Boltzmann Equation Collision Frequency Reformulation

To derive the Enskog s equation for a dense gas, Enskog [37] did departure from the kinetic theory of dilute mono-atomic gases that is described by the Boltzmann 

The scattering cross section is expressed in terms of the variable k, simply by defining a function g) by [109, 117]  

The deflection angle 0 is defined as the angle between the fined eind initial velocities 0 = -k — Ill ll appeeirs that the phrase scattering is used for the case of Maxwelliein billiard ball particles, whereas the phrase deflection is used in the more general case considering a center of force. 

It appears that the variable k will be convenient shortly calculating the collision term because it allows us to write down the number of scattered particles in a form independent of the particular system of coordinates used in the previous calculations (i.e., the laboratory frame). Trigonometrical considerations imply that  

It follows that the number of particles scattered into the differential area element per unit time, given by (2.646), can be rewritten using (2.648)-(2.650). The result is  

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