  Quantum systems typically reach thermal equilibrium rather quickly when
coupled to a thermal environment. The usual way of bounding the speed of this
process is by estimating the spectral gap of the dissipative generator. However
the gap, by itself, does not always yield a reasonable estimate for the
thermalization time in many-body systems: without further structure, a uniform
lower bound on it only constrains the thermalization time to grow polynomially
with system size.
  Here, instead, we show that for a large class of geometrically-2-local models
of Davies generators with commuting Hamiltonians, the thermalization time is
much shorter than one would na\"ively estimate from the gap: at most
logarithmic in the system size. This yields the so-called rapid mixing of
dissipative dynamics. The result is particularly relevant for 1D systems, for
which we prove rapid thermalization with a system size independent decay rate
only from a positive gap in the generator. We also prove that systems in
hypercubic lattices of any dimension, and exponential graphs, such as trees,
have rapid mixing at high enough temperatures. We do this by introducing a
novel notion of clustering which we call "strong local indistinguishability"
based on a max-relative entropy, and then proving that it implies a lower bound
on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour
commuting models.
  This has consequences for the rate of thermalization towards Gibbs states,
and also for their relevant Wasserstein distances and transportation cost
inequalities. Along the way, we show that several measures of decay of
correlations on Gibbs states of commuting Hamiltonians are equivalent, a result
of independent interest. At the technical level, we also show a direct relation
between properties of Davies and Schmidt dynamics, that allows to transfer
results of thermalization between both.
