Poincare recurrence theorem
An example I heard about in a lecture on time and entropy mentioned an example of a melting ice cube. You put an ice cube in a hot cup of water and the ice cube melts. The water molecules could randomly recreate the ice cube exactly how it was at the start. At any time, we could simply put the film in reverse. Perhaps the electron cloud distortion problem, mentioned above, precludes simply reversing the film.
Perhaps it can never happen. Lets make an extremely simple system. Our environment is a cirlce with the thickness of 1 atom and we have only 1 atom in the circle. The atom impinges in the circle at an angle of 60° to the tangent at the collison point. It hits again and again is back to it's original position.
Next we start with an irrational number of degrees then we never return to our original position. An irrational times a rational is an irrational. We would never have N x 360°.
Ok, what about random collisions in three dimensions. We can reduce this visually to two dimensions. Just picture yourself on one of the 3d axis, far off, away from the action. Picture a gas cloud loosely held together by gravity, it can be massive, it doesn't matter because we need only focus on one atom or molecule in the nebula(we'll say atom). We, only have to focus on one atom because if we can prove that it will never get back to it's starting condition(speed and direction), then none of the atomes or molecules will ever get back to their original state.
- 1 divided by infinity equals zero.
- If you select a number uniformly at random from any compact interval on the real line, say [0, 360], then the probability of getting a rational number is zero and the probability of getting an irrational number is one.
- Each irrational has one example(if we identify it by the mantissa, the mantissa is what comes after the deciaml point.) that when added to it, sums to an integer. Let X>0 be irrational, then 1-x is also irrational and their sum is 1 which is rational.
So, with this information we want to show that our atom will never get back to its orignal postion by examining the angles each random collision causes.
You can probably see the beginnings argument already. Lets make it really, really easy for the atom to get back to its orignal position. We imagine an atom colliding with it in such a way that it moves exactly backwards. Then we imagine another knocking it exactly back at the right speed. We had 2 collisions and our original is in exactly the same position with exacly the same speed and direction. Well, because 1/infinity = 0, this could never happen. Not that it is unlikely, but that it is impossible. The odds of either of these two atoms to hit it exactly right are zero. Not close to zero, the odds are zero.
Ok, so we know that we are going to have to change the direction of this atom through various angles. And the sum of these angles must end up to be a multiple of 360°.
Since the probablity of choosing a rattional number from our compact [0,360] interval is zero, we know that the first angle will be an irrational number. Now, there is one mantissa that when added to this angle will create a rational number. What are the odds of picking an angle with this special mantissa? 360.mantissa/infinity = zero. So, we will never, ever get that magic complement to make a rational number angle.
The final result is that our original atom will never get to its original state angle of movement.This argument works for speed and postion too, I think
This whole argument comes fro the idea that you can't pick an integer out of the bucket of Real Numbers. EVER.
Now, this is not to say that you can't get close to the original state. I think it says that each state of the universe can only occur exactly once.