Collision-Number Statistics for Transport Processes

Physical observables are often represented as walkers performing random displacements. When the number of collisions before leaving the explored domain is small, the diffusion approximation leads to incongruous results. In this Letter, we explicitly derive an explicit formula for the moments of the number of particle collisions in an arbitrary volume, for a broad class of transport processes. This approach is shown to generalize the celebrated Kac formula for the moments of residence times. Some applications are illustrated for bounded, unbounded and absorbing domains.