How would physics explain why I can't un-fold paper? This seems like a silly question, but I would love to see a serious answer.
It's known that almost all fundamental laws of physics are time-reversible, and time-asymmetry only comes about when we consider increases in entropy. It's easy to think about this in the case of gas in a box. We know that even if we start out with the gas concentrated in one corner of the box, it will spread out to the entire box.
However, I wonder how these ideas can be applied in other contexts. If I fold a piece of paper, it will obtain a crease. I can create creases in a piece of paper, but I can't remove them.
How could one give an explanation for the apparent irreversibility? Is there an increase in entropy occurring when I fold the paper? Does it even make sense to talk about entropy in this scenario?
Edit: This question is not a duplicate of the post about origami, because I'm specifically asking how does paper folding relate to the usual formalism of thermodynamics/statistical mechanics, namely time-irreversibility and entropy. The other question is not asking for that.
The answer there is very helpful, but it doesn't answer my question.
 A: The first thing we need to realise is that we cannot that easily apply the typical formulation of entropy as the number of microstate realisations of a macrostate as the distinction between micro- and macrostate is less clear in paper.
Like for every solid, the paper’s structure is fixed on some rather microscopic level:
Unless you tear it, crease it, or similar, its fibres (molecules) stay in the same relative position.
However, they can wiggle slightly around their position or slightly bend.
This is how we get temperature in solids and also entropy:
There are bazillions of microstates of specific wiggling that realise a given arrangement of fibres (which here is the macrostate).
For the following, let’s assume a simple model, in which fibres easily stick to each other and are not torn apart.
In reality, the distinction between fibre–fibre connections and chemical bonds within a fibre is much more blurry and creasing and tearing can destroy fibres, but this only complicates the following explanations without yielding any new substance.
Now, before I eventually get to creasing, let’s look at tearing the paper:
Apart from the new tear, the paper’s entropy is not affected:
The fibres are in the same or maybe a slightly different position, but they have as little freedom of movement as before.
So, if anything happens, it must be at the tear.
Here, our fibres have indeed some more freedom of movement, bending to the side, where their movement was previously constrained by a fibre that has been torn out and now resides on the opposing half of the paper.
This is where the entropy is increased.
To see why this freedom of movement really matters, consider what it would need so you could put the paper together again:
This would be easy if the paper’s fibres stayed in their exact relative position (except for being torn apart) and were perfectly rigid.
In that case, we have two perfectly fitting halves of a jigsaw puzzle that we simply put together again.
However, since the fibre on the tear rather dangle than stick out rigidly, each fibre has several possible positions.
If we now account for all fibres at both sides of the tear, we have bazillion of possible combinations of fibre positions (microstates).
And only a vanishingly small minority of these positions are such that we can simply stick the paper together.
Hence the process is irreversible.
Finally, creasing amongst other things introduces a lot of small tears in the paper, to which the above lines of thought apply.
There are also other processes like bulging fibres and so on, but I don’t expect that much insight is gained from treating them as above.
A: If you understand the structure of paper it is obvious that folding or tearing increase entropy
Paper is made by breaking down wood (or for fancy paper, some fabric) into its constituent fibres and then reassembling those fibres in a very ordered way. The fibres are aligned in the plane of the thin layer as a suspension and then dried to preserve the order. So paper is a fairly ordered system.
When paper is bent or torn, some of that order is destroyed. This is obvious in tearing where many individual fibres are broken, a process that destroys the tight alignment between the two halves of each fibre. Even if we assumed that fibres would rejoin if they were all placed into an exact alignment (which ignores the detailed chemistry involved when cellulosic fibres are broken) it should be easy to see that tearing is an entropy increasing process as there are (maybe) millions of independent fibres whose alignment need to be precisely set for this to work. After tearing, the alignment between the haves of each fibre is deranged in a random way compared to their previous alignment in an untorn sheet. That is a big increase in disorder or entropy.
What might be less obvious is that folding has a similar, but less severe effect. When folded, some fibres will break so some degree of tearing will happen. And some fibres will be permanently displaced relative to their more ordered form in unfolded paper because they become untethered to their surrounding fibres in a random way and because their individual molecules are displaced relative to their previous positions.
While any calculation of the entropy change would be humongously hard, it should be obvious that the disorder has increased and therefore the entropy has increased. It is not, in detail, that much different to gas molecules in a box or breaking an egg. There are far more disordered states then there are, original, ordered states in each case.
So folding or tearing paper is like other archetypal cases where we recognise entropy as the driving force for time-irreversibility (breaking eggs, gas molecules expanding to fill a box). The entropy point of view works even for folded paper.
A: Your example is not an ideal one to illustrate the underlying principles, as the factors that prevent you from unfolding the paper involve many trillions of molecular interactions within the paper, which amount to a very complicated mechanism to visualise. Consider instead a simpler set-up with vastly fewer moving parts as follows...
Imagine it is the start of a snooker game. The red balls are arranged in a triangular layout. You are able to scatter the balls with a well-directed shot of the white. The interactions between the individual balls are governed by simple equations that are all clearly time reversible. Consider, however, the difficulty of reversing the effect of the dispersal- how could you, with a single shot, cause all the reds to re-assemble in triangular formation?
By contrast, consider Newton's cradle- the arrangement of identical spheres each suspended to swing in a fixed plane. The interactions of the balls are trivially reversible.
The key difference between the two examples, each of which involves collisions between spheres, is that in the case of the snooker shot a single action can lead to a fantastically large range of possible outcomes- the balls could spread anywhere. When you try to reverse the process, you are imposing a much tighter constraint- there is only one specific outcome that would satisfy you, namely that the balls were rearranged in a triangle.
The principle is analogous to a card trick. The magician asks you to take any card- it does not matter which, so the action is an easy one for you to get right. For the second half of the trick, however, the magician must find the specific one you picked, so the odds of doing so at random are greatly reduced.
You will find that almost all irreversible processes are so because they create a large number of random changes, such that the chances of every one of them being exactly reversed is effectively zero.
In the case of folding paper, the action disturbs the arrangement of countless fibres that had been compressed to lie in a certain plane when the paper was manufactured. Depending upon the materials from which the paper was made, and how sharply the crease was formed, you might be able to eliminate signs of the crease by pressing it with a hot iron, which recreates the heating and compression that were applied in the manufacturing process.
I am not convinced that it is useful to consider entropy to aid one's understanding of why it is difficult to unfold paper perfectly. Consider your example of gas in a box. If you have gas compressed by a piston and the piston is with withdrawn some distance, the gas expands naturally and there is an increase in entropy. The expansion and increase in the entropy of the gas is easy to reverse simply by pushing the piston back to its original position. The difference between that, which is easy, and the perfect unfolding of the paper, which is difficult, is not principally a question of entropy.
Indeed, take a gas in equilibrium in a sealed insulated box. Imagine the position of the molecules at any one instant, then consider the challenge of precisely recreating that arrangement at a later time. The gas is at equilibrium in a sealed insulated box, so there has been no change in entropy, yet the probability of recreating the original arrangement is vanishingly close to zero.
A: I would like to mention some things the other answers do not address:

*

*The way you create a flat, wrinkle-less, crease-less paper is to apply heat (and other processes), but bottom line, you need to add energy to the quantum mechanical system of molecules. By that, you do two things: you create covalent bonding between certain molecules in certain ways, and you set the rotational angle between molecules.


*When you fold the paper, you do two things: you destroy some of these covalent bonds, and you change the rotational angle between molecules. Now when you try to unfold (restore the original) the paper, you cannot manually restore the covalent bonds you destroyed before. You can however, partially, to restore the rotational angle between some molecules, that is how you unfold (try to flatten) the paper. If you want to restore the original paper, you would have to apply extra energy to the QM system of molecules. If you use an iron, you can do pretty good. But it still won't restore the original paper. To do that, you would have to apply extra energy and do changes on the molecular level to the covalent bonds.
A: When you fold a piece of paper to create a sharp crease, you are permanently damaging the structure of the paper, and that damage (disconnection of bonding between adjacent cellulose fibers and breakage of fibers themselves) cannot be repaired by unfolding the sheet and flattening out the crease: it is irreversible.
(This is easily demonstrated by observing that creasing a piece of paper and then unfolding it makes it a lot easier to tear along the crease line because it was weakened by the creasing action.)
Now, creasing a piece of paper is a lot easier to do than actually calculating the associated entropy increase, so the calculation isn't done each time we want to crease a paper sheet, but to be sure the orderliness of the paper sheet has been mechanically disrupted by the performance of work in the region of the crease, and therefore entropy has been increased.
