# Statistical mechanics & the interaction energy

hi everyone!
I am so confused about interaction energy!
as you know, in many statistical mechanics books we see there are equations which are taken in a non-interacting system.

for example for a two particle system, you can see the total energy is:
E = E1 +E2 +E12
while E12 is the interaction energy between particle 1 and particle 2.

What I wanna know is that when can we neglect this interaction energy?
how far is interaction energy's range?

can anybody give me a formula or something for it?!!

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It depends on the nature of the interaction. For example:
_ If the 2 "particles" are Moon and Earth, the interaction between them is gravitation. In this case, it's obvious that when they are very far from each other (say, 1000 times Earth's radius; we know that the actual distance between Earth and Moon is around 60 times Earth's radius), E12 is negligible.
_ Consider the strong interaction between nuclei. If the nuclei, 2 protons for example, are separate at a distance larger than the nuclear radius (around a few fm), strong interaction is negligible, and now instead, we have to consider the electromagnetic interaction between them (protons are charges after all). Again, if we further separate them, say around the distance between surrounding atoms/molecules, the protons are now affected more by the surrounding, and thus, the interaction between the protons is again negligible.
So how far the particles are when their interaction is negligible, it depends on the interaction's nature, ranging from 10^-15m to thousands of km.

But you know, what I really want is the limit
I mean if it's about two nuclei where is the exact limit where I can neglect the interactoin energy?!
does it really have a line as a limit?

But you know, what I really want is the limit
I mean if it's about two nuclei where is the exact limit where I can neglect the interactoin energy?!
does it really have a line as a limit?
I'm no expert, so all I can provide is a general view.

I think it depends on how much you want from the "EXACT limit", or how you define it. Take the notion of atomic orbital as an example. Some people say, it's the region where the probability of finding the electron is about 90%. The number 90% is just a formal number. You can probably find an electron here and there.

The same thing for the limit of interaction. I don't know the exact formula for strong interaction, but it should decrease tremendously with distance, just like gravitation, so I'll take gravitation as an example. It's obvious that gravitational interaction between Earth and Sun is significant, while one between Sun and a person on Earth is negligible, and instead, that between Earth and the person is not negligible. So it's relative.

For strong interaction between nuclei, I would say if the 2 nuclei are taken apart at a distance of about more than nuclear diameter, the interaction is a lot weakened and if it's about atomic radius, it's negligible. But as I said, it's just an estimated number; there is no exact answer.