  In the context of interacting particle systems, we study the influence of the
action of the semigroup on the concentration property of Lipschitz functions.
As an application, this gives a new approach to estimate the relaxation speed
to equilibrium of interacting particle systems. We illustrate our approach in a
variety of examples for which we obtain several new results with short and
non-technical proofs. These examples include the symmetric and asymmetric
exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics").
We also give a new proof of the Poincar\'e inequality, based on coupling, in
the context of one-dimensional Gibbs measures. In particular, we cover the case
of polynomially decaying potentials, where the log-Sobolev inequality does not
hold.
