  In this paper, we prove modified logarithmic Sobolev inequalities for
canonical ensembles with superquadratic single-site potential. These
inequalities were introduced by Bobkov and Ledoux, and are closely related to
concentration of measure and transport-entropy inequalities. Our method is an
adaptation of the iterated two-scale approach that was developed by Menz and
Otto to prove the usual logarithmic Sobolev inequality in this context. As a
consequence, we obtain convergence in Wasserstein distance $W_p$ for Kawasaki
dynamics on the Ginzburg-Landau model.
