A NOTE ON THE RELATION BETWEEN THE HYPOTHESIS OF LOCAL EQUILIBRIUM AND THE CLAUSIUS-DUHEM INEQUALITY

The paper is devoted to the study of certain fundamental aspects of the thermodynamics of irreversible processes. The study systematically distinguishes between quantities defined in the concrete physical space and those defined in the abstract state space. In particular, otherwise known equations are written with this distinction in mind. In them the nonequilibrium entropy SBAR, contact temperature TBAR, and contact forces FBAR, defined in physical space, are contrasted with the corresponding thermodynamic quantities S, T and F defined in state space; they are allowed to be different, as experience suggests. The thermodynamic quantities are those of the accompanying equilibrium state defined for the same values of U, a, alpha (internal energy, external and internal deformation measures) of the extensive parameters which characterize the parent nonequilibrium state. It is pointed out that the consequences of this distinction are treated differently in the formalism of Rational Thermodynamics and under the assumption of "local" equilibrium (HLE, also known as the Principle of Local State). Consequently, it becomes necessary to recognize the existence of two versions of the entropy balance equation (alias Clausius-Duhem Inequality, CDI1 and CDI2). These are formally identical, but differ in the interpretation of entropy and temperature which appear in them. The principal purpose of this paper is to examine whether, or in what measure, these two versions are consistent with each other. The paper advances a line of reasoning which demonstrates that a calculation made in accordance with HLE does not lead to the same result as one based directly on CDI1 formulated without reference to the state space. Nevertheless, the two converge to each other asymptotically under clearly defined conditions. In a digression, the paper suggests a measure for the "distance" of a nonequilibrium state from equilibrium in the form of a suitably defined Deborah number (ratio of two relaxation times). Finally, the paper indicates a correction sigma which subtracted from the entropy S of an equilibrium state associated by an adiabatic projection with a given nonequilibrium state could provide a measure of the entropy SBAR in nonequilibrium. Paradoxically, S is larger than SBAR.