Consider a bunch of nano magnets held together in a clump. A small asteroid hits the clump of magnets and breaks them apart (high entropy). Over time, the nano magnets vibrate due to brownian motion and when they randomly come near each other they clump together again (low entropy). Random brownian motion of nano magnets can cause them to go into their clump once again, even though they were broken apart into a high entropy state.

Or instead of the asteroid let's just consider vigorously stirring the nano magnets or blenderizing them to create a high entropy suspension of nano magnets in a somewhat viscous fluid. If the fluid is not too viscous there will be enough brownian motion that the high entropy nano magnets will eventually clump back together over time...

A classic example books use to describe entropy is an egg breaking apart and not coming back together again. The egg shell peices get scattered when it breaks. Consider if egg shells had nano magnets embedded in them, so that when you break apart an egg shell it sticks back together again if it comes near another egg shell peice. The wind blows an empty hollow egg off the table and it breaks apart the shell. The wind blows the broken pieces around randomly, and voila - over time, the broken egg shells clump together again in a low entropy state if they randomly blow in the wind and come near each other. However, once they clump together again, they don't necessarily end up in the exact same configuration as the original egg shell - they may just end up as a clump of shells piled together, without forming a oval egg. However the pile of magnetic egg shells may form some kind of beautiful artistic piece or something that looks like one of the alphabets, or a symbol.

Clumped together egg shell peices are low entropy compared to scattered ones. Did we reverse time? No we just had egg shells form a low entropy clump without going back to their original low entropy oval shape. Just because the eggs clump together not in a perfect oval doesn't mean it isn't low entropy: it's just a different configuration of information, and still low entropy.

Hence why I don't think the second law is as much tied to time as people think. More importantly information configuration is deeply related to time. Even if the second law weren't true in many situations, time would still exist since low entropy states can be reconfigured to be a different state than before (different configuration of information). You can still go back to a low entropy state without it being the

Another strange issue with information is that entropy could be partly subjective: a non english speaking person may not even know what a dog is, and therefore the information to them is high entropy as it is just scrambled symbols (consider an alien reading a web page, who doesn't even use the same letters as humans - the internet information to an alien would be high entropy scrambled jibber jabber, although he may detect some ordered patterns even if he can't understand what the symbols mean. Even if he sees some order, it is not as ordered to the alien as it is to a human - is information/entropy subjective, or ultimately objective?).

I think the arrow of time is tied to Information configuration, not entropy in general. Information is related to entropy, but it's an important distinction. If the second law were violated (hypothetical) you can lower the entropy of the entire universe to the same low entropy as the big bang state, but have a different configuration of the particles in different locations than before (but still a low entropy state just as low as before, but a different order or configuration of the items or information).

With the blenderized nano magnets becoming ordered again due to brownian motion (not energy input, brownian motion is free) this is a paradox because brownian motion should randomize everything, not clump things together. One could argue that you had to add energy to the system in order to blenderize/randomize the nano magnets first. But this is like arguing that if you blenderized sugar or salt particles, you are adding energy to make them higher entropy, therefore they should clump together again when you dump them in water? The only way salt or sugar would clump together if you added energy once again to the system, to evaporate the water and let the sugar/salt crystallize once again. Whereas with the nano magnets you don't need to add energy again for them to clump back together, the random brownian motion does the work for you and the magnets clump together

Another paradox is it takes energy to make things high entropy (not just low entropy). Consider rolling dice: this takes quite a bit of energy to shuffle and throw the dice. Compare this to brownian motion, which randomizes things for free (but it doesn't maximally randomize things as much as it could, see my other articles on "container information theory"). If brownian motion creates a free random number generator, is this not a machine? In computer programs it takes energy (cpu outputs heat) to create a random number generator. In brownian motion, the machine is created for you at no expense of energy). Although this is interesting (since the second law doesn't allow one to make machines at no cost) it's not as interesting as thinking about the blenderized nano magnet paradox where random brownian motion creates ordered clumps, at no cost of energy (order from chaos?).

Macroscopic magnets will also clump together but they are a bit different since when they collide with each other, heat is given off. In the case of nano scopic magnets, if any heat is given off when nano magnets collide, this heat recycles back into the system perfectly to cause more brownian motion, clumping more magnets together (the heat isn't lost, it recycles back into more brownian motion causing more chance that the magnets will come near each other and become ordered, since more brownian motion increases the probability that two nano magnets will come near each other). Therefore the microscopic world is quite different than the macroscopic world, and more paradoxes occur mesoscopically or nanoscopically.

You can see a video I made with my microscope of nano magnets clumping together into a low entropy state due to random vibrations of brownian motion. If random motion can create ordered clumps, is this a paradox to the second law? Over time, nano magnets should tend to go toward maximum disorder or highest entropy. But when nano magnets collide from random brownian motion, they go into low entropy clumps, not high entropy ones. In order for the magnets to be in this low entropy state, did the universe spend way more energy at some other point in time (during the big bang) to create the magnetism in the first place? But magnets can be reused... over and over, so if an asteroid kept hitting a clump of nano magnets and putting it into high entropy state, the magnets would clump back together over time due to brownian motion (assuming the magnets were in a container close enough together). Or instead of the asteroid analogy, think about blenderizing nano magnets utterly randomly. After time, brownian motion causes them to go into a low entropy clump if you let the fluid rest at room temperature, as long as the nano magnets were in a container (related: container information theory). It would be like dice always rolling six no matter how much you try to make the dice roll random numbers by shuffling the dice.

Related also is the milk demon potential second law violating device which uses gravity and randomness to extract energy from directed motion. Are magnetism and gravity (forces) a key to putting structure into randomness (order into disorder) without expending energy? But what about viscosity, not just magnetism, and what about electrostatic drag.