We now face the real problem: compute the interaction of the colliding particles. If we assume a hard sphere interaction, we compute the new velocities by imposing momentum and (kinetic) energy conservation
If we assume equal masses, we finally arrive at the solution
or 
- that is, the interaction produces a
simple exchange of velocities. Differing masses offers more
interesting interactions, as does dissipation.
Back to the collisions. Only particles j and k are involved in the collision, so only their future (and the future of their collision partners) is affected. So we need to recompute collision times for particles j, k, and their possible partners. Once the collision time list is recalculated, proceed as before with ordering that list, moving particles, and calculating interactions.