Collision operator four particle

The collision term for four particles in Cohen s version and the general Boltzmann operator for n = 4 in the Prigogine formalism (Section VC). 

For slow collisions the transitions occur from the continuum state of the J — 0 (spherical) symmetry to the bound state of the J = 1 symmetry, and are induced by the electric dipole moment of the H — H system. This dipole moment is the expectation value of the dipole moment operator for all four particles with respect to the ground state leptonic wave function, D = where r, denote the positions of... 

We will carry out our program in two steps. In this section we will derive the two-particle density operator Fn in a three-particle collision approximation for the application in the collision integral of Fl. As compared with Section II.2, the main difference will be the occurrence of bound states and, especially, the generalization of the asymptotic condition, which now has to account for bound states too. For the purpose of the application in the kinetic equation of the atoms (bound states) we need an approximation of the next-higher-order density matrix, that is, F 23 This quantity will be determined under inclusion of certain four-particle interaction. 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

The four-particle collision operator is assumed to correspond to one channel only, and reads with (3.40)... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

Fig. 24. Some of the four successive binary collision events taking place among four particles (four-body ring events) that are responsible for the divergence of the four-body collision operator.

This is the model for the overall residence time distribution of the particles in the impinging stream contactor under consideration. The model contains several parameters related to equipment structure and operating conditions, i.e. the mean residence times in the four sub-spaces, 7ac, 7im, 7fai and cs. Among the four parameters, the mean residence time in the impingement zone, t m, and that in the collision-slipping region, fcs, are symmetrical parameters, which have the same influence on the overall residence time distribution. It can be seen from Eq. (3.27) or... 

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