Integral expansion often reducing to the density-gradient expansion, extended to non-Markov stochastic processes: Consequent non-Markovian stochastic equation whose leading terms coincide with Schrödinger's.

Abstract

An integral expansion is obtained that reduces under explicitly given conditions to the density-gradient expansion for the number density of stochastic particles. Explicit coefficients in terms of moments are calculated up to and including the fourth order, corresponding to the super-Burnett approximation. The expansion is proven to be valid for both Markov and non-Markov processes, including those with infinite memory, typical of a stochastic motion with inertia. An application of this expansion is the relation between local average velocities in Markovian stochastic processes and the drift velocity of a probability cloud. As a further application we obtain a non-Markovian stochastic equation whose leading terms coincide with the Schr\"odinger equation. The additional terms could be interpreted as corrections that produce contributions to atomic spectra of the fine-structure type. In the case of scattering, where the density gradients can be locally high, the correction terms become relevant and they could perhaps explain the differences between the calculated and measured rotational and/or vibrational cross sections for hydrogen molecules.