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We consider a velocity field with linear viscous interactions defined on aone dimensional lattice. Brownian baths with different parameters can becoupled to the boundary sites and to the bulk sites, determining differentkinds of non-equilibrium steady states or free-cooling dynamics. Analyticalresults for spatial and temporal correlations are provided by analyticaldiagonalisation of the system's equations in the infinite size limit. Wedemonstrate that spatial correlations are scale-free and time-scales becomeexceedingly long when the system is driven only at the boundaries. On thecontrary, in the case a bath is coupled to the bulk sites too, an exponentialcorrelation decay is found with a finite characteristic length. This is alsotrue in the free cooling regime, but in this case the correlation length growsdiffusively in time. We discuss the crucial role of boundary driving forlong-range correlations and slow time-scales, proposing an analogy between thissimplified dynamical model and dense vibro-fluidized granular materials.Several generalizations and connections with the statistical physics of activematter are also suggested.

Long range correlations and slow time scales in a boundary driven granular model