A nonlinear Coulomb interaction term appears in the equations describing the motions of a collection of Paul-trapped particles. A realistic description may also include a damping term which arises from collisions with a background gas and/or from laser cooling. The motion of the ion in a collection of N ions is given by:
where , , and are the
damping forces in the x-,
y-, and z- directions respectively, is the distance between
particles j and i, and is the unit vector pointing from
particle j to particle i.