  The main objective of this paper is to show that, within the present
framework of the kinetic theoretical approach to irreversible thermodynamics,
there is no evidence that provides a basis to modify the ordinary Fourier
equation relating the heat flux in a non-equilibrium steady state to the
gradient of the local equilibrium temperature. This fact is supported, among
other arguments, through the kinetic foundations of generalized hydrodynamics.
Some attempts have been recently proposed asserting that, in the presence of
non-linearities of the state variables, such a temperature should be replaced
by the non-equilibrium temperature as defined in Extended Irreversible
Thermodynamics. In the approximations used for such a temperature there is so
far no evidence that sustains this proposal.
