One source is popping back the forces they experienced during their approach, the other is the forces they experienced as they separated. Since they are at their initial separation again, the stack is empty, so there is zero potential energy; and there's no reason the kinetic energy should be what it was initially. So energy has been added or subtracted.
You got it backwards. The stack reads in from infinity, not from 0 separation. As they approach, they're pushing, not popping. Plus, the contents of the stack are included in the potential energy, so either way you cut it, it adds up. If the randomness is on the side you don't integrate from, you won't have changes.
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As for the magnetic forces thing, having a preferred frame of reference is quite different from laws no longer being invariant under translation. What you mean is that the laws are no longer invariant under boosts.
Noether's theorem applied to that symmetry yields something to do with the center of mass which I don't quite understand, but seems to amount to the notion that the center of mass doesn't deviate from its nominal trajectory. This seems to me to be awfully similar to the conservation of momentum, but must be technically distinct.
As they approach, they're pushing, not popping.
Yes. Then as they separate, they pop those forces back out again. When they reach separation X, which can be infinity if you like (or we can just define potential energy relative to that point) they have zero potential energy and a kinetic energy which cannot in general be equal to what they started with. The simplest way of seeing this is to have the coupling be constant A on the way in, then change to B at the point of closest approach. Then their total energy on again reaching the starting point is A int...
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