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We consider a family of models having an arbitrary positive amount of mass oneach site and randomly exchanging an arbitrary amount of mass with nearestneighbor sites. We restrict to the case of diffusive models. We identify aclass of reversible models for which the product invariant measure is known andthe gradient condition is satisfied so that we can explicitly compute thetransport coefficients associated to the diffusive hydrodynamic rescaling.Based on the Macroscopic Fluctuation Theory \cite{mft} we have that the largedeviations rate functional for a stationary non equilibrium state can becomputed solving a Hamilton-Jacobi equation depending only on the transportcoefficients and the details of the boundary sources. Thus, we are able toidentify a class of models having transport coefficients for which theHamilton-Jacobi equation can indeed be solved. We give a completecharacterization in the case of generalized zero range models and discussseveral other cases. For the generalized zero range models we identify a classof discrete models that, modulo trivial extensions, coincides with the classdiscussed in \cite{FG} and a class of continuous dynamics that coincides withthe class in \cite{FFG}. Along the discussion we obtain a completecharacterization of reversible misanthrope processes solving the discreteequations in \cite{CC}.

On a class of solvable stationary non equilibrium states for mass exchange models