Rigorous Theory of Transport Properties

Thus the net flux of molecules will be the difference in the upward and downward crossing rates 

Comparing Eqs. 12.3 and 12.62, we can write the kinetic gas theory expression for the diffusion coefficient D as 

The two expressions differ only by the leading constant, with Eq. 12.64 about 15% smaller than the rigorous formula. 

The previous section gave a simple treatment of transport properties based on the kinetic theory of gases. That approach has the advantages that it is very intuitive and mathematically tractable, and the final results are in approximate agreement with experiment and with more rigorous theory. To go beyond that treatment requires much more complex and specialized theory in statistical mechanics, molecular interactions and collisions [60,178, 269], 

This section will give an introduction to the more rigorous theory of transport properties, but by necessity will not go through detailed derivation of all of the results. The goal here will be to sketch the physical and mathematical framework of the formulations. We then present results in enough detail to connect with the rigorous formulas used in computational evaluation of transport properties [211], which are summarized in the following section. 

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The difficulties that surround the formulation of a rigorous theory of the transport properties of dense pure fluids set out in earlier sections pertain equally strongly to mixtures of fluids with an arbitrary number of components. For that reason the formal theory is not pursued in this section. Instead, attention is concentrated upon an approximate theory that has led to a valuable means of predicting the properties of multicomponent mixtures from a limited amount of experimental information on the pure components. 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

We begin with a simple, physical derivation of the Boltzmann equation, which is the starting point in obtaining the rigorous transport properties. Following this discussion, the theory of Chapman and Enskog [60,114] is presented. 

The latter serves as a reminder that the kinetic theory predicts the cross effects like the transport of mass resulting from a temperature gradient (thermal diffusion). It can also be shown that the theory predicts transport of energy resulting from a concentration gradient (the diffusion-thermo effects). These second-order effects are often referred to as the Soret - and Dufour effects. Unfortunately, no shortcuts are available as these terms do not appear when applying simple kinetic theory, only the more rigorous solution methods resolve these properties. 

The word prediction is defined to imply the generation of values of a transport property of a fluid or fluid mixture, by means of a method based upon a rigorous theory, in a region of state where direct measurements do not exist. 

The absence of a rigorous theory for the transport properties of fluids in the intermediate-density range means that it has been necessary to employ methods of evaluation based upon an approximate theory, the principle of corresponding states (Chapter 12) or empiricism (see Section 5.3.3). The only approximate theory to have been used to any extent is the Enskog theory, outlined in Section 5.1 and discussed in a modified form in Section 5.2 in the context of the initial density dependence of the transport properties. 

The theory proposed by Chapman and Enskog nses the rigorous kinetic theory of monatomic gases and expresses the transport properties in terms of intermolecnlar potential energy q>(r), where r is the distance between a pair of molecules undergoing a collision. The expression for q>(r) is most closely represented by the Lermard-Jones (6-12)... 

Introduction. In this paper we discuss time-dependent neutron transport in a medium. Our interest centers on problems in which the neutrons interact with the medium without affecting it, and in which the neutrons do not interact with each other. Equations describing such processes are not too difficult to write down [1]. In the classical formulation they are linear, although certain new approaches to the theory have produced non-linear versions [2 3 4]. In any event, the structure of these equations usually makes it impossible to find explicit solutions, and extremely difficult to determine rigorously what properties the solutions possess. From the mathematician s viewpoint, even the existence of the solution is often not obvious. 

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