Quasi-diagonal Inhomogeneous Closure for Classical and Quantum Statistical Dynamics

The Quasi-diagonal Direct Interaction Approximation (QDIA) closure equations are formulated for inhomogeneous classical and quantum fields interacting through dynamical equations with quadratic nonlinearity and with first or second order time derivatives. Associated more complex inhomogeneous DIA and Self-Energy closure equations are expounded as part of the derivation. The QDIA employs a bare vertex approximation and is only a few times more computationally intensive than the homogeneous DIA. Examples of applications to turbulent classical geophysical and Navier Stokes fluids, including non-Gaussian noise, to classical and quantum Klein Gordon equations with g phi^3 Lagrangian interaction, and to coupled field-auxiliary field equations associated lambda phi^4 Lagrangian interaction, are presented.