The Enskog Approximation

The self-diffusion coefficient is related to the velocity autocorrelation function 

To calculate the velocity autocorrelation function we therefore only need sfor fc = 0. This simplifies matters considerably compared to a full calculation of the van Hove self-correlation function S ik, a ). 

Before discussing the contribution of S(f s to K we first note the contribution of f sE Fo a good approximation ts diagonal on the state 2) and the matrix element is 

Since the Enskog expression for the self-diffusion coefficient 

Inverting the Laplace transform gives the simple exponential decay 

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Applying the Enskog approximate solution method, consistent flux vectors are derived for dense gases accounting for the finite size of the molecules. Because of the inherent corrections there are two contributions to the dense gas fluxes, one flux associated with the collisional transfer which is important in dense gases only, and another flux due to the motion of molecules between collisions as for dilute gases. The total flux of the property ip is determined by the sum of these two contributions ... 

The pair kinetic equation in Section VII.D follows directly from these results if the dynamic memory function " xbs.abs neglected, and the static structural correlations in (D.3) to (D.6) are approximated so that all binary collisions are calculated in the Enskog approximation. [This is the singly independent disconnected (SID) approximation, which is discussed in detail in Ref. 53.] We have also used the static hierarchy to obtain the final form involving the mean force, given in (7.32). This latter reduction involving the static hierarchy is carried out below in the context of a comparison of the singlet and doublet formulations. 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

The Chapman-Enskog theory was developed for dilute, monatomic gases for pure substances and for binary mixtures. The extension to multicomponent gas mixtures was performed by Curtiss and Hirschfelder (C12, Hll), who in addition have shown that the Chapman-Enskog results may also be obtained by means of an alternate variational method. Recently Kihara (K3) has shown how expressions for the higher approximations to the transport coefficients may be obtained, which are considerably simpler than those previously proposed by Chapman and Cowling these simpler formulas are particularly advantageous for calculating the coefficients of diffusion and thermal diffusion (M3, M4). 

Here ji(qa) is the spherical Bessel function of order l,g(a) is the radial distribution function at contact, and f = /fSmn/Anpo2g a) is the Enskog mean free time between collisions. The transport coefficients in the above expressions are given only by their Enskog values that is, only collisional contributions are retained. Since it is only in dense fluids that the Enskog values represents the important contributions to transport coefficient, the above expressions are reasonable only for dense hard-sphere fluids. Earlier Alley, Alder, and Yip [32] have done molecular dynamics simulations to determine the wavenumber-dependent transport coefficients that should be used in hard-sphere generalized hydrodynamic equations. They have shown that for intermediate values of q, the wavenumber-dependent transport coefficients are well-approximated by their collisional contributions. This implies that Eqs. (20)-(23) are even more realistic as q and z are increased. 

In this section a detailed investigation of the rate is presented by using a fully microscopic calculation of the friction which refrains from approximating the short-time response by the Enskog form. A similar calculation has been carried out for viscosity. As the short-time friction is expected to be a sensitive function of the interatomic potential, the comparison between the present calculation for continuous potential and the previous one by Biswas and Bagchi [164] could provide valuable insight into the problem. 

Various attempts have been made to obtain approximate solutions to the Boltzmann equation. Two of these methods were suggested independently by Chapman [10] [11] and by Enskog [24] giving identical results. In this book emphasis is placed on the Enskog method, rather than the Chapman one, as most modern work follows the Enskog approach since it is less intuitive and more systematic, although still very demanding mathematically. 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

An analytical expression for the heat flux vector can be derived in a similar manner using the Enskog approach. That is, we introduce the first order approximation of the distribution function from (2.246) into the heat flux definition (2.72) and thereafter substitute the partial solution for flux vector integrand as follows [39] ... 

A more rigorous derivation of these relations were given by Curtiss and Hirschfelder [16] extending the Enskog theory to multicomponent systems. FYom the Curtiss and Hirschfelder theory of dilute mono-atomic gas mixtures the Maxwell-Stefan diffusivities are in a first approximation equal to the binary diffusivities, Dgr Dsr- On the other hand, Curtiss and Bird [18] [19] did show that for dense gases and liquids the Maxwell-Stefan equations are still valid, but the strongly concentration dependent diffusivities appearing therein are not the binary diffusivities but merely empirical parameters. 

By use of the Chapman-Enskog approximate solution method [11], the kinetic pressure tensor can be given by [22] ... 

The applicability of the Enskog theory for high pressures is explained by the vortical character of the thermal motion of molecules. For molecular motions presented in Fig. 1 the relative motion of two neighboring molecules is only essential. In this case all molecules being on some sphere (circle) interact with their neighbors on the next spheres (circles) identically. So, the conditions for the applicability of two-particle approximation arise. 

The specific approximations to the static structural correlations that were made to obtain the result for are described in detail in Ref. 53. We have presented the results in a form that describes the binary collision events at the Enskog level of approximation. 

The viscosity coefficient establishes the proportionality between the friction force in the direction of flow and the velocity gradient in the orthogonal direction. Calculation of this coefficient is made with the pressure tensor in the first approximation of Chapman-Enskog... 

We illustrate how the present formalism leads to practical microscopic calculations by considering two applications. In Section 5 we study the density correlation function in the Boltzmann-Enskog approximation, and in Section 6 the recollision effects on the velocity autocorrelation function and the selfdiffusion coefficient are analyzed. In both cases, we assume, for simplicity that the fluid is a system of hard spheres. Although the calculations themselves are relatively crude and the numerical results have limited significance, these problems are of interest because they serve to indicate the level of complexity of current microscopic calculations and lead to a discussion of the areas where further calculations are needed. The chapter then closes with summary and discussions in Section 7. 

Second, we know that the Boltzmann-Enskog transport theory gives a reasonable description of transport coefficients and the short-time properties of correlation functions, " so the approximation for G should be formulated in such a way that the leading term corresponds to the Boltzmann-Enskog approximation. Third, in deriving the correction to the leading term we must consider the effects of recollision processes. ... 

The term will lead to a generalized Boltzmann-Enskog approximation. This... 

Boltzmann-Enskog approximation consists of recollision terms that describe correlated binary collisions. The effects of these processes on the behavior of time correlation functions have not yet been fully studied because the calculations involved are considerably more complicated. The problem where the effects of recollisions have been most extensively investigated is that of the velocity autocorrelation function, which is a simpler function than the dynamic structure factor. From this problem we can already see the kind of analysis involved in treating correlated collisions. ... 

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