**Thermodynamic variables such as temperature and pressure are ill-defined out of thermal equilibrium. However, the relation between entropy and the information contained in the statistics of the system's microstates is assumed to hold regardless of how far the system is from equilibrium. We proved, based on first principles, a universal inequality relating the entropy of a system at steady state and the diffusion coefficient of its constituents. The relation is valid arbitrarily far from equilibrium. It can be used to obtain a lower bound for the diffusion coefficient from the calculated thermodynamic entropy or, conversely, an upper bound for the entropy based on measured diffusion coefficients. We demonstrate the applicability of the relation in several examples. We derived a functional which takes as input measurable pair-correlations (such as the structure factor) and gives a useful upper bound for the entropy. We use it to pin-point and characterize dynamic transitions in several experimental and computational systems, including driven and active particles.
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