Three-particle collision

Two and Three Particle Collision Operator for the FHP LG Let us look more closely at the form of the LG collision operator for a hexagonal lattice. Conceptually, it is constructed in almost the same manner as its continuous counterpart. In particular, we must examine, at each site, the gain and loss of particles along a given direction. 

Handling the triple-collision gain-lo.ss term in the same way, the complete two- and three- particle collision term for the Cp direction can be written in the following form [hciss88b] ... 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

Frequency of Collisions between Particles. Particles in suspension collide with each other as a consequence of at least three mechanisms of particle transport ... 

Fig. 56. Simple binary collision events in liquids, (a) Patti of two particles, 1 and 2, undergoing a single collision only, (b) Three particles undergoing two binary collisions, (c) Three particles undergoing three binary collisions where the second collision of particles 1 and 2 is correlated with the first collision between these particles this is the simplest ring graph. After R ibois and De Leener [490].

Ternary and Other Induced Spectra. Three-particle induced dipoles and the associated ternary collision-induced absorption spectra and dipole autocorrelation functions have been studied for fluids composed of mixtures of rare gases, and for neat fluids of nonpolar molecules — that is for systems that are widely thought to interact with radiation only by virtue of interaction-induced properties. A convenient framework is thus obtained for understanding the variety of experimental observations. The computer simulation studies permit an insight into the involved basic processes, but were not intended for direct comparison with measurements [57]. Methods have been developed for computer... 

In the above expression, C (pi z) is the finite frequency generalization of the Boltzmann-Lorentz collision operator. Cq1 (pi z) can be described by the finite frequency generalization of the Choh-Uhlenbeck collision operator. [57]. This operator describes the dynamical correlations created by the collisions between three particles. Using the above-mentioned description the expression of (pi z) can be shown to be written as [57]... 

B. Binary Density Operator in Three-Particle Collision Approximation— Boltzmann Equation for Nonideal Gases... 

The second problem is the decoupling of the chain of equations for the formal solutions. It is necessary to make approximations in order to truncate this chain. The simplest approximation is the binary collision approximation. That means, we neglect three-particle collisions. Then, we obtain... 

The second approximation that we will consider is the three-particle approximation. In the case that three-particle collisions are taken into account we must start with the more general expression (2.34). Using the identity... 

As can be seen the difference between F,(r) and F,(t0) is of the order n. If we take into account three-particle collisions, it is therefore not possible to neglect the retardation. Using the expansion (2.39) it is easy to eliminate Fx(t0). Thus, we obtain the import density expansion13-15... 

The first term in this expansion describes the binary collision and leads, as we have shown, to the Boltzmann collision integral. Among the terms of the order n, the first describes the three-particle collision and has, as will be shown, the same structure as the Boltzmann collision term. 

The second term arose from the elimination of the retardation. It describes two successive binary collisions of three particles. This term plays an essential role, since the three-particle collision term also contains such special three-particle collisions, the contribution of which is thus canceled. This is very important since processes of such types produce... 

As mentioned in Section 2.1, the usual Boltzmann equation conserves the kinetic energy only. In this sense the Boltzmann equation is referred to as an equation for ideal systems. For nonideal systems we will show that the binary density operator, in the three-particle collision approximation, provides for an energy conservation up to the next-higher order in the density (second virial coefficient). For this reason we consider the time derivative of the mean value of the kinetic energy,12 16 17... 

In order to discuss the fundamental problems that are connected with the bound states in kinetic theory, we first restrict ourselves to systems with two-particle bound states only. The states of the two-particle system are determined by Eq. (2.12). Furthermore, we remark that to describe the formation of two-particle bound states by a collision, at least three particles are necessary in order to fulfill energy and momentum conservation. Thus, it is necessary to consider the quantum mechanics of three-particle systems. 

As in Section II. 1, we get from (3.25) the usual shape of the Boltzmann equation for free particles. Obviously, such equations are of no interest for systems with bound states. Equation (3.15a) for atoms is collisionless, and the equation for the free particles does not contain contributions that account for the formation and the decay of bound states. In order to derive such equations we must take into account three-particle and higher-order collisions. Subsequently, this problem will be dealt with. 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

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 this connection, Fm is given in the three-particle collision approximation by... 

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... 

With Eq. (3.37) for Fn it is possible to write a kinetic equation for F, that describes the formation and the decay of two-particle bound states in three-particle collisions. Introducing (3.37) into the first equation of the hierarchy (1-29), we obtain in a similar way as in Section III.2 a kinetic equation for the density operator of free particles. This equation may be written in the following form ... 

On behalf of the projection (1 Pn the bound-state contribution (12) of the last term drops out. Equation (3.47) contains for k = 0 (scattering of three unbound particles) the collision integral (2.42), and for k = 1,2,3 it describes the formation and the decay of bound states in the three-particle collision. Here we have... 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

As already outlined, it is useful to express the collision operators in terms of the scattering T matrix. For the three-particle collision operator we may use the formulas given by (3.49-3.58). Then we arrive at... 

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. 

All these calculations can handle two-particle collisions, but none attempts to deal directly with the three-particle coalescence. Schwartz[20] recognized... 

Collision theory, as its name might suggest, focuses on the collisions between particles. The collisions must be frequent, and the colliding particles must have sufficient energy to form an activated complex. The transition-state theory focuses on the behavior of the activated complex. According to the transition-state theory, there are three main factors that determine if a reaction will occur ... 

The temperature-dependent second and third virial coefficient describe the increasing two- and three-particle collisions between the gas molecules and their accompanying increase in gas density. The virial coefficients are calculated using a suitable intermolecular por-tential model (usually a 12-6 Lennard-Jones Potential) from rudimentary statistical thermodynamics. 

In this section we derive a set of regularized equations of motion and a triple collision manifold (TCM) for the Coulomb three-body system. Three particles (electron, nucleus, and electron) have masses mi = mg, m2 = m and m3 = mg and charges —e, Ze, and —e. We consider the Coulomb three-body system whose Hamiltonian is... 

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