  Fundamental aspects of inverse kinetic theories for incompressible
Navier-Stokes equations concern the possibility of defining uniquely the
kinetic equation underlying such models and furthermore, the construction of a
kinetic theory implying also the energy equation. The latter condition is
consistent with the requirement that fluid fields result classical solutions of
the fluid equations. These issues appear of potential relevance both from the
mathematical viewpoint and for the physical interpretation of the theory. In
this paper we intend to prove that the non-uniqueness feature can be resolved
by imposing suitable assumptions. These include, in particular, the requirement
that the kinetic equation be equivalent, in a suitable sense, to a
Fokker-Planck kinetic equation. Its Fokker-Planck coefficients are proven to be
uniquely determined by means of appropriate prescriptions. In addition, as a
further result, it is proven that the inverse kinetic equation satisfies both
an entropy principle and the energy equation for the fluid fields.
