Inclusion of a chemical reaction into kinetic theory

1 INCLUSION OF A CHEMICAL REACTION INTO KINETIC THEORY 

Before introducing the chemical reaction into the picture of the system, it is necessary to explore the motion of one or two particles in a little more detail. Consider a collision between particles a and jS, and the time a little before and afterwards, such that several other collisions between the particle a which has co-ordinates r, v, and all other particles (jS ce) may occur. All the other particles which do not collide with a may be ignored. Averaging over all the positions of the N — 2) particles removes their influence (which does not affect the motion of particle a) and leads to a doublet distribution, f2. Then averaging over all positions of the particles (3 (which are colliding with particle a) gives an equation for the singlet distribution [541,542] 

Hie singlet distribution ff changes when a collision occurs between the particle ot and any one of the remaining 3 = (N — 1) particles (labelled 2 during the collision process). The probability of particles ot and /3 being able to collide depends upon their mutual position in space and their velocity, which is described by the doublet density f 0. Now, in the Boltzmann equation analysis, this leads to a factorisation of the doublet density in terms of the singlet density of both particles. Because any correlation in the position and motion of these particles is lost by this procedure, an alternative approach must be tried to estimate the doublet density. 

Collision of particle a with causes f 23 to change because of the velocity change of v3 and v2s etc. But, in addition, collision of either a or 0 with another particle fi (labelled with co-ordinates r3 and v3) can also change f . This arises if a and (u or j3 and jti are so positioned and moving that they collide it is determined by f. Averaging the Liouville equation over the co-ordinates of (N — 3) particles (not a, 3 or p) then averaging over the possible position and velocity of theja particles shows [541, 542] 

This equation for the doublet density if involves the triplet density f. It is a typical problem in statistical mechanics. To make progress, the hierarchy must be broken and this is usually done with a superposition approximation. The manner by which this is done is discussed in fee next sub-section. 

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