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Background (and much of the argument for the question)

The second law of thermodynamics says(as my book states it):

If a process occurs in a closed system, the entropy of the system increases for irreversible processes and remains constant for a reversible process. It never decreases.

Now while looking at some educational videos on YouTube I found that entropy is sometimes related to the amount of disorder in the system. They say this as follows (not the exact statement) :

There are many more disordered states than ordered states and therefore it is much more likely for the entropy to increase or remain the same. Also it is not necessary that entropy cannot decrease, rather entropy in some part of the system do can decrease but the only but only necessity is that increase in entropy in some other part of the system should compensate for it so that there is a net increase in the entropy of the whole system.

An example of this is that of the formation of crystals where the crystals do become ordered (and hence decrease the entropy) but the heat released in due process compensate for this and increases the total entropy of the system by the following formula:

$$\Delta S = \int_i^f \frac {dQ}{T}$$

Question

It is usually introduced to the concept of entropy by giving the example of breaking of egg. Now the following bothers me:

  • Is it possible to revert back the broken egg into the original state by whatsoever process that can do it? Given that the net entropy of such a system increases.

  • If possible, then what is that process that can revert back a broken egg into it's original state?

Note that example of video of the egg being rewinded is not in the list of possible answers.

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    $\begingroup$ Non-fertilised or fertilised? In the latter case it requires a strike of lightning! What I mean to say is that you could have taken a simpler example. $\endgroup$ – my2cts Feb 8 at 17:22
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    $\begingroup$ You should make up your mind, if you're asking about thermodynamics, or magical egg repairing... $\endgroup$ – Mithoron Feb 8 at 23:55
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    $\begingroup$ Grind the egg up and mix it in with your chickens' feed? $\endgroup$ – user253751 Feb 9 at 12:51
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    $\begingroup$ The process to restore the egg would be some extremely involved high-tech egg surgery. There's nothing magical about it, you'd just have to do exactly what you'd imagine having to do in order to repair a broken egg. What thermodynamics says is that no matter how you do this egg repair and reconstruction, there's a certain amount of energy that you can't avoid spending. $\endgroup$ – Nathaniel Feb 9 at 13:08
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    $\begingroup$ @JohanLiebert I think the second bullet made it a lot more interesting! $\endgroup$ – Superfast Jellyfish Feb 9 at 19:49
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Theoretically, it is possible, at least if by 'original state' you mean 'macroscopically identical' - if you want the microscopic state to be identical, you encounter a problem, that it is impossible to precisely measure the microscopic state, especially after it was altered, so the 'original state' is unknown.

However, practically, we don't have the technological capabilities to merge all the pieces of the eggshell or to fix the organic membranes and separate the mixed contents of the egg.

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    $\begingroup$ Feed it to a chicken. $\endgroup$ – Joshua Feb 9 at 16:27
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    $\begingroup$ @Joshua That's actually a good point. Living organisms are notoriously good at reducing their local entropy (at the cost of a large increase in external energy, of course). That's one of life's defining features. $\endgroup$ – gardenhead Feb 9 at 19:41
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    $\begingroup$ @descheleschilder Emitted photons are not a part of the original egg, as you say it yourself, they were produced in the process of breaking. We allow interaction and manipulation, which would include energy transfer from external sources, so we don't need these photons. We only use the original matter, but the energy used to put it back together is a different story. $\endgroup$ – Adam Latosiński Feb 9 at 20:37
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    $\begingroup$ @Joshua I was thinking about the chicken solution, but this method doesn't produce the egg in the same state as the original one. It produces another very similar egg, but there will be differences. $\endgroup$ – Adam Latosiński Feb 9 at 20:40
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    $\begingroup$ @AdamLatosiński All the energy released (photons) have their origin in the egg. So the energy taken away by these photons was part off the egg. You need these photons because other photons produced by whatever source changes the surrounding of the broken egg. You can't isolate the breaking process from the broken egg. It's a continuous line, as the whole history of the universe. You have to consider everything. However you look at it, it's just impossible to bring the egg back to its original state. Just as it is impossible to bring the universe back to the state it was in 1 second ago. $\endgroup$ – descheleschilder Feb 10 at 1:36
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Let us first consider what exactly happens when an egg breaks. Chemical bonds are broken in the egg shell (mainly calcium carbonate) and the energy is converted to heat and sound. The interior of egg once exposed evaporation takes place and some chemical reactions might degrade the yolk.

If we are only concerned about the exterior of the egg going back into the initial form we would need all the energy lost in form of heat and sound to be returned to the shell in exactly the reverse of how it was released. Effectively we want the scenario of what would happen if we played the video of the egg breaking in reverse to actually happen.

Now if we want the interior if the egg also to go back to the initial form then we’d need for the water lost to be gained and the chemical reactions to be reversed. This also has to happen in the exact reverse order.

Those are if you want to go to the exact state as before. However, if you’re satisfied at getting an egg back you can probably conduct “surgery” on the broken egg to bring it back into a functioning egg state.

I find it easier to think about this in terms of a drop of ink in water. Initially, the drop is concentrated around one region. In case of the image, it’s more like a layer of ink. Soon the ink molecules will collide with the water molecules and each other and their velocities completely randomized.

enter image description here

Now if we want to go back to the initial state where the ink and water are separated, then the collisions have to be exactly in the reverse order and direction as to how it got to the current state. Let us say with probability $p$ (consider independent of time for simplicity) a molecule reverses its direction in the time interval $dt$ such that it reaches the state it was at $-dt$. Now for this particle to reverse back to a state at time $-T$, the probability would be $p^{T/dt}$. That was for one particle. Now if you have $N$ particles in your system, that means the probability to reverse state would be given by $p^{N(T/dt)}$. Even if we took $p=0.999$, which in itself is crazy, the total probability would still be ridiculously close to $0$ due to just the number of particles being $N\sim 10^{23}$. Plug it in a calculator and check it out for yourself!

This is the microscopic view of entropy. The disordered/mixed/homogeneous state where things are spread out is way way way more likely (read always) than the initial state where we had separated liquids. However notice that this statement is true for any microscopic state. Even if we began our microscopic state with the ink and water molecules mixed the probability to return to this particular mixed state is zero. The difference being if we look at the macroscopic view a mixture will still look like a mixture. So macroscopically once the mixture appears it’ll keep looking like a mixture even if many collisions are going on in the microscopic world.

As @gardenhead pointed out, this is not the only way for us to get back to the macroscopic initial state of separated ink and water. However, the number of the set of motions that lead to a well separated state are still much less compared to arbitrary motion. Basically what we’re saying is that $p$ is still (much) less than $1$.

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    $\begingroup$ Going in reverse is not the only way to get back to the initial state. There are many other paths through phase space that should bring the system back to its initial state. $\endgroup$ – gardenhead Feb 9 at 1:38
  • $\begingroup$ However if you’re satisfied at getting an egg back you can probably conduct “surgery” on the broken egg to bring it back into a functioning egg state. Why do you say probably?-1 $\endgroup$ – descheleschilder Feb 10 at 1:40
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    $\begingroup$ Because I am unaware if we can do so with our current state of expertise. $\endgroup$ – Superfast Jellyfish Feb 10 at 3:48
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    $\begingroup$ Then maybe you could have done some more research. On top of that, too great a part of your answer is dedicated to the laws of thermodynamics. In my viewpoint. So that's why I downvoted. It's not clear to me if the OP question is answered too. $\endgroup$ – descheleschilder Feb 10 at 11:48
  • $\begingroup$ I think it's a good answer, in that it describes entropy as a statistical concept, which is a more accurate description. The OP's problem was that he was trying to use "breaking an egg" to understand entropy, which is understandable given that it's a terrible example. The ink example is much better. $\endgroup$ – Richter65 Feb 10 at 15:00
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I assume a chicken egg. A hen can create a new egg that is macroscpically identical to the old one. Elementary particles are indistinguishable so even the fact you're holding the remains of the old egg doesn't matter.

Hens manage to create eggs because they absorb low entropy food and excrete high entropy feces (this way they also manage to keep their bodies in a low entropy state we call "being alive"). This is the other part of the system you're asking about.

So yes, we can revert back a broken egg into the original one. It's lot easier if we're hens.

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    $\begingroup$ @descheleschilder How can you prove whether an object presented at time $t_1$ is different or the same as an object presented at time $t_0$? For instance, are you the same "person" as the entity claiming your identity 60 minutes ago? How do we know that? $\endgroup$ – Lawnmower Man Feb 9 at 7:28
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    $\begingroup$ @LawnmowerMan One difference is that the cumulative total number of eggs will be different in the two cases. $\endgroup$ – Superfast Jellyfish Feb 9 at 8:55
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    $\begingroup$ If you feed the broken egg to the hen, the broken egg isn't reversed to the non-broken state it was in before. Rather, a new egg is created. And sure, some of the newly produced egg's ingredients were part of the broken egg. It's just a fact that eggs are produced by the hen. Broken eggs though are not repaired to their non-broken state. $\endgroup$ – descheleschilder Feb 9 at 11:51
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    $\begingroup$ @LawnmowerMan What I mean by an egg is the thermodynamic quantity made of $N$ molecules of $M$ constituents at temperature $T$, volume $V$, pressure $P$. $\endgroup$ – Superfast Jellyfish Feb 9 at 19:47
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    $\begingroup$ @LawnmowerMan en.wikipedia.org/wiki/Ship_of_Theseus $\endgroup$ – jawheele Feb 10 at 6:13
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Others have written excellent answers, but I just wanted to make an analogy using a jigsaw. Imagine if you had a 10,000 piece jigsaw, made from a picture of an egg.

If that jigsaw was jumbled up, then it is extremely unlikely that any amount of continued jumbling would return it to its completed state.

However, by adding external energy (in the form of a person who has eaten food), you could sit down and do the jigsaw and return it to it's original state.

The difference between the egg jigsaw and the egg itself is not just that the egg has many many more "pieces" (thinking of all of the molecules of protein from the white and yolk, and fragments of shell) but that the technology exists to join jigsaw pieces together, and the technology does not exist, at this time to reassemble broken shell fragments into a whole shell.

However, this is purely a practical problem - an engineering problem, you might say. In theory, just according to physics, reassembling the egg is equally possible as reassembling the egg jigsaw. In both cases, we reverse entropy by pumping energy into the process, and in both cases it is extremely unlikely to happen by chance.

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  • $\begingroup$ However, this is purely a practical problem. Purely? Why? $\endgroup$ – descheleschilder Feb 10 at 19:01
  • $\begingroup$ I just meant that it's not possible in practice, but that it's theoretically possible. We can imagine that it might be possible, now, if, say, we devoted trillions of dollars to a kind of Manhattan Project where the ultimate aim was to reassemble an egg. But, that would be a waste of money so it's not going to happen. $\endgroup$ – Max Williams Feb 11 at 11:44
  • $\begingroup$ How do you know that it is possible in practice? Maybe in science fiction movies...I know it's not possible in practice, however great a project you would use and however much money you would spend. $\endgroup$ – descheleschilder Feb 12 at 8:38
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    $\begingroup$ I said it might be. If you're 100% certain that it isn't, then I'll make sure you're on the fictional committee for this fictional "Rebuild the egg" Manhattan Project, and you can perhaps save us all a lot of fictional money. $\endgroup$ – Max Williams Feb 12 at 9:37
  • $\begingroup$ Haha. That's a good one. But to be serious. If life on Earth would depend on building a whole new egg from the remains od a broken one the answer (theoretical and practical) all life will be doomed. Only a chicken can make an egg. Without a chicken (which is what the OP means) it's impossible to recreate the exact circumstances to recreate an egg from the remnants of a broken egg. Besides, the are many thins heve escaped the egg in the process of breaking. $\endgroup$ – descheleschilder Feb 12 at 11:43
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The other answers to your question should provide you with sufficient insight, summarized as follows: The energy in the system is so partitioned and dissipated following the egg's breakage that the probability of energetic fluctuations perfectly reversing every mechanistic step is practically zero, even though classical dynamics are time-reversible.

To answer your latter question regarding the mechanism of such a "egg un-breaking," it would be the exact same as reversing $t$ in every equation describing the classical dynamics of the system, as classical dynamics are time-symmetric. However, bear in mind that the process is statistically negligible and possibly even mathematically impossible; there is evidence that certain $N$-body problems are irreversible, and that a loss of information occurs as time progresses, though I confess I am not an expert in this field and therefore cannot defend that statement with certainty.

I highly encourage you the ponder the correct interpretation of entropy, as a measure on the [number of] microstates of a macrostate. The reason the entropy of a broken egg is higher than that of an intact egg is because there is only one possible configuration satisfying our definition of an intact egg (namely, that the shell is in one piece), and a innumerably large number of possible configurations satisfying our definition of a broken egg. Note that we don't even need to define the relative energies of either state to make a statement on the qualitative differences in entropy.

The reason the system remains in a higher entropy macrostate is because it is vastly more probable. It is far more likely that the energy fluctuations are small in magnitude and keep the system in the same macrostate, though the microstate changes continuously (for example, the system evolves through different microstates as time progresses to reach the ultimate equilibrium, with the yolk drifting randomly across the floor, but the egg remains in the "broken" macrostate).

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    $\begingroup$ I believe there are many more than one microstates corresponding to an unbroken egg - though of course still vastly less than a broken one. $\endgroup$ – gardenhead Feb 9 at 2:18
  • $\begingroup$ @gardenhead I agree. I tried to avoid the complexities of the various microstates (e.g. the position of the yolk inside the egg, the microscopic structure of the shell, etc.) to try and keep the explanation simple. $\endgroup$ – hydro Feb 12 at 19:01
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As a trivial exercise, break the egg, scoop up the contents, use glue to stick the fragments of eggshell together, and insert the contents. Barring any chemical changes to the contents due to exposure to the atmosphere, you've unbroken the egg, having increased your own entropy considerably.

Now you may argue that the glue joints aren't the same as the original eggshell, so you can instead use chemical processes to reconstitute the chemical bonds. (In principle: I think we'd need some improvements in nanotech to actually do it.) This involves increasing your own entropy even further. Likewise any chemical changes in the contents can in principle be reversed, given suitably advanced handwavium.

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  • $\begingroup$ This is complete nonsense. $\endgroup$ – descheleschilder Feb 9 at 20:02
  • $\begingroup$ @descheleschilder Why? I think james is examining the pragmatic understanding of the question, and the answer is correct: With enough technical prowess, time and energy there should not be a principle problem to re-instate the egg to almost arbitrary levels of similarity. $\endgroup$ – Peter - Reinstate Monica Feb 10 at 2:19
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    $\begingroup$ Compare it to "invisible mending" of holes in clothes, or other artful restorations. $\endgroup$ – Peter - Reinstate Monica Feb 10 at 2:24
  • $\begingroup$ You can't compare it with that. And even if you can, how would you mend a broken vase? With glue? But then the breaking lines are still visible and you add something new to the broken vase: the glue. How will you put together the broken pieces, without seeing is in the vase? $\endgroup$ – descheleschilder Feb 12 at 11:46
  • $\begingroup$ @descheleschilder: To invisibly mend your broken vase, you use appropriate nanobots to replace the ceramic molecules along the fractures. I didn't say that it was going to be easy, did I? The higher the arbitrary level of similarity you insist on, the more your entropy increases. $\endgroup$ – jamesqf Feb 13 at 6:24
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To make a broken egg in reality (so not in a video) return to the non-broken state you have to reverse all the momenta of all the particles that are part of the broken egg, and you have to include all the surrounding particles which are affected by the breaking as well. Including the motions of all particles that constitute you if you are looking at the egg when it breaks. While the egg breaks also photons are emitted. These radiate away at the speed of light, so we can't get a hold of these (or the ones absorbed by matter in the surroundings of the broken egg, e.g. the surface it breaks on). Neither can you use photons that are produced by some source since this alters the surrounding of the broken egg.

Some kind of future surgery is pure fiction. The broken egg is part of a continuous process in spacetime, including the breaking itself, and you can't isolate the broken egg from that process (governed by the second law of quantum statistical mechanics at the microlevel and the laws of classical chemistry or classical mechanics at the macrolevel). You need to reverse this continuous process, which is an impossibility.

I can see no means to accomplish this, without changing the broken egg itself, so this will be impossible (and we haven't even taken into account quantum mechanics).

What you ask is somewhat like asking if we can make an egg without the aid of a chicken (even if you let a chicken eat the broken egg the newly created egg isn't the same egg as the broken egg was before it broke). This is obviously impossible like it is impossible to create a living baby without a woman (who has a womb) and man.

Or take the "simpler" question if you can reverse a flash of lightning. In that case, you have to reverse the increase in entropy (for the whole universe) into a decrease in entropy. This is the main point here. Irreversible processes are...well.. irreversible.

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  • $\begingroup$ You wouldn't have to REVERSE all the particle motions, you'd just have to have them follow paths that would put them back in their original positions. You also have to ask what is meant by "unbreaking" the egg: do you want to reverse time's arrow, putting every particle back in its original position, or is sufficient to have molecules of calcium carbonate forming a shell, even if they aren't the original molecules? $\endgroup$ – jamesqf Feb 9 at 18:46
  • $\begingroup$ @jamesqf you'd just have to have them follow paths that would put them back in their original positions. Isn't that the same as reversing their momenta? And you have to include all the photons send out while the egg broke. How you wanna do that? You can only reverse the arrow of time by reversing all particles momenta (naively spoken). $\endgroup$ – descheleschilder Feb 9 at 19:31
  • $\begingroup$ Your answer is based on a certain understanding of the question which I'm not sure is intended by the OP: That the "reinstated egg" is atom for atom identical with the one before the breaking. This interpretation has the problem that there is a fuzziness to the microscopic state which makes the requirement somewhat fuzzy (you'd need to specify a probability that the reinstated egg state is a possible later state of the original one). $\endgroup$ – Peter - Reinstate Monica Feb 10 at 2:13
  • $\begingroup$ The other problem is that in order to do that, if you are serious, you need to reverse the entire light cone which is impossible precisely because of thermodynamics! You can only decrease entropy locally, but that does not reinstate the egg perfectly on the microscopic level because you'll have e.g. stray photons resulting from the breaking, or a different electron alignment because of the thunderstorm resulting from the broken egg ... nature is non-linear. $\endgroup$ – Peter - Reinstate Monica Feb 10 at 2:15
  • $\begingroup$ As an aside, the reason we cannot reinstate the egg resembles the reason we cannot reinstate Monica: We would need to "rewind" all of Stack Exchange Inc. to the state it was a couple years ago. $\endgroup$ – Peter - Reinstate Monica Feb 10 at 2:21
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If we allow an external environment to interact with our "egg" system, putting our egg back to its original state is the same thing as creating an egg which is identical to the original. The crux of the question is to clarify what we mean by "identical".

In the context of thermodynamics, we cannot possibly mean microscopically identical: if we are to talk about thermodynamic entropy, we must be doing some `coarse-graining', meaning that we identify a single macroscopic state with many ($N$, let's say) sufficiently similar microscopic states. The entropy can then be defined by counting how many many microstates give the same macrostate, $S=\log N$. A state has high entropy if there are lots of microscopic arrangements that look the same macroscopically.

One answer to the question is to start an egg farm, and spend your time checking all the eggs to see if they're the same as the original. Eventually, if you don't run out of resources and patience, you'll find an egg that's sufficiently close to the original that you can't tell the difference. At that point, you have in effect restored the egg to its original state.

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  • $\begingroup$ From a physics perspective, your egg farm proposal is an incredibly inefficient way of unbreaking the egg, even if it's fairly decent from an engineering point of view (with current technology). $\endgroup$ – wizzwizz4 Feb 10 at 15:45

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