Abstract:

In this talk I am going to consider a Boltzmann-type equation for the description of a collision dynamic which is not instantaneous. This new class of kinetic equations has been introduced by Kanzler, Schmeiser, and Tora to model ensembles of living agents, where the changes of state are the result of complicated internal processes, and not simple mechanical interactions. We extend their work introducing a first-order approximation to the instantaneous equation, where non-binary collisions are included. This is motivated by the fact that during an extended collision period there is a positive probability that a colliding pair is joined by additional particles. The interaction kernel is of alignment type, where the states of the particles approach each other. For this spatially homogeneous approximation, we check that the formal properties of the system are kept. Furthermore, existence and uniqueness of solutions are examined.