Extension of the R-Sigma method to any order

The modeling of nanoscale devices requires the introduction of quantum-mechanical features into the transport equations used to describe the collective motion of the carriers within the crystal. For this reason a number of models have been devised, under the general heading of “quantum corrections,” with the aim of embedding additional terms into the standard transport equations without substantially modifying their form. Recently, a method indicated with the “R-” acronym has been introduced, which overcomes the Ehrenfest approximation while keeping the Newtonian form of the single-particle dynamics. This makes it possible to consistently include, into the derivation of the transport equation from the Liouville theorem, higher-order moments of the wave function. In the original formulation, the R- equations were derived up to the second order of the expansion into moments. In this paper the derivation is carried out to any order.