  We prove that spin chains weakly coupled to a large heat bath thermalize
rapidly at any temperature for finite-range, translation-invariant commuting
Hamiltonians, reaching equilibrium in a time which scales logarithmically with
the system size. Our main result is a generalization to the quantum setting of
a seminal result of Holley and Stroock for classical spin chains and represents
an exponential improvement over bounds based on the non-closure of the spectral
gap. From a physical point of view, our result rigorously establishes the
absence of dissipative phase transition for Davies evolutions over
translation-invariant spin chains. The result also applies in the case of
Symmetry Protected Topological phases where the evolution is respecting the
symmetry of the phase. This has wide-ranging applications to the study of
many-body in and out-of-equilibrium quantum systems.
