Abstract
In this work, we are interested in a molecular theory (i.e., statistical mechanics) of time- and space-dependent nonequilibrium (irreversible) processes in matter regarded as composed of many discrete particles.
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Notes
- 1.
This viewpoint toward volume (per mole) should be modified because it does not have a molecular representation free from macroscopic variables in its definition. It will be found more appropriate to regard it as a mean volume allowed to molecules in a randomly distributed assembly of particles. Especially, if the fluid is away from equilibrium the latter concept is more appropriate since specific volume is not the same as the mean volume per molecule. For the notion of molecular representation of mean volume per molecule, see Ref. [7] and also Ref. [8]. Modification of manifold \(\mathfrak {P}\) will be discussed in a later chapter in which volume transport phenomena are discussed.
- 2.
A conservation law can be best stated by means of a cyclic process since the state of the system must be exactly restored on completion of the cyclic process and so must be the value of internal energy of the system. Vanishing cyclic integrals consequently play important roles in formulation of thermodynamics of irreversible processes.
- 3.
In some textbooks in thermodynamics inexact differentials are denoted with a bar on the differential sign, e.g.,
or 
. In this work we do not adopt such a notation. - 4.
This modification of the viewpoint can be implemented if the notion of volume allotted to a molecule is adopted [7] via the Voronoi volume. This notion will be more easily understood if the theory is formulated as a molecular theory as will be done in a later chapter on irreversible processes in condensed phase where volume transport is explicitly taken into account. In phenomenological formulation, the notion must be formally accepted, but unfortunately would remain abstract.
- 5.
It should be noted that fluid particle velocity is not the velocity of a molecule comprising the fluid. It is the velocity of the packet of fluid in an elementary volume of fluid at position \(\mathbf {r}\) and time t. From the molecular theory viewpoint it is the ensemble average of molecular velocities in a sufficiently small volume around the position \(\mathbf {r}\) at time t. For its molecular theory definition, see Chaps. 3, 5, 6, and 7 of this work.
- 6.
This theorem is proved in Chap. 4, Ref. [3]. See also H. Poincaré, Thermodynamique (Georges Carré, Paris, 1892).
- 7.
It must be recognized that the presence and notion of unavailable work in an irreversible cycle is not as apparent in the Clausius and Kelvin principles as in the Carnot theorem, although they are equivalent to the Carnot theorem. In the end, it was the Carnot theorem that Clausius used to formulate the inequality named after him.
- 8.
In this connection, I would like to quote a passage in an article by I. Prigogine in Science 201, 777 (1978) where he states “150 years after its formulation the second law of thermodynamics still appears to be more a program than a well-defined theory in the usual sense, as nothing precise (except the sign) is said about the S production”. In view of the fact that entropy was not defined for irreversible processes by Clausius, we interpret that the S production here is meant for the uncompensated heat despite the notation S used for entropy in the aforementioned paper. In fact, Prigogine’s use of terminology appears to be oblivious to the genesis of the notion of Clausius entropy S, as is evident from the discussion given below.
- 9.
Reversible processes are traditionally defined as those of quasistatic processes which are in continuous equilibrium with the surroundings, but it is more mathematically precise to define them as quasistatic processes for which \(\mathrm {N}=0.\) This will be the definition of reversible processes used throughout in this work.
- 10.
- 11.
There are basically two different versions of extended irreversible thermodynamics: one class of versions can be found in Refs. [36, 37, 41, 43, 44] and the other in Refs. [6, 35, 45, 46] and this work. In the former, it is assumed that there exists a nonequilibrium entropy which is a state function in the thermodynamic space and the nonequilibrium entropy is statistically represented by approximations of the Boltzmann entropy or its dense fluid generalization or the information entropy for dynamical systems, without a support of a kinetic equation. In the latter class (i.e., in Refs. [6, 45, 46]), it is shown from the second law of thermodynamics that there exists a quantity called compensation function (renamed to calortropy in this and earlier work) and its differential is an exact differential in the thermodynamics space by virtue of the second law. The compensation function, however, is not the same as the Boltzmann entropy appearing in the kinetic theory of dilute gases by Boltzmann. Since thermodynamics of irreversible processes must be securely founded on the thermodynamic laws, it is crucial to show that the basic equations are consistent with the thermodynamic laws and, for example, the extended Gibbs relation is equivalent to the second law of thermodynamics. An assumption for such a basic equation is not acceptable if the resulting theory will have anything to do with the thermodynamic laws in accounting for macroscopic processes in nature. Neither can the thermodynamic laws afford approximate representations. The basic thermodynamic equation in the formulation made in Refs. [6, 46] and in this work is a rigorous consequence of the second law of thermodynamics which mathematically extends equilibrium thermodynamics that is certainly endowed with a physical basis supported by the second law; it is not an assumption as in Refs. [35–44].
- 12.
A simple, albeit somewhat ideal, example would be a cyclic process consisting of the irreversible segment where \(dN<0\) and a reversible segment for the remainder of the cycle in which \(dN=0\). This cycle, if possible to construct, would violate the second law.
- 13.
In his work, Clausius made no mention of embedding the states A and B in the equilibrium part of the thermodynamic manifold. He simply constructed a cycle consisting of an irreversible segment starting from state A and ending at state B and a reversible segment starting from state B and restoring the system to state A. Since states A and B are also part of a reversible process, they must be embedded in the equilibrium part of the thermodynamic manifold. It now may be stated that there was no clear notion of thermodynamic manifold in the Clausius formulation of the second law of thermodynamics.
- 14.
Inequality (2.60), together with (2.54) and (2.55), means that for an isolated system \(\Delta S_{\text {e}}\ge 0\) as is generally taken in the thermodynamics literature. However, this inequality for an isolated system can be misleading and cause confusion. In applying this inequality, it must be remembered that \(\Delta S_{\text {e}}\) is for the reversible segment complementary to the irreversible step making up the cycle in question and the inequality (2.60) simply means that \(\Delta S_{\text {e}}\) for the reversible segment is always larger than the irreversible compensated heat change. If the system is isolated, the compensated heat \(dQ=0\) everywhere in the interval of integration and hence \(\Delta S_{\text {e}}=0\) identically. This conclusion is also consistent with the definition of \(dS_{\text {e}}\) in (2.44). It is clearly convenient to think in terms of \(\Delta \Psi \ge 0\) instead of \(\Delta S_{\text {e}}\ge 0\).
- 15.
- 16.
Although (2.73) and (2.74) are sufficient for \(\mathbf {J}_{\text {c}}\) and \(\Xi _{\text {c}}\) to yield a one-form for \(d_{t}\hat{\Psi }\), it is not unique. For example, a vector \(\mathbf {A}\) and its divergence \(\nabla \varvec{\cdot A}\) can be added to \(\mathbf {J}_{\text {c}}\) and \(\Xi _{\text {c}}\), respectively, with no effect at all on the one-form that can be obtained from the balance equation for the calortropy. In fact, the vector \(\mathbf {A}\) can be taken with \(\sum _{a=1}^{r}\mathbf {J}_{a}/m_{a}\) as can be seen later in Chap. 3. The formulas for \(\mathbf {J}_{\text {c}}\) and \(\Xi _{\text {c}}\) proposed in Proposition 4 therefore are minimal in the sense that they are sufficient to produce a one-form for \(d_{t}\hat{\Psi }.\)
- 17.
It must be noted that a Pfaffian differential form is not necessarily an exact differential unless it satisfies a set of integrability conditions [50–53]. In the case of thermodynamics, the second law of thermodynamics preempts the integrability conditions which are partial differential equations not easy to solve in general.
- 18.
These phenomenological transport coefficients may be identified with their moecular theory (i.e., kinetic theory) formulas in the forms of collision bracket integrals, which appear in the kinetic theory chapters.
- 19.
Originally, the Rayleigh dissipation function was quadratic with respect to velocities. Here we are generalizing not only the original definition for fluxes (velocities) to \(\Phi _{a}^{(s)}\), but also the flux dependence of the dissipation function to a highly nonlinear form from a quadratic function of velocities.
- 20.
In actual applications of the present theory, the nonconserved variables require closure relations. For example, in the case of single-component fluids it will be found useful to take the first thirteen moments (i.e., density, pressure, three velocity components, three components of heat flux, two normal stresses, three components of shear stress) for macroscopic variables and then assume the closure relations \( {\varvec{\psi }}_{a}^{(1)}= {\varvec{\psi }}_{a}^{(3)}=0\). Applications discussed in a later chapter take this kind of closure relations.
or 
. In this work we do not adopt such a notation.References
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Eu, B.C. (2016). Thermodynamic Theory of Irreversible Processes. In: Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-41147-7_2
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