The Enskog Theory and Its Modifications

Consider a dense gas of hard spheres, all with mass m and diameter a. Since the collisions of hard-sphere molecules are instantaneous, the probability is zero that any particle will collide with more than one particle at a time. Hence we still suppose that the dynamical events taking place in the gas are made up of binary collisions, and that to derive an equation for the single-particle distribution function /(r, v, /) we need only take binary collisions into account. However, the Stosszahlansatz used in deriving the Boltzmann equation for a dilute gas should be modified to take into account any spatial and velocity correlations that may exist between the colliding spheres. The Enskog theory continues to ignore the possibility of correlations in the velocities before collision, but attempts to take into account the spatial correlations. In addition, the Enskog theory takes into account the variation of the distribution function over distances of the order of the molecular diameter, which also leads to corrections to the Boltzmann equation. 

To construct Enskog s extension of the Boltzmann equation to higher densities, we consider only the change in /(r, v, /) with time due to collisions, since only this term is affected by the density of the gas. Let us first consider the change in /(r, v, /) due to the direct collisions. For a dilute gas we have argued that the number of direct collisions taking place in 8t in time 8t between molecules with velocity v and molecules with velocity Vi, with apse line in direction dk about k is 

Enskog proposed that since the gas is not in equilibrium we should multiply Eq. (142) by a factor [ (r,r+ak), which is the nonequilibrium pair correlation function for two particles in contact. Since the form of this function is not known, he made the following assumption The analytic dependence of (r, r+ak) on density is exactly the same as that of g(a), but the density to be used in computing if (r, r+afc) is the local density at a point midway between the two spheres, i.e., n(r+5ak). Since g(a) has the density expansion  

This expression for the nonequilibrium pair correlation function is based on the supposition that even in a nonequilibrium state the pair correlation function should be determined solely by the physical requirement that hard spheres cannot overlap. Thus t(r, r+ak) should take into account the same kind of excluded volume effects as are incorporated in the equilibrium quantity g(a). To do this, Enskog assumes that t(r, r+ak) has the same dependence on the local density at the point of contact, r+Jak, as g(a) has on the equilibrium density Hq. The point of contact r+jak is used, since the pair correlation function should be invariant upon interchange of the two particles. 

A similar analysis can be given for the restituting collision and we can immediately write the modification of the Boltzmann equation proposed by Enskog  

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The contribution of statistical mechanics is not, of course, limited to the kind of results shown in Figures 3-7. Several approximate theories have been proposed which are based on statistical mechanics but which require some assumption, or assumptions, to avoid a detailed description of a fluid in nonequilibrium. A good example is the Enskog theory and its modifications ( 1). As remarked, however, such theories will not be discussed here with one exception, namely the theory of corresponding states. One specific application is outlined as follows. 

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