  We consider a velocity field with linear viscous interactions defined on a
one dimensional lattice. Brownian baths with different parameters can be
coupled to the boundary sites and to the bulk sites, determining different
kinds of non-equilibrium steady states or free-cooling dynamics. Analytical
results for spatial and temporal correlations are provided by analytical
diagonalisation of the system's equations in the infinite size limit. We
demonstrate that spatial correlations are scale-free and time-scales become
exceedingly long when the system is driven only at the boundaries. On the
contrary, in the case a bath is coupled to the bulk sites too, an exponential
correlation decay is found with a finite characteristic length. This is also
true in the free cooling regime, but in this case the correlation length grows
diffusively in time. We discuss the crucial role of boundary driving for
long-range correlations and slow time-scales, proposing an analogy between this
simplified dynamical model and dense vibro-fluidized granular materials.
Several generalizations and connections with the statistical physics of active
matter are also suggested.
