  The Second Law of Thermodynamics asserts that the physical entropy of an
adiabatic system is an increasing function in time. In this paper we will study
a more stringent version of this law, according to which the entropy should not
only increase in time, but the rate of increase is optimal in absolute value
among all possible evolutions. We will establish this property in the framework
of non-linear scalar hyperbolic conservation law with strictly convex fluxes.
