Collision theory binary, hard-sphere

In the real world, however, the interaction potential between molecules cannot be described by the hard-sphere potential. It is continuous in nature. This makes the calculations difficult, and even an exact calculation of the binary collision term for a continuous potential is numerically formidable [46]. Sjogren and Sjolander have developed a repeated ring kinetic theory for a one-component system where the interaction is described by a continuous potential [9]. They have also included the effect of the full many-body propagators in describing the intermediate propagation. 

In the simplest version of the kinetic theory of gases, molecules are treated as hard spheres of diameter d which make binary collisions only. In this approximation the mean distance traveled by a molecule between successive collisions, the mean free path I, is related to the collision diameter by ... 

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. 

Data representation can be considered truly satisfactory only when it has a molecular basis. The first such successful approach was that of Enskog (Enskog 1922) for a system of hard spheres in which he made empirical modifications to the Boltzmann theory to account for the finite size of the molecules. Use of the Boltzmann equation, which considers only binary collisions, is valid for this model, since multiple collisions have a low probability. Enskog obtained expressions relating the diffusion, viscosity and thermal conductivity for the dense system, subscript E, to the dilute-gas values, superscript (0),... 

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