Recent efforts have shown that the dynamic properties of a wide class of liquids can be mapped onto semi-universal scaling laws and constitutive relations that are motivated by thermodynamic analyses of much simpler models. In particular, it has been found that many systems exhibit dynamics whose behavior in state space closely follows that of soft-sphere particles interacting through an inverse power repulsion. In the present work, we show that a recently-developed coarse-graining theory

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... es a natural way to understand how arbitrary liquids can be mapped onto effective soft-sphere models and hence how one might potentially be able to extract underlying dynamical scaling laws. The theory is based on the relative entropy, an information metric that quantifies how well a soft-sphere approximation to a liquid's multidimensional potential energy landscape performs. We show that optimization of the relative entropy not only enables one to extract effective soft sphere potentials that suggest an inherent scaling of thermodynamic and dynamic properties in temperature-density space, but that also has rather interesting connections to excess-entropy based theories of liquid dynamics. We apply the approach to a binary mixture of Lennard-Jones particles, and show that it gives effective softsphere scaling laws that well-describe the behavior of the diffusion constants. Our results suggest that the relative-entropy formalism may be useful for "perturbative" type theories of dynamics, offering a general strategy for systematically connecting complex energy landscapes to simpler reference ones with better understood dynamic behavior.