can't) experience the whole state that results, accoring to unitary quantum theory, in the wake of a measurement. We only experience part of it (as though it had "collapsed"), and if we discard the rest, then we've sacrificed much of the information that would be needed in order to run the movie backward and un-break the egg. In this sense, egg-breaking is practically irreversible in an even stronger sense than classical physics would suggest. However, to argue that egg-breaking is truly irreversible, we would have to argue that the state really does "collapse" — and then we'd be obliged to speculate about exactly when and how that occurs. The important point here is that this would necessarily involve speculation, because we don't yet have any compelling or broadly-applicable theory of exactly how that process should work.

Thinking carefully about what quantum theory might be missing is a valuable thing to do, and I'm not trying to discourage that at all. I think about it a lot (and I still don't have any novel insights!). However, if we stick with quantum theory as-is rather than speculating about what it might be missing, then I think the appropriate answer is that we could un-break an egg as long as we didn't discard information by applying Born's rule before reversing the movie.

• So the collapse of a wavefunction is symmetric under time reversal?
– Ben
Feb 8, 2019 at 20:33
• Feb 8, 2019 at 20:57
• @Ben That's a deep question, and I don't think it has a completely satisfying answer yet. What we can say is this: nothing in quantum theory says that the wavefunction has to collapse, and although we do eventually need to apply Born's rule in order to make predictions, we can defer application of Born's rule arbitrarily far into the future and still get the same (correct) predictions. In that sense, quantum physics is reversible, because there is no point in time at which we must apply Born's rule; we can always wait a little longer, and everything remains reversible up until then. Feb 8, 2019 at 21:09
• @Ben ...even if we wait until a year after the egg is broken. Feb 8, 2019 at 21:10
• $\def\l{\rlap{<} \hspace{0.5em}} n \l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l\l \hspace{-0.5em} \phantom{<} N \, ?$ =P
– Nat
Feb 8, 2019 at 21:14

tl;dr- It depends on how perfectly you "reverse time"; a perfect reversal would unbreak the egg, while imperfect reversals might not. Any system in which information is conserved, i.e. in which earlier states can be perfectly predicted from later states, can describe a perfect time reversal.

I had a coworker bring up time reversibility during a lunchtime conversation the other day and how physical systems would behave. Sparing the unimportant details of the conversation, his position was that: if you were to break an egg it would unbreak by running the clock backward.

This seems like an issue of semantics more than physics. Basically, what do you mean by "running the clockbackward"?

Fundamentally, time is an ordering construct: it defines some sequence along which things can be placed. If you consider any system that can be said to evolve along some timeline, then reverse that timeline, you should see it go back to its earlier states. Which isn't a statement about physics, but just simply what "reversing time" would mean.

Still, we can construct weaker notions of "reversing time" in which this wouldn't hold.

For example, consider a table that has 100 coins on it, all initially at heads. Every second, someone picks up a coin and flips it. We'd expect that, as time goes to infinity, the state of the table to tend toward 50 coins on heads and 50 on tails – though, over the course of infinity, the table should eventually return to 100 heads arbitrarily many times; just, it won't tend to be the preferred state.

Then, you can define "reversing time" two different ways:

1. Literally rewinding the coin flips perfectly, such that once we arrive at the initial time, all of the coins are necessarily on heads.

2. Rewinding the coin flips by someone watching a video of the original coin flip happening, then doing their best to flip it in the reverse manner.

Same deal with your egg, where we could define "reversing time" in two different ways:

1. Literally rewinding the egg's breaking perfectly, such that once we arrive at the initial time, the egg is whole.

2. Rewinding some of the breaking in a way that we'd consider the reverse, e.g. throwing broken egg pieces back in the general direction from which they came.

So, if you mean "reversing time" in a complete sense, then, yeah, the egg would unbreak. But if you mean "reversing time" in an incomplete sense, where things kinda undergo reversal but not necessarily exactly, then the egg might not unbreak.

As for physics, any physical model in which transforms are fully time-reversible, i.e. in which information isn't destroyed, we can define "reversing time" as the system going to the state that it would've had to progress from to arrive at the current state.

By contrast, any physical model in which information is destroyed would, by definition, not provide a general mechanism to predict an earlier state from viewing the current state. Such a system wouldn't provide a mode for describing time-reversal, though since it would necessarily be an incomplete physics, it wouldn't prohibit it, either.

I've never studied this, but my knee-jerk reaction was that it this was incorrect, as it would violate the 2nd law of thermodynamics.

The second-law of thermodynamics is cool with time reversal.

Specifically, the second-law says that entropy tends to increase as time increases. So if time's decreasing, it's fine for entropy to also decrease. At least, if you're actually reversing time.

• This answer seems very similar to the earlier one, but I can't up-vote it as well, because of the reference to time as an "ordering construct", which makes it sound artificial. At least since the concept of inflation has surpassed the earlier and unadorned version of the Big Bang theory in its potential for explaining observed phenomena plausibly, I haven't heard of any evidence for time's artificiality: Even the field-based version of inflation fails, according to Guth, to exclude any pre-inflationary period possibly eternal to the past. Sep 6, 2019 at 19:36