Kinetic theory with angle–action variables

We present a kinetic theory for one-dimensional inhomogeneous systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory valid at order 1/N in a proper thermodynamic limit N→+∞, we obtain a closed kinetic equation describing the relaxation of the distribution function of the system as a whole due to resonances between different orbits. This equation is written in angle–action variables. It conserves mass and energy and increases the Boltzmann entropy (H-theorem). Using a thermal bath approximation, we derive a Fokker–Planck equation describing the relaxation of a test particle towards the Boltzmann distribution under the combined effects of diffusion and friction. We mention some analogies with the kinetic theory of point vortices in two-dimensional hydrodynamics. We also stress the limitations of our approach and the connection with recent works.