The aim of this contribution is to explore the basis and consequences of the formalism known in the literature as the method of local equilibrium. The view is taken that contemporary controversies regarding the foundations of thermodynamics are rooted not only in different sets of concepts and principles, but also in semantics. Therefore, an attempt is made here to use a consistent group of terms, each of whose dictionary meaning corresponds to its physical nature as closely as possible. Since the intention is to study irreversible processes in systems in which even locally there prevails a state of nonequilibrium, the term local equilibrium is abandoned in favor of the phrase principle of local state. In defining the thermodynamic state of a system, a distinction is made between the intensive parameters which appear in the physical space (Kontaktgroessen) and those which describe states of constrained equilibrium in the Gibbsian phase (state) space. The latter consists of a set of extensive variables (internal energy U, external deformation parameters a and the internal deformation variables alpha) because, in contrast with intensive variables, they can be measured in equilibrium as well as in nonequilibrium. The properties and uses of the internal variables are outlined following P.W. Bridgman's proposal. The principle of local state is applied by associating with every nonequilibrium state n an accompanying equilibrium state e of equal values of U, a, alpha, and by asserting that the entropy SBAR assignable in physical space and temperature TBAR measured in it can be approximated by the values S and T calculated in the Gibbsian phase space by standard, classical methods. A continuous sequence of accompanying equilibrium states (curve R in phase space) is called an accompanying reversible process, it is conceived as an adiabatic projection of the continuous 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 production THETA by eliminating the rate du/dt between it and the energy balance equation. The nature of the approximation involved in this procedure is made explicit. Hence, the preference to speak of the local state approximation. The essential part of the method consists in the formulation of the Gibbs equation for the accompanying reversible process in the phase spice. This is obtained from the knowledge of the physics of the situation and leads to the identification of the internal deformation variables and the hypothetical virtual (i.e., reversible) work done against them. The Gibbs equation forms the basis for the derivation of an explicit form of the local rate of entropy production and a rational formulation of the rate equations between the generalized forces and fluxes which appear in it. The union between the rate equations and the fundamental equation relating the extensive variables of the constrained equilibrium states in phase space yields the constitutive law for the system. The constitutive law with the rate equations inserted into the appropriate conservation laws produces the set of partial differential equations which govern the process. Their solution, in the form of a set of time-dependent fields, subject to the appropriate boundary conditions constitutes the irreversible process under study. It is noted that the local-state approximation, made explicit in this paper, has been used and tested in fluid mechanics though its validity is contested in contemporary continuum mechanics and mechanics of solids. The author takes the view that these principal theories of irreversible processes, as they occur in engineering applications, fit into the same formalism, as expounded in the text.
