On time reversible description of the process of coagulation and fragmentation

The process of coagulation is associated with scalar conservation laws, where the adhesion particle dynamics results from shock waves. Conversely, the fragmentation of a massive particle into a number of smaller ones, or into a continuous (dust) distribution, is associated with refraction waves. It is generally agreed that a reversible solution of a conservation law cannot include neither shock waves, nor the spontanus emergence of refraction waves. The present paper is an attemp to demonstrate that both coagulation and fragmentation may coexist for a reversible solution, under a natural generalization of the system of conservation law. This is done by the introduction of an action principle which includes, in addition to the inertial (kinetic energy) term, also an appropriatly defined internal energy. The above generalization of the system of conservation law appears as the Euler-Lagrange equations for this action.