The Boltzmann Equation Collision Term Approximation

To determine the rate of change of / caused by molecular collisions a number of simplifying assumptions concerning the molecular motion and collisions are introduced. 

The chaos assumption thus allows the pair correlation to be neglected outside the range of molecular interaction, thus /correlation 0 when r2 — ri ro. 

A result of the Boltzmann H-theorem analysis is that the latter statistical assumption makes Boltzmann s equation irreversible in time (e.g., [39], Sect. 4.2). 

The conventional collision term ( )coiiision closure is further discussed in the paragraph called generalized collision term in the current subsection, presenting an outline of the derivation of the approximate constitutive relations for ( )coiiision 1 

It is required to express either pair of molecular velocities before and after the interaction in terms of the other pair, and of two independent geometrical variables b and (j ) in order to complete the specification of the encounter. The original Boltzmann equation derivation considers elastic collisions in free space between two spin-less molecules of equal mass. However, due to the major interest in multicomponent mixtures, the theory outline consider elastic collisions between two spin-less mono-atomic molecules in an ideal gas mixture. The theory may be useful even if the molecules are not mono-atomic, provided that their states of internal motion (i.e., rotation and vibration) are not affected by the collisions. The two molecules under consideration are treated as point particles with respective masses m and m2. In the laboratory frame, the incoming molecule positions are denoted by ri and r2, and the particle velocities are indicated by ci and C2. The corresponding positions and velocities after the encounter are r j, and c, c, respectively. The classical trajectories for two interacting molecules presented in the laboratory system frame are viewed in Fig. 2.1. It is supposed that the particle interaction is determined by conservative potential interaction forces only. Any external forces which might act on the molecules are considered negligible compared to the potential forces involved locally in the collision. The relative position vectors in the laboratory frame are defined by  

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