Abstract
In Chap. 6 we have confined the formulation of the kinetic theory to a pure dense fluid for simplicity of formulation. To develop a theory of irreversible thermodynamics in a general form covering liquid mixtures it is now necessary to generalize the theories formulated in the previous chapters. It is possible to achieve this goal if we first formulate a kinetic theory of a mixture of dense simple fluids by using a grand canonical ensemble method and then develop therewith a thermodynamic theory of irreversible transport processes and attendant generalized hydrodynamics of a fluid mixture.
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Notes
- 1.
If chemical reactions are allowed, \(N_{a}\) will change also owing to the chemical reactions occurring within the system. However, this case is excluded in the present consideration for simplicity of discussion. This restriction, however, is easy to remove.
- 2.
For a schematic illustration of the equilibrium grand ensemble, see Fig. 4.2 in Chap. 4. A similar schematic representation may be used for a nonequilibrium grand ensemble.
- 3.
- 4.
In this respect, the subsystems may be regarded as if they are giant polyatomic molecules that preserve their chemical compositions but can change their states, as they interact with each other in the course collisions.
- 5.
Since the interactions between subsystems (petit ensembles) are through the particles in the peripheries of the subsystems and hence are relatively smaller compared to the total interaction energies, a perturbation treatment of the Lippmann-Schwinger equation for \(\mathcal {T}_{ss^{\prime }}^{(N)}\left( \epsilon \right) \) is quite justifiable and permissible. Nevertheless, we will pursue the present formulation, deferring such an approximation to the last stage where the transport coefficients are calculated.
- 6.
In this connection, it should be recognized the transition operators are defined in the complex eigenvalue plane, and in fact in the upper positive plane \(\lambda +i\epsilon \), whereas the Liouville operator \(\mathcal {L}^{(\mathcal {N})}\) has real eigenvalues.
- 7.
It should be recalled that the manifold \(\mathfrak {P\cup T}\) is of nonequilibrium and a projection of the \(6\mathbf {N}\)-dimensional phase space of particles.
- 8.
If there are angular momenta for the particles, then they must be included to the list shown. Since the external force is assumed to remain unchanged over the elementary volume around the position \(\mathbf {r}\), the external potential energy \(U_{a}^{(ex)}\) per molecule may be added to the particle Hamiltonian \(H_{ja}^{\prime \prime } \).
- 9.
It should be noted that this factor c can be disposed of with no effect on the theory formulated in the present formulation.
- 10.
In the traditional description of hydrodynamic processes the concept of molar volume or, simply, volume assignable to each particle appears in the form of specific volume \(v_{sp}=1/\rho \), where \(\rho \) is mass density. However, this specific volume is not equal to the molar volume per particle at all densities of the fluid [11]. It becomes the molar volume at the limit of low density. Therefore it is reasonable to count the molar volume among the macroscopic variables at the same level as density, momentum, energy, etc. used for hydrodynamic description of flow processes.
- 11.
- 12.
The histocial evolution of the concept of energy and heat has taken a considerably winding path, which is intimately tied with the development of thermodynamics. For histocial account of the concept of energy and heat, see S.G. Brush, The Kind of Motion We Call Heat (North-Holland, Amsterdam, 1976), vols. 1 and 2.
- 13.
A remark on terminology renormalization: In the mathematical procedure of recasting some of terms in \(\mathcal {Z}_{a}^{(1)}\), the divergence of vectors or tensors \(\mathbf {\nabla } \cdot \varvec{\phi }\) arising from intermolecular forces, is combined with the divergence term in the evolution equation \(\mathbf {\nabla } \cdot \varvec{\psi }^{(q)}\) to a form similar to the original form. This procedure will be henceforth called renormalization for a want of a better terminology. This procedure produces a physically better balanced system of evolution equations, especially, when they are approximated in the linear regime of thermodynamic driving forces of the processes, and for the purpose of formulating the theory of irreversible thermodynamics presented later.
- 14.
In the following expression for \(\mathcal {H} _{ja}^{(\mathbf {N})}\) the nonequilibrium terms do not involve more than one species. This is a special model for the nonequilibrium contribution to which we will confine the theory. It may be generalized with a more complicated structure of the theory resulting thereby. Such a complication is not warranted at this point in development.
- 15.
At this point in development of theory it would be more appropriate to regard T, \(\widehat{\mu }_{a}\), and \(X_{a}^{(q)}\), respectively, as parameters conjugate to energy, mass, and moments characterizing nonequilibrium, but we are using the term temperature, nonequilibrium chemical potential, and generalized potentials in anticipation of their identification as such through thermodynamic correspondences made between the statistical mechanical and phenomenological quantities.
- 16.
It should be kept in mind that by using the nonequilibrium statistical thermodynamics formalism developed earlier, the nonequilibrium distribution function \(f^{(\mathbf {N})}\) is necessarily replaced by \(f_{\text {c}}^{(\mathbf {N})}\). Therefore, fluctuations inherent to \(f^{(\mathbf {N})}\) are neglected. This means the generalized potentials are without fluctuations.
- 17.
The second-order cumulant approximation is not suitable owing to the possibility it produces a negative calortropy production in some region of thermodynamic manifold, but the third-order cumulant approximation is always positive. It, in fact, offers an intriguing possibility of producing a second minimum in \(\sigma _{\text {c}}\) away from equilibrium, so that the system self-organizes around the state of the second minimum removed away from equilibrium. For a study of the third-order cumulant approximation in the case of a dilute gas transport phenomena, see B.C. Eu, J. Chem. Phys. 75, 4031 (1981).
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Eu, B.C. (2016). Kinetic Theory of a Dense Simple Fluid Mixture. In: Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-41147-7_7
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