Completeness of the Lattice-Boltzmann IKT approach for classical incompressible fluids

Despite the abundant literature on the subject appeared in the last few years, the lattice Boltzmann method (LBM) is probably the one for which a complete understanding is not yet available. As an example, an unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper aims at investigating discrete inverse kinetic theories (IKT) for incompressible fluids. We intend to show that the discrete IKT can be defined in such a way to satisfy, in particular, the requirement of \emph{completeness}, i.e., {\it all} fluid fields are expressed as moments of the kinetic distribution function and {\it all} hydrodynamic equations can be identified with suitable moment equations of an appropriate inverse kinetic equation IKE.