  Inspired by one--dimensional light--particle systems, the dynamics of a
non-Hamiltonian system with long--range forces is investigated. While the
molecular dynamics does not reach an equilibrium state, it may be approximated
in the thermodynamic limit by a Vlasov equation that does possess stable
stationary solutions. This implies that on a macroscopic scale, the molecular
dynamics evolves on a slow timescale that diverges with the system size. At the
single-particle level, the evolution is driven by incoherent interaction
between the particles, which may be effectively modeled by a noise, leading to
a Brownian-like dynamics of the momentum. Because this self-generated diffusion
process depends on the particle distribution, the associated Fokker-Planck
equation is nonlinear, and a subdiffusive behavior of the momentum fluctuation
emerges, in agreement with numerics.
