  Relating thermodynamic and kinetic properties is a conceptual challenge with
many practical benefits. Here, based on first principles, we derive a rigorous
inequality relating the entropy and the dynamic propagator of particle
configurations. It is universal and applicable to steady states arbitrarily far
from thermal equilibrium. Applying the general relation to diffusive dynamics
yields a relation between the entropy and the (normal or anomalous) diffusion
coefficient. The relation can be used to obtain useful bounds for the late-time
diffusion coefficient from the calculated steady-state entropy or, conversely,
to estimate the entropy based on measured diffusion coefficients. We
demonstrate the validity and usefulness of the relation through several
examples and discuss its broad range of applications, in particular, for
systems far from equilibrium.
