Heat conduction and the nonequilibrium stationary states of stochastic energy exchange processes

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.