Nonequilibrium thermodynamics. II. Application to inhomogeneous systems

Abstract

We provide an extension of a recent approach to study non-equilibrium thermodynamics [Phys.Rev. E 81, 051130 (2010), to be denoted by I in this work] to inhomogeneous systems by considering the latter to be composed of quasi-independent subsystems. The system Σ along with the (macroscopically extremely large) medium Σ forms an isolated system Σ 0 . Starting from the Gibbsian formulation of the entropy for Σ 0 , which is valid even when Σ 0 is out of equilibrium, we derive the Gibbsian formulation of the entropy of Σ, which need not be in equilibrium. We show that the additivity of entropy requires quasi-independence of the subsystems, which requires that the interaction energies between different subsystems must be negligible so that the energy also becomes additive. The thermodynamic potentials of subsystems such as the Gibbs free energy that continuously decrease during approach to equilibrium are determined by the field parameters (temperature, pressure, etc.) of the medium and exist no matter how far the subsystems are out of equilibrium so that their field variables may not even exist. This and the requirement of quasiindependence make our approach different from the conventional approach due to de Groot and others as discussed in the text. As the energy depends on the frame of reference, the thermodynamic potentials and Gibbs fundamental relation, but not the entropy, depend on the frame of reference. The possibility of relative motion between subsystems described by their net linear and angular momenta gives rise to viscous dissipation. The concept of internal equilibrium introduced in I is developed further here and its important consequences are discussed for inhomogeneous systems. The concept of internal variables (various examples are given in the text) as variables that cannot be controlled by the observer for non-equilibrium evolution is also discussed. They are important because the internal equilibrium in the presence of internal variables is lost if internal variables are not used in thermodynamics. It is argued that their affinity vanishes only in equilibrium. Gibbs fundamental relation, thermodynamic potentials and irreversible entropy generation are expressed in terms of observables and internal variables. We use these relations to eventually formulate the non-equilibrium thermodynamics of inhomogeneous systems. We also briefly discuss the case when bodies form an isolated system without any medium to obtain their irreversible contributions and show that this case is no different than when bodies are in an extremely large medium.

On Equivalence of Nonequilibrium Thermodynamic and Statistical Entropies

“…Our first hurdle is about measuring or computing S for nonequilibrium states. We have shown earlier [35][36][37][38][39] that the thermodynamic entropy can be determined or measured for a special class of nonequilibrium states, which we have recently introduced as being in internal equilibrium. These states are of considerable interest in this work and are revisited in Section 3.4.…”

“…The sample approach will play a central role in our statistical formalism. We consider thermodynamic entropy S to be the physical entropy of the body alone, even though we recognize that it can only be determined experimentally using calorimetry for equilibrium states and possibly with some effort for states in internal equilibrium introduced previously [35][36][37] and to be explained below in Section 3.4. The entropy S, being an intrinsic property of the system alone, depends explicitly on the current state of the body but not that of the medium.…”

“…which using Equation (4) can be written as an extremely useful identity in the form of an equality [35][36][37][38][39] dQ(t) = T (t)dS(t)…”

“…Here, we are interested in its application to nonequilibrium classical thermodynamics [1][2][3][4][5][6][7]9] and its statistical formulation for a macroscopic system shown in Figure 1 as developed recently by us [35][36][37][38][39]. The system is contained within a clearly defined boundary (real or imaginary); the latter specifies a volume V associated with the system, which may change in time.…”

“…We briefly review the notion of a thermodynamic state or simply a state and entropy in equilibrium thermodynamics and several statistical entropies in the next section. We discuss the concept of a NEQS, state variables and state functions in Section 3 using the approach developed recently by us [35][36][37][38][39]. We briefly review the vitrification process and the residual entropy in Section 4.…”