Perspective: Measuring physical entropy out of equilibrium

Entropy is one of the key thermodynamic variables reflecting changes in the state of matter. Unlike other thermodynamic variables, it is well-defined also for nonequilibrium steady states through its relation to information. Applying this relation to physical systems is an ongoing challenge, as it requires knowledge of microscopic high-dimensional continuous distributions which is generally unattainable. A set of new approaches for the measurement of entropy in nonequilibrium steady or absorbing states have been developed and successfully applied to identify dynamic structures and transitions in diverse systems, ranging from jammed packings to swarming bacteria. We briefly review these approaches, emphasizing why applications to physical systems, including those out of equilibrium, is substantially different from the general statistical challenge of entropy estimation and inference. We point at promising current and future directions.