In the following an evolution problem posed by an initial condition is considered, where the boundaries of the simulation domain are assumed to be far enough away to have no influence on the evolution. If a bounded domain is considered, the initial condition must be replaced by a boundary condition when a particle reaches the boundary (refer to [114] for a rigorous treatment). Injecting boundary conditions, which are used in stationary problems, are not considered here, but only boundaries of a computational nature, i.e. the finite range of the phase space. The case for stationary problems is analysed in [63].
First the WBE is written as a Fredholm integral equation. The adjoint equation of the latter is then used to express the mean value of an arbitrary physical quantity as a Neumann series. The value of this series represents the solution to the computational problem and can be determined by a stochastic sampling. This sampling procedure is performed using numerical particles, which follow Newton trajectories and are scattered between different states as determined by the Wigner potential and other scattering mechanisms.