The Physical Entropy of Single Conservation Laws

Abstract

By means of the comparisons with the formulas in statistical mechanics and thermodynamics, in this paper it is demonstrated that for the single conservation law $\partial_{t} u + \partial_{x} f(u) = 0$, if the flux function $f (u)$ is convex (or concave), then, the physical entropy is $S = -f (u)$; Furthermore, if we assume this result can be generalized to any $f (u)$ with two order continuous derivative, from the thermodynamical principle that the local entropy production must be non-negative, one entropy inequality is derived, by which the O.A. Olejnik's famous E-condition can be explained successfully in physics.