Diffusion-entropy scaling across dimensions

A quantitative relationship between the diffusion coefficient $D$ of a tagged particle in a liquid and the entropy $S$ of that liquid has long been sought, as it would allow entropy to be inferred directly from diffusion measurements and transport properties to be predicted from thermodynamic information. Here, we employ extensive computer simulations to independently compute both $D$ and $S$ for Lennard-Jones (LJ) liquids and for water across a wide range of thermodynamic state points. Our study covers two and three dimensions for both systems, and additionally explores one-dimensional confinement for water. We find that the ratio of diffusion coefficients between two states follows an almost perfect exponential dependence on their entropy difference. For LJ liquids, the exponential prefactor exhibits a pronounced dependence on dimensionality $d$, consistent in trend but quantitatively distinct from theoretical predictions. In contrast, water shows a strikingly weak dimensionality $d$ dependence, deviating from theory, which we attribute to the dominant role of jump diffusion. Remarkably, the exponential diffusion-entropy relationship persists even when translational and rotational contributions to entropy are separated and considered individually. This robustness suggests that entropy provides a unifying measure governing particle mobility in liquids, largely independent of microscopic mechanisms or dimensional constraints.