  We discuss the application of numerical linked cluster expansions (NLCEs) to
study one dimensional lattice systems in thermal equilibrium and after quantum
quenches from thermal equilibrium states. For the former, we calculate
observables in the grand canonical ensemble, and for the latter we calculate
observables in the diagonal ensemble. When converged, NLCEs provide results in
the thermodynamic limit. We use two different NLCEs - a maximally connected
expansion introduced in previous works and a site-based expansion. We compare
the effectiveness of both NLCEs. The site-based NLCE is found to work best for
systems in thermal equilibrium. However, in thermal equilibrium and after
quantum quenches, the site-based NLCE can diverge when the maximally connected
one converges. We relate this divergence to the exponentially large number of
clusters in the site-based NLCE and the behavior of the weights of observables
in those clusters. We discuss the effectiveness of resummations to cure the
divergence. Our NLCE calculations are compared to exact diagonalization ones in
lattices with periodic boundary conditions. NLCEs are found to outperform exact
diagonalization in periodic systems for all quantities studied.
