Second-order point process phase space

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-... 

For second-order point processes such as hard-sphere collisions, the total number of particles remains unchanged. However, the number of particles with a specific phase-space vector will always increase or decrease. 

This PBE is written in a general form and contains the terms representing accumulation, real-space advection, phase-space advection, phase-space diffusion, and second-, first-, and zeroth-order point processes. (See Chapter 5 for more details on these processes.) Let us... 

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