A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension

Abstract

Deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of non-Markovian stochastic evolution equations after an average over realization using a theorem. Examples are given, show that deterministic differential equations that contain derivatives with respect to time higher than or equal to two can be derived after a Taylor series expansion of the dynamical variables. It is shown that the derivation of such deterministic differential equations can be done by solving a set of linear equations that increase in number after increasing the number of previous time steps in the updating rules that define a given model. Two explicit examples, the first containing updating rules that depend on two previous time steps and the second on three, are worked in some detail in order to show some features of the linear transformation that allow one to obtain the deterministic differential equations.