Simulation of charge transport has become a strategically important CAD tool for the semiconductor industry in the development of new technologies.
Due to the increasing miniaturization of the device, Monte Carlo simulations are necessary just on a small portion of the device where strong non-equilibrium phenomena occur; the large part of the the device is accurately described by macroscopic equations (such as drift-diffusion, hydrodynamic models).
In order to determine the domain of validity of the kinetic-macroscopic approaches, we need a criteria to determine whether the Monte Carlo particle ensemble is near enough to a Maxwellian-like distribution or not.
Sobolev Norms in Hs for s<-\frac32; can be used to measure this distance: the norm is exactly computed in the case s=-2. Numerical results for bulk silicon are shown.