  We introduce on any smooth oriented minimal surface in Euclidean $3$-space a
meromorphic quadratic differential, $P$, which we call the entropy
differential. This differential arises naturally in a number of different
contexts. Of particular interest is the realization of its real part as a
conservation law for a natural geometric functional -- which is, essentially,
the entropy of the Gauss curvature. We characterize several classical surfaces
-- including Enneper's surface, the catenoid and the helicoid -- in terms of
$P$. As an application, we prove a novel curvature estimate for embedded
minimal surfaces with small entropy differential and an associated compactness
theorem.
