Derivation of the source term

Let us imagine that one particle has its center of mass at position % and is characterized by phase-space vector ij. Moreover, let us imagine that another particle has its center of mass at the physical point x and has the phase-space vector fj. We define the frequency of this second-order point process, a(x, tj, x, rf), so that the quantity fl(x, / x, i])At represents the fraction of particles undergoing the point process in a time interval df. This frequency is symmetric with respect to permutations in the ordering of the particles, resulting in = a(x,i] x, i] ). We now define a pair number-density func- 

The integral of the frequency appearing between parentheses in Eq. (5.152) is called the kernel  

The relationship among ij, rj, and rj can be derived by relating continuity statements written in terms of pre- and post-event values for the internal coordinates and depends on the second-order point process under investigation. The positive source term can also be written in terms of the frequency (rather than the kernel)  

The negative source term is more readily found  

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The parameterization of the particle collision densities was obviously performed emplo3ung elementary concepts from the kinetic theory of gases, thus the derivation of the source term closures at the microscopic level have been followed by some kind of averaging and numerical discretization by a discrete numerical scheme [f6, 92, 118]. 

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