7 Conclusions

Application of the numerical MC method for solving integral equations has been thoroughly studied for the cases of the transient and stationary Boltzmann equation. The well known Ensemble MC and the Single-Particle MC algorithms have been rederived in a unified way. In addition, specific algorithms which are very seldomly used have been discussed. The backward MC algorithm is well suited for the simulation of rare events, whereas the weighted MC method has been demonstrated to be an effective statistical enhancement technique. The same numerical approach proven to be successful for the semi-classical transport problem has been applied to solve the Wigner-Boltzmann equation. A MC method for the simulation of far-from-equilibrium transport in nano-structures has been developed. Treating the Wigner potential operator as a source of scattering is complicated by the so-called negative sign problem, which would lead to a run-away of variance, unless a proper variance reduction technique is used.