Is it ever physically possible for a broken egg to reassemble into an unbroken one?

[Paleophyte]

OK, check my thinking on this:

n! doesn't represent the probabilities correctly. That's just the number of permutations you'd get if you shuffled the system's components and put them into a linear array like a deck of cards. We need to account for spatial distribution because there's a big difference between two molecules a light year apart and the same two molecules a micrometer apart.

To account for space we need something more like a 'seat n people in m chairs' problem but seating particles in potential locations. That yields m!/(m-n)! combinations, which can be nasty to calculate. Happily, so long as the number of potential locations is much, much larger than the number of particles, the difference between m and m-n is trivial and we can approximate it as m^n, which is a much simpler calculation. It's simpler than that since, as you pointed out, the value of m doesn't influence the uppermost exponent much. The spatial resolution doesn't matter much so long as it's fine enough to satisfy m >> n. In that case the calculations get really simple because a system with 10^x components has roughly 10^10^x+2 possible arrangements. So a pint of water with ~10^25 molecules wouldn't have more than 10^10^27 possible arrangements, which is impressively large but a long way short of a googolplex.

But none of that accounts for time. We need to set this all in motion so what we need is less of a 'seat n people in m chairs' problem and more of a 'musical chairs' problem but with a lot more chairs than people. For this we simply view each iteration as a single m^n snapshot and multiply the results for i iterations. That will be valid so long as each m^n snapshot is unique, which it will be since the laws of thermodymanics preclude take-backsies. That produces m^n^i possible combinations, and that will get very large very quickly.

No, it produces (m^n)^i =m^(ni). For this, you *multiply* exponents. This is very different than m^n^i=m^(n^i). You are doing the parentheses the wrong way and it makes a difference. (x^y)^z=x^(yz) is very different than x^(y^z).

For example, (10^3)^2 =(1000)^2=1000000=10^6, but 10^(3^2)=10^9. Or, another, (10^3)^10=10^30, but 10^3^10 =10^59049, which is considerably bigger.

Thanx, you're right, I got my brackets wrong. mni

Using nanosecond resolution, a pint of water with 10^25 molecules passes the googolplex mark in 4 nanoseconds, beats 10^10^200 in 8 nanoseconds, and passes 10^10^25,000,000,000 in the first second. I know, those are absurdly large numbers, but that's what iterating in 4 dimensions gets you.

No, it passes through the *googol* mark in 4 nanoseconds (10^25)^4=10^100. That is a googol. It gets to 10^200 in 8 seconds. In a second, it gets to (10^25)^(10^9) =10^(25*10^9)<10^(100*10^9)=10^10^11

In a year, it would get to (10^25)^(3*10^16)=10^(75*10^16)<10^(100*10^16)=10^10^18

In 10 billion years, it would get to (10^25)^(3*10^26)<10^10^28

Errr, no. A single mn snapshot gives you 10^10^27, which has left googol so far behind in the dust that it's laughable. That said, this won't get my beer over the googolplex line at mni for any reasonable value of i. At nanosecond resolution you're looking at 10^46 times the age of the universe. Even at Planck time resolution you're looking at a triilion times the age of the universe or so. So I can't get my beer over the googolplex mark, at least not using this simplistic analysis of the odds. I'm still hopeful that factoring in all the other complexities would get it over the line but I don't have the mathing for that.

The universe however gets us ~10^10^82 permutations per snapshot. That means that we need 10^18 iterations to get us over the googolplex mark or, attosecond resolution to achieve googolplex per second. Attosecond resolution is pretty reasonable given that it's the timescale at which helium is fused into carbon and you could hide the entire inflationary epoch inside of it. And since the fundamental structure of our universe was established in those first seconds, we experience events of 1 in a googolplex every moment of our lives.

They spawn an infinity of little horrors.

Some people enjoy such horrors. I'm a professional mathematician. This is my wheelhouse.

Sorry, text fails to convey the admiration that I feel for said little horrors.