A remark on the hydrodynamics of the zero-range processes

The nonequilibrium stationary hydrodynamical properties of the symmetric nearest neighbor zero-range processes are studied: local equilibrium and Fourier's law are proven to hold, and the bulk diffusion coefficient and the equal time covariance of the limiting nonequilibrium stationary density fluctuations field are computed. The result fits with those already known and confirms some conjectures derived from a time-dependent macroscopic analysis. The very simple proof is based on a result already published but may be not so well known in this context.Partially supported by NATO Grant No. 040.82.Partially supported by FAPESP: Fundacão de Amparo à Pesquisa do Estado de São Paulo, Grant No. 82/1719-9.