Dipole interaction potential for mixed species

[shehry1]

Homework Statement

Not a homework question. I need the solution to this to model another problem.

We all know the potential for the electromagnetic dipole dipole interaction (e.g. see Jackson)

I want to know if there is a solution to the problem in which there are several replicas of the two dipoles all mixed up together (as in the case of polarized gases) and were occupying the same volume. e.g. we might have a 1000000 p1 and say 100000 p2 with p1 and p2 being the two dipole moments.

Obviously I would have to make some kind of an integration over all the particles. But this is where I am somewhat unsure. Since the two species are mixed up hence I do not think an integration over the entire volume would do the trick.

As an approximation - I am thinking about ignoring the shielding effect altogether

Homework Equations

Only the dipole interaction potentials
http://en.wikipedia.org/wiki/Magnetic_dipole-dipole_interactionEDIT: To make matters simple the two gases are polarized parallel to each other

The Attempt at a Solution

 

[NateD]

I don't think there is a simple solution to this problem, but I do have a few ideas.1) Use an approximation where the dipole-dipole interaction is assumed to be isotropic. This means that we can assume all of the dipoles interact with each other with equal strength and that the integration can be done over the entire volume.2) Use a Monte Carlo method to simulate the system. This would involve randomly sampling the positions of the two types of dipoles and then calculating the dipole-dipole interactions between them.3) Use a mean field approach to approximate the dipole-dipole interactions. This would involve calculating the average field from all of the dipoles and then using this field to calculate the interactions.4) Use a numerical method such as finite elements or finite differences to solve the problem. This would involve discretizing the volume into small elements or points and then solving for the dipole-dipole interactions between them.