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I'll quote Feynman's Lectures, chapter 52 (Symmetry in Physical Laws) of volume 1:

[...] If we see the egg splattering on the sidewalk and the shell cracking open, and so on, then we will surely say, "That is irreversible, because if we run the moving picture backwards the egg will all collect together, and that is obviously ridiculous!" But if we look at the individual atoms themselves, the laws look completely reversible. [...]

I understand that the laws that govern atoms are reversible, but macroscopic things are made of atoms and macroscopic processes do not look reversible at all. How should I interpret this apparent paradox?

Is it the case that most states (combinations of positions and velocities of the atoms) of a physical system lead to an intuitive ending (e.g. egg splattering) and thus intuitive endings are more likely, but if we reverse all the velocities we would arrive at a higly unlikely state?

Or could it be that states that lead to unlikely endings (e.g. egg collecting itselt back together) are so unstable that small sources of randomness are enough to ensure that unlikely endings indeed will not happen? What kinds of randomness must be accounted for? Influences from outside the system? Would it be correct to say that quantum mechanics is a source of randomness in a closed system?

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  • $\begingroup$ If this looks like too many question marks in a single Question, I'll be glad to hear suggestions on how to split it in two or more questions. $\endgroup$ – fonini Dec 25 '15 at 6:00
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    $\begingroup$ Any feasible clock design requires an energy source and a heat sink. Clocks are, by their very nature, non-equilibrium devices... but if we don't have a clock we can't tell time, hence time itself, by its very definition trough clocks, is a quantity linked to thermodynamic disequilibrium. The reversibility of microscopic laws is not enough to guarantee reversibility of processes, by the way. A trivially reversible dynamics that has an irreversible solution is a classical ball rolling down from a hill into an infinite plain. The openness of the environment guarantees irreversibility. $\endgroup$ – CuriousOne Dec 25 '15 at 6:34
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Is it the case that most states (combinations of positions and velocities of the atoms) of a physical system lead to an intuitive ending (e.g. egg splattering) and thus intuitive endings are more likely, but if we reverse all the velocities we would arrive at a highly unlikely state?

Not intuitive, but consequent, due to the laws of mechanics in this simple picture.

Think of a jig saw puzzle. It can be randomized very fast. It may even take days to be put together again, just with 1000 bits or so.

One mole of matter contains about $10^{23}$ molecules. These molecules, depending on their interactions between them, which individually are time reversible, have an enormous number of permutations . To get back to a previous state is statistically very improbable.

Or could it be that states that lead to unlikely endings (e.g. egg collecting itselt back together) are so unstable that small sources of randomness are enough to ensure that unlikely endings indeed will not happen?

No, it is a matter of probabilities, not stability.

What kinds of randomness must be accounted for? Influences from outside the system? Would it be correct to say that quantum mechanics is a source of randomness in a closed system?

Quantum mechanics introduces extra probable states in addition to simple number counting, as it is a theory that predicts probabilities of interactions.

Thus the probabilities of spontaneously getting the same state can be considered zero within the lifetime of the universe.

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  • $\begingroup$ If it's not a matter of stability, consider this problem: let's say I have arranged all the molecules of a broken egg in the exact state they need to be so that the egg will collect together. I know that this is the state I need, because it's the reversed-velocities state. Now, will it really collecect back together, regardless of the wind? I know that if I drop the egg, it will shatter, regardless of the wind, or people stepping on the floor, or whatever kind of noise. $\endgroup$ – fonini Dec 25 '15 at 22:31
  • $\begingroup$ If you calculate the probability of doing what you describe you will find such a tiny number that any wind or whatever will be a tiny perturbation. Maybe your hand trembled when putting mollecule number 800000000000000000001 in its appropriate variables? that also will be a tiny perubation $\endgroup$ – anna v Dec 26 '15 at 5:44
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All possible outcomes are equally unlikely. The odds of any exact configuration of particle positions and momentums in a splattered egg are all negligible.

However, some of these configurations result in macroscopic outcomes that look identical, and if you count the number of configurations that appear identical let's call that number the "multiplicity" then they can add together to give you a much higher probability of occurring (although remember each individual state in that multiplicity has a very low probability). So the reason some processes are "irreversible" is because they have a negligible multiplicity compared to other processes.

For example consider an egg was already dropped and splattered on a table. There will be a very large multiplicity associated to the egg just lying there broken (because there are many configurations of random velocities of the particles which just cancel out), but there will be a very low multiplicity assigned to the process of the egg pieces coming together to form an unbroken egg (because this process requires very specific velocities and not many configurations will meet that requirement) so that outcome is negligible.

Maybe if you stare at a broken egg until the universe ends you will eventually see it spontaneously form into a whole egg again, but in the course of a normal human lifespan it is considered to be a statistical impossibility.

So in a sense forward or backward in time aren't special because both are just transitioning between possible configurations of the system according to legal laws of physics. It only looks special when you let time go in one direction toward higher multiplicity and then reverse the direction of time, because now you've formed a "history" and you're requiring that the system do something which is normally very unlikely, flow toward lower multiplicity. If that history didn't exist then even moving backward in time statistically it would still go toward higher multiplicity which would look perfectly normal.

By the way, you can replace everything I said above about "multiplicity" with "entropy" because they are related by a natural log. Increasing multiplicity is increasing entropy, I just didn't use the word entropy because I think it's less intuitive in this case than talking about the individual particle configurations.

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    $\begingroup$ This doesn't solve the arrow of time problem from the microscopic perspective because every past has a unique future in classical physics and the only "reverse" of that unique future is the very unlikely past. Statistics is not really enough to solve this as long as the dynamics is causal. You have to throw either quantum mechanics into the mix (in which case the dynamics is open to allow multiple futures, i.e. there is no unique past, either) or you make the system relativistic and open (i.e. humpty dumpty would need the help of the entire past universe to come together, again). $\endgroup$ – CuriousOne Dec 25 '15 at 7:50
  • $\begingroup$ @CuriousOne, I'd be interested in hearing more about these two things. Have you written an answer on them? $\endgroup$ – knzhou Dec 25 '15 at 8:04
  • $\begingroup$ No, it's just the usual criticism to a simple statistical argument. While it is true that there are an infinite number of possible universes, with perfectly causal dynamics there is exactly one path trough them and while it is both narrow and unpredictable, it is still unique. We need some sort of deal breaker that destroys that uniqueness. I know two possible mechanisms, one is radiation into infinity and the other one is quantum mechanics. How much you trust these arguments about the arrow of time... that's up to you. $\endgroup$ – CuriousOne Dec 25 '15 at 8:19
  • $\begingroup$ @CuriousOne But isn't it just a question of initial conditions? In this context reversing time is identical to reversing velocity. So under this definition of "backward time" we can run the clock back to even before time 0 which is also uniquely determined and would still find that the system eventually tends to maximum entropy. At that point you could even then run the clock forward again and observe that "forward time" now behaves strangely while backward time seems normal. So it's the initial conditions (and the fact that time doesn't change direction) that's special, not the arrow of time. $\endgroup$ – voidlike Dec 25 '15 at 8:39
  • $\begingroup$ If you are working in a completely causal dynamics, then reversing time leads you back the same path that you came from. The universe had to start from incredibly precisely determined initial conditions to make the chicken that laid the egg in the first place! Of course it didn't. Instead we needed a very large non-equilibrium system and 4 billion years of evolution to make the egg. Reversibility is not necessary, though and does, IMHO not exist outside of simplifying classical equations. It is therefor not necessary to discuss it away with initial conditions. A random big bang can makes eggs! $\endgroup$ – CuriousOne Dec 25 '15 at 8:52

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