I am interested in a pair_dipole style http://lammps.sandia.gov/doc/pair_dipole.html.
I went through both Allen and Toukmaji references mentioned
at the doc page, but I don't actually see where these (at least in this format)
formulas are discussed. Maybe I overlooked something.

A related question is what is actually a closed system of equations being solved.
In Toukmaji this is obtained by introducing an extended Lagrangian,
is this actually what LAMMPS integrate?

No P3M or PM methods are implemented for this, right?

I would be grateful if someone can recommend other good theoretical papers on the dipole formulation.

The equations in the compute() method are the short-range
dipole/dipole and dipole/point-charge interactions. There is no extra Langevin
term (that could be added by something like fix langevin).

We have an (unreleased) pair dipole/long that adds the damping terms
to the short-range as in A&T, however we have not released it b/c
we do not have the PPPM counterpart. We also have an unreleased
ewald/dipole that has not been integrated/tested.

Could you possible clarify to me what's the evolution equation (if any) usually used for a dipole degree of freedom
associated to each particle?
I mean, the balance of linear momentum dictates the movement of the center of mass of the composite particle
(nuclei + electrons). Torques/moments are terms in balance of angular momentum.
Apart from the explicit expression of a dipole as a moment/distribution of electrons in a composite particle,
is it usually assumed that a dipole is, kind of, rigidly connected to the particle and its change is only dictated
by rotation of the particle? In other words, what are the general assumptions made to use dipole degrees of freedom
and torques/moments?

Maybe people who use pair_dipole can comment of this?