Statistical mechanics & the interaction energy

• shadi_s10
In summary, the limit for neglecting interaction energy between particles is relative to the distance between the particles. It depends on the nature of the interaction.
shadi_s10
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 want to 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?!

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.

Thanks a lot for the answer
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?

shadi_s10 said:
Thanks a lot for the answer
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.

thanks a lot for your help

1. What is statistical mechanics?

Statistical mechanics is a branch of physics that studies the behavior of large collections of particles, such as atoms and molecules. It uses statistical methods to predict the macroscopic properties of a system based on the microscopic interactions between its individual particles.

2. How does statistical mechanics relate to thermodynamics?

Statistical mechanics provides a microscopic explanation for the macroscopic laws of thermodynamics. It helps to understand how the behavior of individual particles gives rise to the thermodynamic properties of a system, such as temperature, pressure, and entropy.

3. What is the interaction energy in statistical mechanics?

The interaction energy in statistical mechanics refers to the potential energy associated with the interactions between particles in a system. It can be attractive or repulsive, and it plays a crucial role in determining the overall behavior and equilibrium state of the system.

4. How is the interaction energy calculated in statistical mechanics?

The interaction energy is calculated using mathematical models that describe the forces between particles in a system. These models can vary depending on the type of interaction, such as gravitational, electric, or van der Waals forces. Statistical methods are then used to calculate the average interaction energy for a large number of particles.

5. What are the applications of statistical mechanics in modern science?

Statistical mechanics has a wide range of applications in various fields, including condensed matter physics, astrophysics, chemistry, and materials science. It is used to understand and predict the behavior of complex systems, such as gases, liquids, and solids, and has also been applied to study biological systems and networks.

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