Stochastic Formulation of Newton’s Acceleration

Abstract

The theoretical equivalence of the Wigner and ballistic Boltzmann equations for up to quadratic electric potentials provides the convenient opportunity to evaluate stochastic algorithms for the solution of the former equation with the analytic solutions of the latter equation - Liouville trajectories corresponding to acceleration due to a constant electric field. The direct application of this idea is impeded by the fact that the analytic transformation of the first equation into the second involves generalized functions. In particular, the Wigner potential acts as a derivative of the delta function which gives rise to a Newtonian accelerating force. The second problem is related to the discrete nature of the Wigner momentum space. These peculiarities incorporate unphysical effects in the approximate Wigner solution, which tends to the Boltzmann counterpart in a limiting case only.