Block Factorization of the Relative Entropy via Spatial Mixing

We consider spin systems in the $d$-dimensional lattice $Z^d$ satisfying so-called strong spatial mixing condition. show that relative entropy functional of corresponding Gibbs measure satisfies a family inequalities which control on given region $V\subset Z^d$ terms weighted sum entropies blocks $A\subset V$ when each $A$ is an arbitrary nonnegative weight $\alpha_A$. These generalize well known logarithmic Sobolev inequality for Glauber dynamics. Moreover, they provide natural extension classical Shearer satisfied by Shannon entropy. Finally, imply modified give quantitative convergence to equilibrium block dynamics heat bath type.