  An interesting issue in fluid dynamics is represented by the possible
existence of inverse kinetic theories (IKT) which are able to deliver, in a
suitable sense, the complete set of fluid equations which are associated to a
prescribed fluid. From the mathematical viewpoint this involves the formal
description of a fluid by means of a classical dynamical system which advances
in time the relevant fluid fields. The possibility of defining an IKT for the
3D incompressible Navier-Stokes equations (INSE), recently investigated (Ellero
\textit{et al}, 2004-2007) raises the interesting question whether the theory
can be applied also to thermofluids, in such a way to satisfy also the second
principle of thermodynamics. The goal of this paper is to prove that such a
generalization is actually possible, by means of a suitable \textit{extended
phase-space formulation}. We consider, as a reference test, the case of
non-isentropic incompressible thermofluids, whose dynamics is described by the
Fourier and the incompressible Navier-Stokes equations, the latter subject to
the conditions of validity of the Boussinesq approximation.

Fluid Dynamics;Classical Physics