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 Behavior of the Specific Entropy in the Hydrodynamic Scaling Limit." Ann. Probab. 29 (3) 1086 - 1110, July 2001. https://doi.org/10.1214/aop/1015345597