Kinetic Theory of a Dense Simple Fluid Mixture
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 the
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Kinetic Theory of a Dense Simple Fluid Mixture
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. The grand canonical ensemble kinetic theory is parallel in structure to the GBE theory of a pure simple fluid—described in Chap. 6. For this reason the general methodology and various concepts remain unaltered from those employed in Chap. 6. Therefore we will not go into details of derivation of the coarse-grained kinetic equation for dense fluid mixtures, but briefly mention the important steps with appropriate definitions of notation used for the fluid mixture necessary to follow the discussion. However, since hydrodynamics of a mixture often involves generalized forms of equations sometimes considerably different from those for a pure fluid we will find it necessary to derive them again. Nevertheless, the generic structures of generalized hydrodynamic equations and irreversible thermodynamics formalism for a mixture will be found to remain basically unchanged, but they reduce to their counterparts for a pure simple fluid discussed in Chap. 6 as the fluid becomes pure, and to the dilute simple gas counterparts as the density diminishes to a dilute gas value.
© Springer International Publishing Switzerland 2016 B.C. Eu, Kinetic Theory of Nonequilibrium Ensembles, Irreversible Thermodynamics, and Generalized Hydrodynamics, DOI 10.1007/978-3-319-41147-7_7
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7 Kinetic Theory of a Dense Simple Fluid Mixture
7.1 Generalized Boltzmann Equation for a Simple Fluid Mixture 7.1.1 Nonequilibrium Grand Ensemble We consider a fluid of a nonreactive mixture of r species of N1 , N2 , . . ., and Nr monatomic molecules contained in a fixed volume V . The density and composition of the fluid mixture are arbitrary. This volume of the fluid exchanges matter as well as energy with its surroundings. Therefore the numbers of different species particles of the system are not fixed and the dimension of the phase space accordingly varies for the system over time in the course of exchange processes. For this reason the canonical ensemble method [1] discussed in Chap. 6 cannot be applied in a straightforward manner, if a statistical mechanical theory is sought. It is necessary to resort to the grand canonical ensemble method devised by Gibbs [2] to overcome the difficulty of the varying phase space dimension as the particle exchange process progresses with time. To discuss this method we would like to define the notation to be employed. The different species will be distinguished by subscript a, b
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