Abstract The diffusion process of a passive contaminant in the atmospheric boundary layer can be modelled in terms of a Eulerian description or in terms of a Lagrangian description. Here the relation between the two methods is investigated. It is shown that the Lagrangian motion of a particle, which may take only n distinct velocities, can be connected to an n-th order closure model of the Eulerian equations. This procedure is elaborated for a third order closure model, where it is possible to investigate non-Gaussian turbulence. An interpretation is given of the Eulerian turbulent fluxes in terms of particle motion. It turns out that the diffusion process is modelled by a hyperbolic set of equations. The Eulerian moments of the concentration distribution agree with Taylor's theory. A consequence of the hyperbolic model is that the contaminant diffuses with a finite velocity. The width of the concentration distribution grows linearly in the vicinity of the source. This is in contrast with the results from the usual gradient transfer theory.