Abstract
The Vlasov equation of kinetic theory is introduced and the Hamiltonian structure of its moments is presented. Then we focus on the geodesic evolution of the Vlasov moments. As a first step, these moment equations generalize the Camassa-Holm equation to its multi-component version. Subsequently, adding electrostatic forces to the geodesic moment equations relates them to the Benney equations and to the equations for beam dynamics in particle accelerators. Next, we develop a kinetic theory for self assembly in nano-particles. Darcy's law is introduced as a general principle for aggregation dynamics in friction dominated systems (at different scales). Then, a kinetic equation is introduced for the dissipative motion of isotropic nano-particles. The zeroth-moment dynamics of this equation recovers the classical Darcy's law at the macroscopic level. A kinetic-theory description for oriented nano-particles is also presented. At the macroscopic level, the zeroth moments of this kinetic equation recover the magnetization dynamics of the Landau-Lifshitz-Gilbert equation. The moment equations exhibit the spontaneous emergence of singular solutions (clumpons) that finally merge in one singularity. This behaviour represents aggregation and alignment of oriented nano-particles. Finally, the Smoluchowski description is derived from the dissipative Vlasov equation for anisotropic interactions. Various levels of approximate Smoluchowski descriptions are proposed as special cases of the general treatment. As a result, the macroscopic momentum emerges as an additional dynamical variable that in general cannot be neglected.