INTERNAL VARIABLES IN THE LOCAL-EQUILIBRIUM APPROXIMATION

The aim of this contribution is to explore the basis and consequences of the formalism known in the literature as the method of local equili brium. The view is taken that contemporary controversies regarding the foundations of thermodynamics are rooted not only in different sets o f concepts and principles, but also in semantics. Therefore, an attemp t is made here to use a consistent group of terms, each of whose dicti onary meaning corresponds to its physical nature as closely as possibl e. Since the intention is to study irreversible processes in systems i n which even locally there prevails a state of nonequilibrium, the ter m local equilibrium is abandoned in favor of the phrase principle of l ocal state. In defining the thermodynamic state of a system, a distinc tion is made between the intensive parameters which appear in the phys ical space (Kontaktgroessen) and those which describe states of constr ained equilibrium in the Gibbsian phase (state) space. The latter cons ists of a set of extensive variables (internal energy U, external defo rmation parameters a and the internal deformation variables alpha) bec ause, in contrast with intensive variables, they can be measured in eq uilibrium as well as in nonequilibrium. The properties and uses of the internal variables are outlined following P.W. Bridgman's proposal. T he principle of local state is applied by associating with every noneq uilibrium state n an accompanying equilibrium state e of equal values of U, a, alpha, and by asserting that the entropy SBAR assignable in p hysical space and temperature TBAR measured in it can be approximated by the values S and T calculated in the Gibbsian phase space by standa rd, classical methods. A continuous sequence of accompanying equilibri um states (curve R in phase space) is called an accompanying reversibl e process, it is conceived as an adiabatic projection of the continuou s sequence of nonequilibrium states which constitute the irreversible process I. This allows us to cast the classical Gibbs equation in rate form and to derive explicit expressions for the rate of entropy produ ction THETA by eliminating the rate du/dt between it and the energy ba lance equation. The nature of the approximation involved in this proce dure is made explicit. Hence, the preference to speak of the local sta te approximation. The essential part of the method consists in the for mulation of the Gibbs equation for the accompanying reversible process in the phase spice. This is obtained from the knowledge of the physic s of the situation and leads to the identification of the internal def ormation variables and the hypothetical virtual (i.e., reversible) wor k done against them. The Gibbs equation forms the basis for the deriva tion of an explicit form of the local rate of entropy production and a rational formulation of the rate equations between the generalized fo rces and fluxes which appear in it. The union between the rate equatio ns and the fundamental equation relating the extensive variables of th e constrained equilibrium states in phase space yields the constitutiv e law for the system. The constitutive law with the rate equations ins erted into the appropriate conservation laws produces the set of parti al differential equations which govern the process. Their solution, in the form of a set of time-dependent fields, subject to the appropriat e boundary conditions constitutes the irreversible process under study . It is noted that the local-state approximation, made explicit in thi s paper, has been used and tested in fluid mechanics though its validi ty is contested in contemporary continuum mechanics and mechanics of s olids. The author takes the view that these principal theories of irre versible processes, as they occur in engineering applications, fit int o the same formalism, as expounded in the text.