  I revisit the exactly solvable Kipnis--Marchioro--Presutti model of heat
conduction [J. Stat. Phys. 27 65 (1982)] and describe, for one-dimensional
systems of arbitrary sizes whose ends are in contact with thermal baths at
different temperatures, a systematic characterization of their non-equilibrium
stationary states. These arguments avoid resorting to the analysis of a dual
process and yield a straightforward derivation of Fourier's law, as well as
higher-order static correlations, such as the covariant matrix. The
transposition of these results to families of gradient models generalizing the
KMP model is established and specific cases are examined.
