(untitled) (Journal of Computational Mathematics, 1998)
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 _t u + _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.