An H-theorem for incompressible fluids
  A basic aspect of the kinetic descriptions of incompressible fluids based on an inverse kinetic approaches is the possibility of satisfying an H-theorem. This property is in fact related to the identification of the kinetic distribution function with a probability density in an appropriate phase space. Goal of this investigation is to analyze the conditions of validity of an H-theorem for the inverse kinetic theory recently proposed by Ellero and Tessarotto [2004, 2005]. It is found that the time-dependent contribution to the kinetic pressure, characteristic of such a kinetic model, can always be uniquely defined in such a way to warrant the constancy of the entropy. 
M.Tessarotto and M.Ellero