In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cut-off to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cut-off Boltzmann equation in the spirit of [21, 23]. We show that the kernel modes that define the spectral method have the correct grazing collis...

A quantum theory of a two and three-level laser with injected atomic coherence is developed by using a density operator method, to the best of our knowledge, for the first time. The initial atomic coherence plays an essential role. At steady state, the equation of motion for the density operator yields to exhibit laser without inversion and a phase locking but no threshold for the laser fie...

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator techniques, we derive a Fokker-Planck equation describing the relaxation of a "test" vortex in a bath of "field" vortices at statistical equilibrium. The relaxation is due to the combined effect of a diffusion and a drift. The drift is shown to be responsible for the organization ...

Two-frequency Wigner distribution is introduced to capture the asymptotic behaviour of the space–frequency correlation of paraxial waves in the radiative transfer limits. The scaling limits give rises to deterministic transport-like equations. Depending on the ratio of the wavelength to the correlation length the limiting equation is either a Boltzmann-like integral equation or a Fokker–Planck-...