  We offer a new point of view on the (Modified) Log-Sobolev inequality and
lower bounds on the Ricci-curvature in the setting where the dynamics are
obtained as the limit of Markov processes. In this setting, the large deviation
rate function of the stationary measures of the Markov processes, plays the
role of entropy. We define an entropy-information inequality (EII) that
generalizes the (MLSI) and is equivalent to exponential decay of the rate
function along the flow, and define an entropy-convexity inequality (ECI) that
serves as an analogue of a lower bound on the Ricci-curvature in this setting.
