We consider nonconservative, reversible spin systems, with unbounded discrete spins. We show that for a class of these dynamics in a high temperature regime, the relative entropy with respect to the equilibrium distribution decays exponentially in time, although the logarithmic-Sobolev inequality fails. To this end we prove a weaker modification of the logarithmic-Sobolev inequality.
"Entropy inequalities for unbounded spin systems." Ann. Probab. 30 (4) 1959 - 1976, October 2002. https://doi.org/10.1214/aop/1039548378