The paper studies the behavior of the specific entropy for one-dimensional simple exclusion
processes under the hydrodynamic scaling of time and space. It is shown that if the initial
configurations possess a macroscopic profile then for each positive macroscopic time the
specific microscopic entropy converges to the macroscopic entropy. The latter is defined in
terms of the solution of the corresponding hydrodynamic equation.

The paper studies the behavior of the specific entropy for Ginzburg-Landau type models
under the hydrodynamical scaling of time and space. It is shown that if the initial
configurations possess a macroscopic profile then for any fixed positive macroscopic time
the specific microscopic entropy converges to the macroscopic entropy. The latter is defined
in terms of the solution of the corresponding hydrodynamical equation, which is a non-linear
diffusion equation. The above result is equivalent to the following statement: under the …