From long-range interaction to collective behaviour and from Hamiltonian chaos to stochastic models

Assemblies of charged particles are the prototype of systems with long range interactions. Their behaviour involves effective degrees of freedom of two types. Collective components emerge, commonly classified as modes or waves, which are (often) nonlinearly coupled, or resonant, with a small subset of the initial assembly of particles. The reduction from many bodies to the effective model for this self-consistent dynamics may be performed in the Lagrangian setting, and the resulting dynamics is Hamiltonian.For regular interactions (e.g. regularized Coulomb) the many-body limit leads to the Vlasov equation, but its validity for long times is questionable.The self-consistent dynamics generates chaotic behaviour, leading to some transport in phase space. However, this transport is generally not a diffusion process, and when it is diffusion-like its parameters may have unexpected values. In the strongly chaotic limit (with many overlapping resonances), however, the transport does tend to a diffusion process, with parameters determined by a straightforward prescription—though the physical elementary process underlying short-term transport is not the same process which underlies long-time transport, due to the interplay of chaos and granularity (finite N) effects.