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In this note, I summarise and comment on joint work with C. Bernardin, V.Kannan and J. L. Lebowitz concerning two harmonic systems with bulk noiseswhose nonequilibrium steady states (NESS) are nearly identical (they share thesame thermal conductivity and two-point function), but whose hydrodynamicproperties (convergence towards the NESS) are very different. The goal is todiscuss the results in the general context of nonequilibrium properties ofdynamical systems, in particular, what they tell us about possible effectivemodels, or predictive approximations, for such systems.

Energy fluctuations, hydrodynamics and local correlations in harmonic systems with bulk noises