  This is a review based on the presentation done at the seminar Laurent
Schwartz in December 2021. It is announcing results in the forthcoming
[Menegaki-Mouhot-Marahrens'22]. This work presents a new simple quantitative
method for proving the hydrodynamic limit of a class of interacting particle
systems on lattices. We present here this method in a simplified setting, for
the zero-range process and the Ginzburg-Landau process with Kawasaki dynamics,
in the parabolic scaling and in dimension $1$. The rate of convergence is
quantitative and uniform in time. The proof relies on a consistence-stability
approach in Wasserstein distance, and it avoids the use of the ``block
estimates''.
