Extended Irreversible Thermodynamics: Non-equilibrium Equations of State

Abstract

In Chap. 2 , we postulated the existence of a generalised entropy which is compatible with some classes of evolution equations for the fluxes. Otherwise stated, our formalism aims to describe the class of processes which are compatible with the existence of a non-equilibrium entropy whose rate of production is non-negative. Once the expression of the entropy is known, there is no difficulty in deriving the corresponding equations of state, which are directly obtained as the first derivatives of the entropy with respect to the basic variables. A natural question concerns the physical meaning of these equations of state, which, of course, depend on the fluxes and therefore differ from their analogous local-equilibrium expressions. In classical thermodynamics, it is known that the derivative of the entropy with respect to the internal energy (by keeping fixed the volume and the composition of the system) is the reciprocal of the absolute temperature; the derivatives with respect to the volume and to the number of moles yield the equilibrium pressure and (with a minus sign) the chemical potentials respectively (divided by the absolute temperature). It may then be asked whether the derivatives of the generalised entropy introduced in extended irreversible thermodynamics (EIT) still allow an absolute temperature to be defined, as well as a non-equilibrium pressure and a non-equilibrium chemical potential. Another important problem is to determine whether the non-equilibrium temperature and pressure are measurable by a thermometer and a manometer. These are subtle and unsolved problems which have however received partial answers during the last years stimulated by recent developments in glasses, granular matter, flowing suspensions, nuclear collisions, nano-systems, molecular dynamics and computer simulations, or in the analysis of fluctuations. The objective of the present chapter is to better apprehend the physical meaning of the generalised entropy and to pay detailed attention to the nature of the corresponding equations of state.