The free flight step analysis in the Monte Carlo Method. Test case: charge transport in graphene

Monte Carlo methods are now a definitively well established methods for studying the transport of charges in semiconductors. A lot of drawbacks are however present about the correct reconstruction of the distribution function, which has to be bounded between $0$ and $1$. The standard Ensemble Monte Carlo method is not able to properly include the Pauli exclusion principle and the numerical distribution functions exceed the maximum value. A lot of attempts were made to solve the problem and recently a new DSMC scheme has been developed to overcome the problem. All the improvements are generally based on some approximations or computational techniques involving the distribution function and the scattering terms. Only an attempt was devoted to opportunely change the formulation of the free flight step but there is still a lack in the literature regarding a both theoretical and computational analysis of corrected Monte Carlo methods based on the free flights. In this paper we would like to fulfill such a gap by analyzing deeper the role of the free flight step into a Monte Carlo simulation, its physical properties and the conditions that allow us to preserve the physical consistency of the method.