We argue that a proper treatment of material dispersion should be based on individual particle tracking using realistic size and density. The effect of turbulent diffusion and the scavenging of particles by precipitation are shown to be treatable as stochastic perturbations of the deterministic Newtonian equation of motion. This approach enables one to investigate the chaotic aspects of particle dispersion by means of dynamical systems concepts. Topological entropy is shown to be in this context the growth rate of material lines, which can be considered to provide a novel characterization of the state of the atmosphere. The deposition process is found to be well characterizable by the escape rate (being a measure of the strength of the exponential decay of the number of particles not yet reached the surface), which might depend on local turbulence and rain intensity. The variability of the dispersion process due to the difference between different meteorological forecasts within an ensemble forecast are also illustrated. Examples are taken from volcanic eruptions and the Fukushima accident.