Integral Expansion often Reducing to the Density Gradient Expansion, Extended to Nonmarkov Stochastic Processes. Consequent Stochastic Equation for Quantum Mechanics more Refined than Schrodinger’s.

Abstract

An integral expansion is obtained which reduces under explicitly given conditions to the density gradient expansion for the number density p(r,t) 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 proved to be valid for both Markov and nonMarkov 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. In a nonMarkovian stochastic process with inertia the average local velocity <v> of a spherical cell having a radius equal to the mean free path λ depends on all the preceding history and on the local diffusion velocity v_D.