# Can a broken egg spontaneously reassemble itself (as in the video)?

According to the fluctuation theorem, the second law of thermodynamics is a statistical law. Violations at the micro scale, therefore, certainly have a non-zero probability. However, the application of the theory, in particular the Jarzynski inequality indeed, in principle, extends to systems of particles comprising macro objects.

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Does the fluctuation theorem imply that there is a non-zero probability that a broken egg may spontaneously reassemble itself, in effect implying that (at least some) other laws of physics (such as linear momentum conservation) are in fact statistical statements?

Up to the limits of our theoretical understanding, yes, there is nothing in principle wrong with seeing what happens in the video happen for real in the sense that you can formulate this entire scenario in principle in such theory without running into any contradiction, whereas something like a perpetual motion machine would (namely, that would require physical laws sensitive to the time at which something happens, by the contrapositive of Noether's theorem.).

The actual problem is that it's stupendously super-duper unlikely for it to happen. Generally speaking, the probability for a fluctuation gets exponentially smaller the more atoms you want involved, and here we have on the order of some multiple of Avogadro's number, $$6.022 \times 10^{23}$$, meaning that we can expect probabilities on the order of $$10^{-10^{24}}$$ to see this play out in real life. It is, thus, exceptionally unlikely it will happen before some mechanism - perhaps proton decay - dissolves all matter in the universe.

Depending on your philosophical bent, you could then say that because of that, we cannot empirically verify physical theory to that point, and so maybe it's possible that in fact our theories are wrong on this and something does come up to intervene. But we have no way we can know that to be the case under all possible scenarios for which it could be.

That said, one might also wonder about the possibility of creating this situation artificially, given that one can locally reverse entropy by doing some active work, i.e. expending energy processing information. That might be possible, but getting all the requisite matter into the right state that is exactly the time-reverse of the egg falling and breaking scenario, may be intractable for other reasons. It would require some impressive micro-control over all the movements of atoms to set up the right little "ripples" in the air and board that should all converge "just right" to have that "magic" reassembly power latent within them and "kick the egg back into wholeness".

• Re-creating the chemical bonds that make up the shell aren't just a matter of getting the pieces moving in the right direction. That's why you can't fuse an egg shell back together even by holding the pieces together like a jigsaw puzzle. So it's much harder than "just" macro-scale getting all the pieces to hit together with the reverse of the timing we see for them flying out. – Peter Cordes Mar 16 at 12:25

No, it's not possible.

See, there's a problem with the English word "possible": it's an English word. Even in the best cases it's hard to translate technical, scientific ideas into English sentences. You're asking us to choose between two English words, "possible" and "impossible", to describe an event where a broken egg spontaneously reassembles itself. The only reasonable word to choose is "impossible".

The answers that are saying "well, technically, it's not impossible" are falling into a trap. They're using the word "technically" as if it magically transports us out of the world of English into the divine world of scientific ideas, but it doesn't. We're still speaking English, and of the two English words, "possible" is an awful choice to describe that situation. Yes, the probability of such an event is greater than zero. No, it's not possible.

Let's consider some things that are trillions of trillions of times more likely than an egg spontaneously reassmbling itself:

• William Shakespeare is still alive and has been living in Nashville, Tennessee for the last 150 years.
• Dozens of people actually live in the same house as you and none of you have ever noticed because you just happen to not have been in the same rooms at the same time.
• Everyone on Earth is killed over the next year by independent, freak lightning strikes, even though there were no more lightning storms than usual.

Are those things all possible, according to your definition of the word? All of them are trillions of trillions of times more likely than a broken egg ever spontaneously reassembling itself. The best wording to use for such an event is not "stupendously super-duper unlikely". It's "impossible".

The goal of physics is to understand and describe the universe, and your understanding of the universe will be much worse if you come away thinking "ok, so it is possible" than if you think "no, it is definitely not possible". It is not possible.

• Comments are not for extended discussion; this conversation has been moved to chat. – rob Mar 18 at 1:43

It's possible, but won't happen anywhere within even one universe lifetime, not even close. Physicists often hate saying it is technically possible because the process is so mind bogglingly unlikely, that really it should always be emphasized just how unlikely it is. But yes it is possible.

• I don't think it is even theoretically possible, actually. There are irreversible changes there. For example, the calcium carbonate crystal structure of the shell has been broken and that cannot be fixed without chemical processes being involved. It isn't just statistically unlikely that the molecules involved will adopt the same structure, I don't believe it is physically possible. – terdon Mar 15 at 10:27
• @terdon You're not considering the possibility or those crystal structures spontaneously reforming. Yes, even a single atom of the crystal reforming into the structure is unlikely, but it's not impossible through quantum tunneling. You've fallen prey to what's mentioned in this answer: not understanding just how unlikely even when technically possible. People tend to think about unlikely things in terms of "1 in a million", but these events occurring are more like 1 in 10^(10^(10^10)). – Shufflepants Mar 15 at 12:44
• @terdon Remember that under quantum mechanics, all processes are fully time reversible (technically CPT reversible). The only reason we don't see things like this video actually happen is due to their extreme improbability. – Shufflepants Mar 15 at 12:46
• @Shufflepants I cannot remember what I do not know! :) I'm a biologist, and a theoretical biologist at that, my understanding of quantum mechanics is nonexistent. If what you say is correct, and I am sure it is, and all processes are theoretically reversible under the QM model, then it is indeed a question of unlikely rather than impossible. Thanks! – terdon Mar 15 at 12:50
• There's also a distinction between the possibility of a shattered egg reforming and the possibility of it reforming in the exact sequence shown in the video. The latter is far more unlikely than the former extremely unlikely event. – JimmyJames Mar 15 at 19:23

All these answers that say "yes it is possible ... but very very unlikely" are failing to take into consideration the limits of human knowledge itself. In dealing with something as extraordinary as a broken egg getting reassembled, one misleads oneself if one says either "yes it is possible" or "no it is impossible" because in order to make either claim one would have to assume that what we have discovered so far about physics is sufficiently complete and thorough to justify such an absolute conclusion. But we don't know. For all we know it might be impossible.

A more careful presentation of our state of knowledge would include the following.

1. Physics does not furnish certain knowledge of anything; all we have is knowledge worth betting on.

2. If an event of this kind actually occurred, then what it would show is not that some fantastic coincidence happened in the motions of a mole of particles, but that our knowledge of the nature of reality was less complete than we thought.

• Thanks for the insightful comment. Indeed, the intent of the question is to assess whether or not as far as we know, the current theory of physics will allow such an event to occur. – kbakshi314 Mar 15 at 17:20
• This is exactly what Douglas Adams meant in The Long, Dark Teatime Of The Soul: "The impossible often has a kind of integrity to it which the merely improbable lacks.... The first idea merely supposes that there is something we don't know about, and God knows there are enough of those. The second, however, runs contrary to something fundamental and human which we do know about. We should therefore be very suspicious of it and all its specious rationality." – jez Mar 16 at 15:37

I think a way of looking at the stat mech here is to convert it over to billiards. Take a shot where you launch one ball at two stationary balls. The shot is calibrated just so that the cue ball stops, and the other two balls go off at some angle $$\theta$$ with some speed $$v$$ (we can use conservation of energy and momentum to compute what values $$v$$ and $$\theta$$ have to have relative to the initial velocity, but it's not important to this argument.

A shot like this might be hard, but it's not implausibly impossible. Now, however, imagine trying to set up the reverse situation -- launching two balls at the cue ball, having both of them stop and the cue ball go off with a speed. Again, this is definitely possible, and you could probably even set up an experiment that does this, but it is dramatically harder to do, if just for having to have the timing, speed, and angles of the two balls having to be nearly perfect. An initial state of two moving balls has more relevant degrees of freedom than an initial state of one moving ball.

Now, imagine that rather than two balls flying off, it is every atom in the egg. That's basically why the idea of "this is impossible without being impossible" keeps on coming up in responses to this question.

It is, of course, within the realm of possibility that any of the laws of physics, as we presently understand them, is only true overwhelmingly often, rather than always. However, if we are going to start dealing with an epistemology of absurdly unlikely events, we should also remember David Hume's point that there is no sure knowledge that the laws of nature as we have thus far observed them will continue to hold in the future. No knowledge is certain in science, and that is hardly a novel observation.

However, none of this really has anything to do with the Second Law of Thermodynamics. Fairly early on, it was recognized that there was something that distinguished the Second Law from all other known physical laws; that was that the Second Law describes an irreversible progression into the future, while the other fundamental laws known in the nineteenth century were strictly symmetric under time reversal. In fact, if macroscopic behavior was supposed to be an emergent property, arising out of the behavior of large numbers of microscopic particles interacting in a time-reversal-invariant fashion, it was initially hard to see how the the Second Law, with its fundamental irreversibility, could possibly be correct. Yet empirically, it clearly was correct. It was this fundamental problem that prompted people to look more deeply at the meaning of entropy and the foundation of thermodynamics.

The correct interpretation of the Second Law of Thermodynamics—that it is overwhelmingly likely to be true for macroscopic systems—might lead one to wonder whether some other physical laws might also be merely statistical statements. In fact, there is some interest in this idea. We know that some forces, like the elastic force in rubber (or, to take an even simpler example, the pressure on the wall of a gas-filled container), is a product of entropy and is thus not absolute but merely overwhelmingly likely to have the value predicted by the equation of state. There is interest in applying this same kind of reasoning to other forces, which are currently seen as fundamental, like gravity. So, the ultimate answer to your question is, yes, the possibility that some of the fundamental laws of nature are only statistically valid is a serious possibility that is entertained by a fair number of physicists.

However, there is still a key difference. Unlike the Second Law of Thermodynamics, which was obviously different from (and appeared to be in conflict with) all other known physical laws, there is no a priori reason to expect that gravity needs to be explained as a statistical, rather than fundamental, force.

• Is the purport of your answer and it's contents pointing toward an information theoretic framework for the fundamental laws of the world? If so, these laws too have variational principles complete with Hamiltonian and Lagrangian formulations which are deterministic. The natural interpretation being the fundamental laws are still deterministic in this case and not statistical (it is just that we're looking at the non-fundamental ones which are statistical). – kbakshi314 Mar 15 at 1:58
• @kb314 It could be either. – Buzz Mar 15 at 2:00
• Thanks for the comment and introducing me to the entropic forces. It seems to me then that although not apriori evident, gravity may indeed be a statistical phenomenon. – kbakshi314 Mar 15 at 2:02

To complete, @The_Sympathizer, you can calculate (or, better said, estimate) the entropy with Boltzmann formula $$S=k_B\log N$$, with $$N$$ the number of states. From the initial (i) state to the the broken final state (f), the change in entropy is $$\Delta S=k_B\log N_f/N_i$$ So, $$N_f=N_ie^{\Delta S/k_B}$$ and the probability of the transition from initial to final state can be calculated as $$p_{i\to f}=N_f/(N_f+N_i)\approx 1$$, since there're many more possible microstates in the final configuration. $$p_{f\to i}=1-p_{i\to f}\approx\exp(-\Delta S/k_B)$$ Note that the increase of entropy is a high number, but the Boltzmann constant, $$k_B$$, also adds $$10^{23}$$, so the probability is tremendously low.

I want to establish some baseline definitions. As some have pointed out, english is not very precise.

Definition

1. Possible - A non-zero probability value.
2. Impossible - Probability=0 exactly.
3. Never going to happen, ever - Probability is less than $$10^{-15}$$ (about 1 in 1 quadrillion odds)

Egg spontaneously reassembling itself has a non-zero probability, but I feel pretty safe in saying that the probability is strictly less than $$10^{-15}$$ without doing any fancy calculations.

Conclusion: Possible, but never going to happen, ever.

• Zero probability is not the same as impossible. – Sandejo Mar 17 at 0:15
• For example, if you choose a random point on a plane. Then the probability of a single point to be selected, is zero. But the selection of any point is possible. But I feel some... noice in this. I think the probability of a point to be selected, should not be zero, but $\frac{1}{|\aleph_1|}$. Which is zero in all practical senses. Afaik higher math can somehow handle the difference, but not with the ordinary real numbers. – peterh Mar 17 at 14:08
• Technically, I was affixing my own definition to an English word. Therefore, that definition should be considered axiomatic to the discussion that follows. Furthermore, I feel justified that my definition of the word "impossible" is a reasonable definition given that I was trying to emphasize the difference between "impossible" and "you will have to wait long past the heat death of the universe to ever see it happen". – Ryan Mar 17 at 22:25
• That being said in the strictest mathematical sense in a universe of possibilities, i.e. a set of outcomes, it may contain elements that have zero probability, thus they would be technically possible because they are in the set, but never occur because the probability is zero. Which is what I think you are getting at when you say zero probability is not the same as impossible. – Ryan Mar 17 at 22:26
• And that would mean that we are affixing a different definition to the english word impossible than the one that I supplied, which would, of course, change what I wrote. Again the definitions that I supplied for those words were designed to differentiate almost zero probability from zero probability in the reader's mind. Since we, as humans, are bad at reasoning with very small probabilities. – Ryan Mar 17 at 22:35

There is no way this could happen. Statistical thermodynamics can be applied to free particles (atoms, molecules, but elementary particles are somehow never involved), some chemical interactions (by introducing the chemical potential), but not to complex interactions in which a complex compound of a high variety of bound structures is involved, like in a breaking (reassembling) egg. I was once told too that if you wait long enough, an egg can appear. How stupid I was to believe this!
It is said that the time-reversed breaking of an egg, the spontaneous assemblage of an egg, will for sure occur if we wait long enough. But:
There are phonons released in the process if the egg appears on Earth (or on a planet with a variety of different stuff) all those phonons (sound quanta) must be involved in the reversal of an breaking egg, as well as the photons coming from it. And even a tiny amount of gravitons (gravitational waves). These must be produced by some statistical process. But how will these photons and gravitons be produced in a statistical process? They will not be. That's just impossible. It's even easier to imagine that the right phonon combination can occur spontaneously thereby materializing the time reversed sound coming from the breaking of an egg. This might occur in the atmosphere.
What about the way the egg is broken? I mean, there is a cause for the breaking. It can fall onto a hard surface, it can be hit by a bullet or an airplane, or be crushed in the hand of my strong granny (when she makes me pancakes). This has to be considered too. In fact, everything surrounding the egg has to be considered, including the whole history of the universe (all interacting matter and all developing spacetime, though maybe a Hubble volume will do...).
So the answer most definitely is no.
Stephen Hawking once thought that all physical processes would reverse their direction in time when the universe starts to contract, but he, later on, saw he was mistaken, so no magician will ever live to accidentally make an egg appear...
Can we create a time-reversed breaking egg? Obviously, only in a video in which the process seems easy to occur. But for angg to appear there will always have to be a chicken (if it's a chicken egg) involved.
If the egg contains a little chicken then the spontaneous creation of life could occur (Boltzmann brains). Luckily, it can't.Why luckily? I rather have parents, ancestors, and more down the line. It would be very lonely!

There is something I miss in the other answers: they do not take into account that the egg did actually spontaneously assemble itself in the first place. However, it did not assemble in a way that looks anywhere close to what the video shows, but something that involves chicken, reproductive organs, nutrition, etc.

This is the point where most will probably raise their hands and say: yes, of course, but this was only possible because the chicken that produced the egg was far from thermodynamic equilibrium. But does this really explain why the chicken was able to produce an egg? If it did, one could also rightfully argue, that it is only a matter of creating the right non-equilibrium boundary conditions in order to see something that looks like the process in the video. Experience shows, however, that the latter is not true. Hence, non-equilibrium thermodynamics does not explain anything about structure formation. Instead, it is just a wordy form of "we don't know" combined with the belief that things work out somehow, if physics laws are applied consequentially.

If non-equilibrium cannot explain structure formation, it also cannot explain why an egg being created from a chickens reproductive organs has a higher probability than an whole egg being created from scrambled eggs. It also can't explain why we don't ever see an egg return to its mother's womb, shrink and eventually vanish, followed by a chicken running backwards with a reverse cackling. Notice, that the latter scene does not at all involve any obvious form of undone statistical damage. And yet it is not observed.

It all boils down to the fact that we don't know at all why certain world paths are "allowed" when moving forward in time while they are not allowed when played in reverse. Even when taking into account that the standard model of elementary particle physics is violating time reversal invariance (only CPT is invariant), this won't explain why chicken lay eggs. After all, a chicken and egg is not a gas.

Put it this way. According to quantum mechanics, yes, it could happen. But something else that could happen is that the constituent goop rearranges itself into two eggs of half the volume. And that's overwhelmingly more likely. Still ridiculously unlikely, but way more likely than going back to a single egg.

Why? Because if there are $$n$$ molecules of shell then there are about $$2^n$$ ways to split them into two halves, and all of those possibilities count. The same goes for the "yolk molecules" and so on (yes, I know, but this is physics and the biology details are irrelevant).

Moreover, the two-egg probability is the same if you don't break the egg first. It's much more likely that an egg, just sitting there minding its own business, will spontaneously split into two perfectly formed smaller eggs than that a broken egg will transform itself back into an egg again.

A similar scenario is where people talk about a person diffracting through a doorway or quantum tunneling through a wall. Yes, it could happen, but another outcome which is overwhelmingly more likely is that half the person diffracts or tunnels. Luckily, this too is ridiculously unlikely.

There's a decent chance that all forces are statistical in nature. The odd-ball case is gravity, but see Entropic Gravity. At the moment, the smart money is that:

1. Conservation laws are exact.
2. The standard forces (weak, strong, electric) are caused by particle interactions in quantum field theory and are therefore statistical in nature.
3. Gravity might or might not be statistical in nature, and until we have a decent theory of quantum gravity it's too early to be sure either way.

Waves always roll onto the shore (rather than, say, along the shore-line) because bending towards the shore is the quickest path, when you take into account that waves travel slower in shallow water. Each water molecule can do what it likes, but in bulk, waves always bend towards the shore because all the other possibilities cancel out.

Similarly, light only appears goes in a straight line because when you observe it, it turns out that it went in a straight line. Not because it had to, but because not going in a straight line is much less likely due to the various neighboring paths interfering. If the speed varies in different materials, light takes a shortest path. This is refraction.

All these laws are statistical in nature.

• I'd like to understand the downvote. If someone believes that what I wrote was incorrect, irrelevant to the question, or something else, then I'd like to know so that I can correct it. Thank you. – Adam Chalcraft Mar 18 at 6:37

I think there is two ways to interpret this question. The way I interpret it, is whether the shell can reassemble itself in a way that the old shell is indistinguishable from the new reassembled shell even on the atomic level. Emphasis is on the indistinguishability on the atomic level. This is impossible.

The collapse of the wavefunction is generally attributed to decoherence. This is time asymmetric in the same way the second law of thermodynamics is time asymmetric. I suppose it's theoretically possible for a wavefunction to uncollapse, but this is like saying it's theoretically possible for a broken egg to reassemble itself.

Is the collapse of the wave function inherently time asymmetric?

The reason is the very thing that holds the shell together, covalent bonding. Covalent bonding is a QM phenomenon. Contrary to popular belief, QM is not always time symmetric. now once the covalent bondings are broken (decoherence happens and information leaks to the environment), there is no way to rebuild them the same exact way (because this process is time asymmetric).

Explanation of covalent bond from physics point of view?

So the answer to your question is, that the shell might reassamble itself (very unlikely), but it won't be the same shell, and the reassembled shell and the old shell will be distinguishable on the atomic level.

Something is off about the answers that say it is possible but unlikely. They all seem to assume that the universe evolves probabilistically, otherwise what does "unlikely" mean? But as of now there is no evidence for that. And attempting to make such claims about "likelihood" would make it clear that in such eggsplicitly ridiculous scenarios they are actually ill-defined.

Not only is there a lack of justification for those answers, we can actually somewhat justify the claim that it is simply impossible. The sheer amount of energy that is required to achieve a state (within the reachable future of this universe) that evolves as shown (all the molecules of the egg coming together and re-bonding and pushing the trap hammer backwards with such precise force that it resets itself...) is simply too big to be possible as a matter of 'chance', simply because the universe doesn't try out all possible states. And nobody can hope to set such a state up.

Ultimately the point is: Unless you can convincingly show that the universe evolves probabilistically and can reach all logically possible states from the current one, you cannot talk about probability of some incredible event occurring. And is there any reason to believe that the universe would naturally evolve to a state from which a broken egg would reassemble itself? No. So one's default belief should be that it is impossible, not just unlikely. This belief may be false, but it is better than belief in ill-defined probabilities...

(Note that quantum mechanics does not save the probabilistic evolution idea, since any hypothesis on things like wavefunction collapse are nothing more than speculation, and there is as yet no compelling explanation that requires the universe to have true randomness.)

• Regarding non-determinism, you say "as of now there is no evidence for that." The evidence is in (1) exponential sensitivity in classical dynamics, together with a possible incoherence in the notion that anything is ever perfectly precise in the natural world; (2) quantum physics, where a non-deterministic interpretation is logical, as simple as any other, and matches the empirical data; (3) some issues in epistemology which I won't go into . This does not amount to a proof, but to claim "there is no evidence" is simply bizarre. – Andrew Steane Mar 16 at 10:54
• @AndrewSteane: I have no idea why on earth you think that exponential sensitivity in anything provides any shred of evidence at all. Have you not heard of the logistic map? Concerning quantum mechanics, all the experts failed to provide any coherent explanation of purported decoherence, which is needed for a non-deterministic evolution to be even slightly sensible. It is illogical to claim that it is logical if you cannot provide a single explanation. And, epistomology, of all things?! Totally ridiculous, for reasons I don't even need to go into. – user21820 Mar 16 at 13:52
• 1. The point about the amplifying effect of classical chaos is that a very small lack of precision is sufficient to have a significant impact. If one assumes that the concept of a logistic map holds to infinite precision then one has assumed as a premise the very thing which is under discussion. 2. When one listens to the clicking of a Geiger counter or similar, one hears clicks at some times and not others, and quantum physics does not determine when the times will be. This is true whether or not one brings in ideas such as multiverse. – Andrew Steane Mar 16 at 14:08
• @AndrewSteane: (1) No, we are not at all talking about amplifying a lack of precision; we are talking about underlying reality. You are making a very common basic logical fallacy by conflating between scientific hypotheses and humans' approximate models of reality and reality itself. There is absolutely no evidence of randomness in reality. (2) You can choose to approximately model the geiger counter clicks using a statistical model based on probability, but nothing in quantum mechanics even yields that model in the first place, without some inexplicable 'collapse'. – user21820 Mar 16 at 14:21
• Indeed it is the underlying reality which is under discussion. You think it is deterministic and you interpret the evidence accordingly. But the evidence is also consistent with a mapping from past to future which is not 1:1, in both classical and quantum case. Suppose a Geiger click occurs just as my watch reads noon. Quantum physics implies that nothing beforehand is able to constrain the evolution so as to guarantee that my experience will be, when noon arrives, that I hear a click. To assert that some other version of me hears a click at some other time does not change this. – Andrew Steane Mar 16 at 14:40

It's impossible to observe such a thing in Nature. I will quote from the book of Landau and Lifshitz on statistical physics and maybe advise people to read this book if they need to revise these things:

''In speaking of the ''most probable'' consequence, we must remember that in reality the probability of transition to states of higher entropy is so enormous in comparison with any appreciable decrease in entropy that in practice the latter can never be observed in Nature''

From Section 8: The law of increase of entropy

Actually, even on a 'technically not impossible' level... I don't think this is possible, even under the '.0000000000000000000000000000000000000000000000001% probability' type of stuff... I don't think it has any probability at all. I think it's 0 probability exactly (without some sort of magic force not indicated in the video)

So... first of all... gravity. At the end of the video, the egg spontaneously gains a huge amount of kinetic energy and flies upward. In real life, this energy was initially dissipated on impact, but gravity was still pulling the egg downwards after impact. The egg sped up towards impact. In the fake world, the gravity would pull upwards instead of downwards, so the the part where only the bottom is cracked kinda makes sense... maybe... But then the egg gets a huge boost of speed, which the table simply could not give it, and then slows down... so that doesn't track with upward-facing gravity. So maybe gravity pulls downward in Fake World like it does in the real world. But then the slow sag when the bottom of the egg is cracked would be there, rather than the slow raise that you see in Fake World.

Then there's the spring. There's nothing pushing the spring backwards in Fake World. You may claim that it's the egg, but the egg isn't pushing on it when it begins resetting.

Then of course there's the other forces inside the egg: momentum, cohesiveness, adhesiveness, internal pressure, a million other factors.

So... because of that, even if all the pieces spontaneously had the exact same state (position and velocity) as they do in the beginning of the Fake World snapshot, it would not end in the same state.

Can any one state in the image spontaneously exist? Sure. Maybe. Again, probably with magic, but whatever. Any one state in the image has an extremely small probability of existing, but we wouldn't be surprised by any one state existing (Every one of those states has indeed existed at least once! Well... the reverse of each state. But still.), and could probably be recreated. And the next ten nanosecond states would probably exist in a reasonable approximation to the original, but would not be precisely exactly the same as the original. But if you set it all up exactly the same way as the end state, but in reverse, with 0 randomness, you would not end up with a whole egg flying upwards at the end of it.

Now... can a broken egg become a whole egg 'spontaneously' (NOT like in the image). Sure. Maybe... Define 'spontaneous'? Foreseeable-future technology probably has a way to glue an egg back together with egg-like substance and make it look like it was never broken in the first place. Food replicators. Do these count as spontaneous? I think so. It's the series of random events that made humans get smart and start inventing things and become humans. That's spontaneous. Personally, I think every thing humans do is as about as natural as termites destroying a house or a tree. It's all the result of a big chain of random events.

Can a broken egg become a whole egg in mid-air without anything touching it (or without anything touching it besides what's in that picture), without magic or nanobots or metal inside the egg and a magnet outside or some other unseen force. No. The events of that image are impossible without some unseen/unknown force.