  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 non-homogeneous energy
injection for long-range correlations and slow time-scales, proposing an
analogy between this simplified dynamical model and recent experiments with
dense vibro-fluidized granular materials. Several generalizations and
connections with the statistical physics of active matter are also suggested.
